$\mathbb{E}\left[\lambda_{B}^{4}\right]=\left(\sum_{i}h_{i}I_{i}^{B}\right)^{4}\leq\left(\sum_{i}h_{i}1\right)^{2}\left(\sum_{i}h_{i}I_{i}^{B}\right)^{2}=\lambda^{2}\mathbb{E}\left[\lambda_{B}^{2}\right].$
(211)

This means we can re-use the above computation as

$\mathbb{E}\left[Q(t)^{2}\right]\leq\frac{2t^{2}\lambda^{2}}{N_{B}}\mathbb{E}\left[Q(t)\right].$
(212)

∎

See 13

###### Proof.

To simplify this we will use intermediate steps from the calculation of
$P_{\max}(t)$ that bound the $\alpha_{comm}$ factors, namely equations (87),
(90) which are repeated below

$\displaystyle\alpha_{comm}(A,2k)$ $\displaystyle\leq
2^{2k}\lambda_{A}^{2k+1}$ (213)
$\displaystyle\alpha_{comm}(\left\\{A,B\right\\},2k)$ $\displaystyle\leq
2^{2k}\sum_{l=1}^{2k}\lambda_{A}^{l}\lambda_{B}^{2k+1-l}$ (214) $\displaystyle
P(t)$
$\displaystyle=\frac{2^{2}(t\Upsilon)^{2k+1}}{2k+1}(\Upsilon\alpha_{comm}(A,2k)+\alpha_{comm}(\left\\{A,B\right\\},2k))$
(215)

We will use these to compute a useful upper bound on
$\mathbb{E}\left[P(t)\right]$. Since our random variables are more easily
described for the $I_{i}^{B}$ variables, we will convert all powers of
$\lambda_{A}$ into functions of $L$ and $\lambda_{B}$ as well as upper bound
both by $\lambda_{A}$ and $\lambda_{B}$ by $\lambda$. This results in an upper
bound on $\mathbb{E}\left[P(t)\right]$ as

$\mathbb{E}\left[P(t)\right]\leq\frac{2^{2+2k}(t\Upsilon)^{2k+1}}{2k+1}\mathbb{E}\left[\Upsilon\lambda_{A}^{2k+1}+\sum_{l=1}^{2k}\lambda_{A}^{l}\lambda_{B}^{2k+1-l}\right].$
(216)

We will simplify the expectation value using the facts that
$\lambda_{A},\lambda_{B}\leq\lambda$ and that in
$\sum_{l=1}^{2k}\lambda_{A}^{l}\lambda_{B}^{2k+1-l}$ each term has at least
one factor of $\lambda_{A}\lambda_{B}$. This results in the following
simplifications

$\displaystyle\mathbb{E}\left[\Upsilon\lambda_{A}^{2k+1}+\sum_{l=1}^{2k}\lambda_{A}^{l}\lambda_{B}^{2k+1-l}\right]$
$\displaystyle\leq\lambda^{2k-1}\mathbb{E}\left[\Upsilon\lambda_{A}^{2}+(2k)\lambda_{A}\lambda_{B}\right]$
(217)
$\displaystyle\leq\Upsilon\lambda^{2k-1}\mathbb{E}\left[\lambda_{A}^{2}+\lambda_{A}\lambda_{B}\right]$
(218)
$\displaystyle=\Upsilon\lambda^{2k-1}\mathbb{E}\left[\lambda_{A}(\lambda_{A}+\lambda_{B})\right]$
(219)
$\displaystyle=\Upsilon\lambda^{2k}\left(\lambda-\mathbb{E}\left[\lambda_{B}\right]\right)$
(220)
$\displaystyle=\Upsilon\lambda^{2k}\left(\lambda_{\mathcal{S}}-\chi|\mathcal{S}|\right),$
(221)

where we used an exact expression for $\mathbb{E}\left[\lambda_{B}\right]$
that is a straightforward computation in addition to the fact that
$2k\leq\Upsilon$ for $k\geq 1$. Combining the above expressions (221) and
(216) our final expression for an upper bound on $\mathbb{E}\left[P(t)\right]$
is

$\mathbb{E}\left[P(t)\right]\leq\frac{(2\Upsilon)^{2+2k}}{2k+1}t^{2k+1}\lambda^{2k}\left(\lambda_{\mathcal{S}}-\chi|\mathcal{S}|\right)$
(222)

∎


# On symmetry of $q$-Painlevé equations and associated linear equations

Kouichi Takemura  Address of K.T., Department of Mathematics, Ochanomizu
University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan
<EMAIL_ADDRESS>
###### Abstract.

We investigate the symmetry of the linear $q$-difference equations which are
associated with some $q$-Painlevé equations. We apply it for adjustment of the
expression of the time evolution on the $q$-Painlevé equations in terms of the
Weyl group symmetry.

###### Key words and phrases:

$q$-Painlevé equation, Lax pair, affine Weyl group

###### 2020 Mathematics Subject Classification:

33E17,39A13

##  1\. Introduction

The Painlevé-type equations are non-linear generalizations of the
hypergeometric-type equations. Several members of the discrete analogue of the
Painlevé equation had been discovered individually in the 1990’s, and a
comprehensive list of the second order discrete Painlevé equations was
provided by Sakai [3]. Each member of the $q$-difference Painlevé equation was
essentially labelled by the affine root system from the symmetry. Sakai [3]
realized the time evolution of the discrete Painlevé equations by the symmetry
of the Weyl group, which originates from the Cremona action on a family of
surfaces which are related with the space of initial conditions.

The $q$-Painlevé equation of type $D^{(1)}_{5}$, which is also denoted by
$q$-$P(D^{(1)}_{5})$, is given by

(1.1)
$f\overline{f}=\nu_{3}\nu_{4}\,\frac{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}{(g-1/\nu_{1})(g-1/\nu_{2})},\quad
g\underline{g}=\frac{1}{\nu_{1}\nu_{2}}\frac{(f-\kappa_{1}/\nu_{7})(f-\kappa_{1}/\nu_{8})}{(f-\nu_{3})(f-\nu_{4})}.$

Here, $\overline{f}$ (resp. $\underline{f}$) is the consequence of the time
evolution $t\mapsto qt$ (resp. the inverse time evolution $t\mapsto t/q$) to
$f$. The time evolution for the parameters is given by
$\overline{\kappa}_{1}=\kappa_{1}/q$, $\overline{\kappa}_{2}=q\kappa_{2}$ and
$\overline{\nu}_{i}=\nu_{i}$ $(i=1,2,\dots,8)$. Note that the equation
$q$-$P(D^{(1)}_{5})$ was originally introduced by Jimbo and Sakai [1] as a
$q$-analogue of the sixth Painlevé equation, and it was obtained by
considering a $q$-analogue of monodromy preserving deformation. Yamada studied
the Lax pairs of the discrete Painlevé equations extensively [7, 8], and a
list of the Lax pairs was provided in the survey [2] by Kajiwara, Noumi and
Yamada. A Lax pair of the $q$-Painlevé equation $q$-$P(D^{(1)}_{5})$ was given
as

(1.2) $\displaystyle L_{1}$
$\displaystyle=\Bigl{\\{}\frac{z(g\nu_{1}-1)(g\nu_{2}-1)}{qg}-\frac{\nu_{1}\nu_{2}\nu_{3}\nu_{4}(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}{fg}\Bigr{\\}}$
$\displaystyle+\frac{\nu_{1}\nu_{2}(z-q\nu_{3})(z-q\nu_{4})}{q(qf-z)}(g-T^{-1}_{z})+\frac{(z-\kappa_{1}/\nu_{7})(z-\kappa_{1}/\nu_{8})}{q(f-z)}\Bigl{(}\frac{1}{g}-T_{z}\Bigr{)},$
$\displaystyle L_{2}$
$\displaystyle=\Bigl{(}1-\frac{f}{z}\Bigr{)}T+T_{z}-\frac{1}{g}.$

Here $T_{z}$ represents the transformation $z\mapsto qz$ and $T$ represents
the time evolution such as $T(f)=\overline{f}$. The equation
$q$-$P(D^{(1)}_{5})$ is obtained as the compatibility condition for the Lax
operators $L_{1}$ and $L_{2}$. In this paper we investigate the symmetry of
the linear $q$-difference equation $L_{1}y(z)=0$, where $L_{1}$ (see Eq.
(1.2)) is the Lax operator of the $q$-Painlevé equation of type $D^{(1)}_{5}$.
Typical symmetries of the equation $L_{1}y(z)=0$ are given by the gauge
transformations $y(z)\mapsto p(z)y(z)$ for some specific functions $p(z)$. We
also investigate the symmetry of the associated linear $q$-difference equation
with other $q$-Painlevé equations denoted by $q$-$P(E^{(1)}_{6})$ and
$q$-$P(E^{(1)}_{7})$. The symmetry of the linear $q$-difference equation may
induce the transformation of the parameters and some of them are described in
terms of the affine Weyl group of the $q$-Painlevé equation. The gauge
transformation $y(z)\mapsto z^{\lambda}y(z)$ for $\lambda\in\mathbb{C}$ and
the dilation $z\mapsto az$ for $a\in\mathbb{C}\setminus\\{0\\}$ also induce
the transformation of the parameters.

It is known that the symmetry of each member of the $q$-Painlevé equation was
described by using the affine Weyl group, and a systematic description was
provided in [2]. However, it seems that some adjustment would be necessary to
fit the time evolution from the Lax pair with the Weyl group symmetry
described in [2]. In this paper we describe the expression of the time
evolution of $q$-Painlevé equations $q$-$P(D^{(1)}_{5})$, $q$-$P(E^{(1)}_{6})$
and $q$-$P(E^{(1)}_{7})$ in terms of the Weyl group symmetry and the symmetry
coming from the gauge transformation and the dilation.

This paper is organized as follows. In section 2, we review the Weyl group
action for the $q$-Painlevé equations $q$-$P(D^{(1)}_{5})$ by following [2],
and investigate the symmetry for the linear $q$-difference equation
$L_{1}y(z)=0$. We realize the time evolution of $q$-$P(D^{(1)}_{5})$ in terms
of the actions of the generators of the Weyl group and the symmetry coming
from the linear $q$-difference equation. In section 3, we introduce the
$q$-Painlevé equations $q$-$P(E^{(1)}_{6})$. We also review the Lax pair and
its Weyl group action of $q$-$P(E^{(1)}_{6})$ by following [2], and
investigate the symmetry for the corresponding linear $q$-difference equation.
We also investigate the time evolution of $q$-$P(E^{(1)}_{6})$, which was
inspired by the study of the difference Painlevé equation by Tsuda [6]. In
section 4, we give similar results for the $q$-Painlevé equations
$q$-$P(E^{(1)}_{7})$.

##  2\. The $q$-Painlevé equation of type $D^{(1)}_{5}$

We review the Weyl group symmetry of the $q$-Painlevé equation
$q$-$P(D^{(1)}_{5})$. For this purpose, we recall the action of the operators
$s_{0},\dots,s_{5}$, $\pi_{1}$ and $\pi_{2}$ for the parameters
$(\kappa_{1},\kappa_{2},\nu_{1},\dots,\nu_{8})$ and $(f,g)$ introduced in [2].

(2.1) $\displaystyle s_{0}:$
$\displaystyle\>\nu_{7}\leftrightarrow\nu_{8},\quad
s_{1}:\>\nu_{3}\leftrightarrow\nu_{4},\quad
s_{4}:\>\nu_{1}\leftrightarrow\nu_{2},\quad
s_{5}:\>\nu_{5}\leftrightarrow\nu_{6},$ $\displaystyle s_{2}:$
$\displaystyle\>\nu_{3}\rightarrow\frac{\kappa_{1}}{\nu_{7}},\;\nu_{7}\rightarrow\frac{\kappa_{1}}{\nu_{3}},\;\kappa_{2}\rightarrow\frac{\kappa_{1}\kappa_{2}}{\nu_{3}\nu_{7}},\;g\rightarrow
g\,\frac{f-\nu_{3}}{f-\kappa_{1}/\nu_{7}},$ $\displaystyle s_{3}:$
$\displaystyle\>\nu_{1}\rightarrow\frac{\kappa_{2}}{\nu_{5}},\;\nu_{5}\rightarrow\frac{\kappa_{2}}{\nu_{1}},\;\kappa_{1}\rightarrow\frac{\kappa_{1}\kappa_{2}}{\nu_{1}\nu_{5}},\;f\rightarrow
f\,\frac{g-1/\nu_{1}}{g-\nu_{5}/\kappa_{2}},$ $\displaystyle\pi_{1}:$
$\displaystyle\>q\rightarrow 1/q,\>\nu_{1}\rightarrow
1/\nu_{1},\>\nu_{2}\rightarrow 1/\nu_{2},\>\nu_{3}\rightarrow
1/\nu_{7},\>\nu_{4}\rightarrow 1/\nu_{8},\>\nu_{5}\rightarrow
1/\nu_{5},\>\nu_{6}\rightarrow 1/\nu_{6},$ $\displaystyle\>\nu_{7}\rightarrow
1/\nu_{3},\>\nu_{8}\rightarrow 1/\nu_{4},\>\kappa_{1}\rightarrow
1/\kappa_{1},\>\kappa_{2}\rightarrow 1/\kappa_{2},\>f\rightarrow
f/\kappa_{1},\>g\rightarrow 1/g,$ $\displaystyle\pi_{2}:$
$\displaystyle\>q\rightarrow 1/q,\>\nu_{1}\rightarrow
1/\nu_{7},\>\nu_{2}\rightarrow 1/\nu_{8},\>\nu_{3}\rightarrow
1/\nu_{5},\>\nu_{4}\rightarrow 1/\nu_{6},\>\nu_{5}\rightarrow
1/\nu_{3},\>\nu_{6}\rightarrow 1/\nu_{4},$ $\displaystyle\>\nu_{7}\rightarrow
1/\nu_{1},\>\nu_{8}\rightarrow 1/\nu_{2},\>\kappa_{1}\rightarrow
1/\kappa_{2},\>\kappa_{2}\rightarrow 1/\kappa_{1},\>f\rightarrow
1/(\kappa_{2}g),\>g\rightarrow\kappa_{1}/f.$

The omitted variables are invariant by the action, e.g. $s_{2}(f)=f$. Then we
can confirm that these operations satisfy the relations of the extended Weyl
group $\widetilde{W}(D^{(1)}_{5})$ whose Dynkin diagram is as follows.
$0$$1$$4$$5$$2$$3$
Namely we have $s_{i}^{2}=\mathrm{id}$, $(i=0,\dots,5)$,
$\pi_{1}^{2}=\pi_{2}^{2}=\mathrm{id}$, $(\pi_{1}\pi_{2})^{4}=\mathrm{id}$,
$\pi_{1}s_{0}=s_{1}\pi_{1}$, $\pi_{1}s_{j}=s_{j}\pi_{1}$, $(j=2,3,4,5)$,
$\pi_{2}s_{0}=s_{4}\pi_{2}$, $\pi_{2}s_{1}=s_{5}\pi_{2}$,
$\pi_{2}s_{2}=s_{3}\pi_{2}$ and

(2.2) $\displaystyle s_{i}s_{j}s_{i}=s_{j}s_{i}s_{j},\quad$
$\displaystyle\\{i,j\\}=\\{0,2\\},\\{1,2\\},\\{2,3\\},\\{3,4\\},\\{3,5\\},$
$\displaystyle s_{i}s_{j}=s_{j}s_{i},\quad$ otherwise.

Recall that the operator $L_{1}$ is defined in Eq. (1.2) as one of the Lax
pair, and the linear $q$-differential equation $L_{1}y(z)=0$ is written as

(2.3)
$\displaystyle\Bigl{\\{}\frac{z(g\nu_{1}-1)(g\nu_{2}-1)}{qg}-\frac{\nu_{1}\nu_{2}\nu_{3}\nu_{4}(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}{fg}\Bigr{\\}}y(z)$
$\displaystyle+\frac{\nu_{1}\nu_{2}(z-q\nu_{3})(z-q\nu_{4})}{q(qf-z)}(gy(z)-y(z/q))$
$\displaystyle+\frac{(z-\kappa_{1}/\nu_{7})(z-\kappa_{1}/\nu_{8})}{q(f-z)}\Bigl{(}\frac{1}{g}y(z)-y(qz)\Bigr{)}=0.$

Note that Eq. (2.3) was studied in [4] from the aspect of the initial-value
space of $q$-$P(D^{(1)}_{5})$, and it was observed that the $q$-Heun equation
appears.

In this paper, we investigate the action of the symmetry in Eq. (2.1) on Eq.
(2.3). It follows immediately that Eq. (2.3) is invariant under the actions of
$s_{0}$, $s_{1}$, $s_{4}$ and $s_{5}$. It was discussed in [5] that the action
of $s_{3}$ is related to the $q$-middle convolution.

Next, we investigate the action of $s_{2}$ on Eq. (2.3). It follows from Eq.
(2.1) that $\nu_{3}$ and $\kappa_{1}/\nu_{7}$ are exchanged by the action of
$s_{2}$. Thus, we set

(2.4)
$y(z)=\tilde{y}(z)\frac{(q\nu_{3}/z;q)_{\infty}}{(q\kappa_{1}/(\nu_{7}z);q)_{\infty}}=\tilde{y}(z)\prod_{j=0}^{\infty}\frac{1-q^{j+1}\nu_{3}/z}{1-q^{j+1}\kappa_{1}/(\nu_{7}z)}.$

Then we have

(2.5) $\displaystyle
y(qz)=\tilde{y}(qz)\frac{(1-\nu_{3}/z)}{(1-\kappa_{1}/(\nu_{7}z))}\frac{(q\nu_{3}/z;q)_{\infty}}{(q\kappa_{1}/(\nu_{7}z);q)_{\infty}},$
$\displaystyle
y(z/q)=\tilde{y}(z/q)\frac{(1-q\kappa_{1}/(\nu_{7}z))}{(1-q\nu_{3}/z)}\frac{(q\nu_{3}/z;q)_{\infty}}{(q\kappa_{1}/(\nu_{7}z);q)_{\infty}}.$

Therefore, if $y(z)$ is a solution of Eq. (2.3), then $\tilde{y}(z)$ satisfies

(2.6)
$\displaystyle\Bigl{\\{}\frac{z(g\nu_{1}-1)(g\nu_{2}-1)}{qg}-\frac{\nu_{1}\nu_{2}\nu_{3}\nu_{4}(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}{fg}\Bigr{\\}}\tilde{y}(z)$
$\displaystyle+\frac{\nu_{1}\nu_{2}(z-q\nu_{3})(z-q\nu_{4})}{q(qf-z)}g\tilde{y}(z)-\frac{\nu_{1}\nu_{2}(z-q\kappa_{1}/\nu_{7})(z-q\nu_{4})}{q(qf-z)}\tilde{y}(z/q)$
$\displaystyle+\frac{(z-\kappa_{1}/\nu_{7})(z-\kappa_{1}/\nu_{8})}{q(f-z)}\frac{1}{g}\tilde{y}(z)-\frac{(z-\nu_{3})(z-\kappa_{1}/\nu_{8})}{q(f-z)}\tilde{y}(qz)=0.$

It follows from a straightforward calculation that Eq. (2.6) is written as

(2.7)
$\displaystyle\left\\{\frac{z(\tilde{g}\nu_{1}-1)(\tilde{g}\nu_{2}-1)}{q\tilde{g}}-\frac{\nu_{1}\nu_{2}\tilde{\nu}_{3}\nu_{4}(\tilde{g}-\nu_{5}/\tilde{\kappa}_{2})(\tilde{g}-\nu_{6}/\tilde{\kappa}_{2})}{f\tilde{g}}\right\\}\tilde{y}(z)$
$\displaystyle+\frac{\nu_{1}\nu_{2}(z-q\tilde{\nu}_{3})(z-q\nu_{4})}{q(qf-z)}(\tilde{g}\tilde{y}(z)-\tilde{y}(z/q))$
$\displaystyle+\frac{(z-\kappa_{1}/\tilde{\nu}_{7})(z-\kappa_{1}/\nu_{8})}{q(f-z)}\Bigl{(}\frac{1}{\tilde{g}}\tilde{y}(z)-\tilde{y}(qz)\Bigr{)}=0$

by setting $\tilde{\nu}_{3}=\kappa_{1}/\nu_{7}$,
$\tilde{\nu}_{7}=\kappa_{1}/\nu_{3}$,
$\tilde{\kappa}_{2}=\kappa_{1}\kappa_{2}/(\nu_{3}\nu_{7})$ and
$\tilde{g}=g(f-\nu_{3})/(f-\kappa_{1}/\nu_{7})$. Therefore, the gauge
transformation defined by Eq. (2.4) induces the symmetry $s_{2}$ of the
$q$-Painlevé equation of type $D^{(1)}_{5}$. The gauge transformation defined
by

(2.8)
$y(z)=\tilde{y}(z)\frac{(q\nu_{3}/z;q)_{\infty}(q\nu_{4}/z;q)_{\infty}}{(q\kappa_{1}/(\nu_{7}z);q)_{\infty}(q\kappa_{1}/(\nu_{8}z);q)_{\infty}}$

induces the symmetry $s_{2}s_{1}s_{0}s_{2}$ of the $q$-Painlevé equation of
type $D^{(1)}_{5}$.

We examine another gauge-transformation. Let $y(z)$ be a solution of Eq. (2.3)
and set $y(z)=z^{d}\tilde{y}(z)$. Then the function $\tilde{y}(z)$ satisfies

(2.9)
$\displaystyle\Bigl{\\{}\frac{z(q^{d}gq^{-d}\nu_{1}-1)(q^{d}gq^{-d}\nu_{2}-1)}{qq^{d}g}$
$\displaystyle\qquad-\frac{q^{-d}\nu_{1}q^{-d}\nu_{2}\nu_{3}\nu_{4}(q^{d}g-q^{d}\nu_{5}/\kappa_{2})(q^{d}g-q^{d}\nu_{6}/\kappa_{2})}{fq^{d}g}\Bigr{\\}}\tilde{y}(z)$
$\displaystyle+\frac{q^{-d}\nu_{1}q^{-d}\nu_{2}(z-q\nu_{3})(z-q\nu_{4})}{q(qf-z)}(q^{d}g\tilde{y}(z)-\tilde{y}(z/q))$
$\displaystyle+\frac{(z-\kappa_{1}/\nu_{7})(z-\kappa_{1}/\nu_{8})}{q(f-z)}\Bigl{(}\frac{1}{q^{d}g}\tilde{y}(z)-\tilde{y}(qz)\Bigr{)}=0.$

Hence, the correspondence of the parameters is written as
$(\nu_{1},\nu_{2},\nu_{5}/\kappa_{2},\nu_{6}/\kappa_{2},g)\mapsto(q^{-d}\nu_{1},q^{-d}\nu_{2},q^{d}\nu_{5}/\kappa_{2},q^{d}\nu_{5}/\kappa_{2},q^{d}g)$,
while the parameters $\nu_{3},\nu_{4},\nu_{7}/\kappa_{1},\nu_{8}/\kappa_{1},f$
are unchanged. We denote by $G[s]$ the transformation of the parameters

(2.10)
$\displaystyle(\nu_{1},\nu_{2},\nu_{3},\nu_{4},\nu_{5}/\kappa_{2},\nu_{6}/\kappa_{2},\nu_{7}/\kappa_{1},\nu_{8}/\kappa_{1},f,g)$
$\displaystyle\mapsto(\nu_{1}/s,\nu_{2}/s,\nu_{3},\nu_{4},s\nu_{5}/\kappa_{2},s\nu_{6}/\kappa_{2},\nu_{7}/\kappa_{1},\nu_{8}/\kappa_{1},f,sg).$

We investigate the dilation $z\mapsto cz$ for the non-zero constant $c$. Set
$z=u/c$ and $y(z)=\tilde{y}(u)$. Let $y(z)$ be a solution of Eq. (2.3). Then
the function $\tilde{y}(u)$ satisfies

(2.11)
$\displaystyle\Big{\\{}u\frac{(\nu_{1}g-1)(\nu_{2}g-1)}{qg}-\nu_{1}\nu_{2}c\nu_{3}c\nu_{4}\frac{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}{cfg}\Big{\\}}\tilde{y}(u)$
$\displaystyle+\frac{\nu_{1}\nu_{2}(u-qc\nu_{3})(u-qc\nu_{4})}{q(qcf-u)}\Big{(}g\tilde{y}(u)-\tilde{y}(u/q)\Big{)}$
$\displaystyle+\frac{(u-c\kappa_{1}/\nu_{7})(u-c\kappa_{1}/\nu_{8})}{q(cf-u)}\Big{(}\frac{1}{g}\tilde{y}(u)-\tilde{y}(qu)\Big{)}=0.$

Hence, the correspondence for the parameters is written as

(2.12)
$\displaystyle(\nu_{1},\nu_{2},\nu_{3},\nu_{4},\nu_{5}/\kappa_{2},\nu_{6}/\kappa_{2},\nu_{7}/\kappa_{1},\nu_{8}/\kappa_{1},f,g)$
$\displaystyle\mapsto(\nu_{1},\nu_{2},c\nu_{3},c\nu_{4},\nu_{5}/\kappa_{2},\nu_{6}/\kappa_{2},\nu_{7}/\kappa_{1}/c,\nu_{8}/\kappa_{1}/c,cf,g),$

which we denote by $D[c]$.

We investigate the symmetry related to $z\leftrightarrow 1/z$. More precisely
we take into account a more general situation $z=c/u$ and
$y(z)=z^{d}\tilde{y}(u)=z^{d}\tilde{y}(c/z)$, where $y(z)$ is a solution of
Eq. (2.3). Then we have

(2.13)
$\displaystyle\Bigl{\\{}\frac{c(g\nu_{1}-1)(g\nu_{2}-1)}{uqg}-\frac{\nu_{1}\nu_{2}\nu_{3}\nu_{4}(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}{fg}\Bigr{\\}}\tilde{y}(u)$
$\displaystyle+\frac{\nu_{1}\nu_{2}(c/u-q\nu_{3})(c/u-q\nu_{4})}{q(qf-c/u)}(g\tilde{y}(u)-q^{-d}\tilde{y}(qu))$
$\displaystyle+\frac{(c/u-\kappa_{1}/\nu_{7})(c/u-\kappa_{1}/\nu_{8})}{q(f-c/u)}\Bigl{(}\frac{1}{g}\tilde{y}(u)-q^{d}\tilde{y}(u/q)\Bigr{)}=0.$

Therefore,

(2.14)
$\displaystyle\Bigl{\\{}\frac{u((1/g)\nu_{5}/\kappa_{2}-1)((1/g)\nu_{6}/\kappa_{2}-1)}{q/(gq^{d})}-\frac{c\nu_{5}\nu_{6}\nu_{7}\nu_{8}(1/g-\nu_{1})(1/g-\nu_{2})}{q\kappa_{1}^{2}\kappa_{2}^{2}/(fgq^{d})}\Bigr{\\}}\tilde{y}(u)$
$\displaystyle+\frac{q^{2d}\nu_{5}\nu_{6}(u-c\nu_{7}/\kappa_{1})(u-c\nu_{8}/\kappa_{1})}{\kappa_{2}^{2}q(c/f-u)}\Bigl{(}\frac{1}{gq^{d}}\tilde{y}(u)-\tilde{y}(u/q)\Bigr{)}$
$\displaystyle+\frac{(u-c/(q\nu_{3}))(u-c/(q\nu_{4}))}{q(c/(qf)-u)}(q^{d}g\tilde{y}(u)-\tilde{y}(qu))=0.$

In the case $q^{d}=\kappa_{2}$ and $c=q\kappa_{1}$, the equation is written as

(2.15)
$\displaystyle\Bigl{\\{}\frac{u(\nu_{5}/(g\kappa_{2})-1)(\nu_{6}/(g\kappa_{2})-1)}{q/(g\kappa_{2})}$
$\displaystyle\qquad-\frac{\nu_{5}\nu_{6}\nu_{7}\nu_{8}(1/(g\kappa_{2})-\nu_{1}/\kappa_{2})(1/(g\kappa_{2})-\nu_{2}/\kappa_{2})}{\kappa_{1}/(fg\kappa_{2})}\Bigr{\\}}\tilde{y}(u)$
$\displaystyle+\frac{\nu_{5}\nu_{6}(u-q\nu_{7})(u-q\nu_{8})}{q(q\kappa_{1}/f-u)}\Bigl{(}\frac{1}{g\kappa_{2}}\tilde{y}(u)-\tilde{y}(u/q)\Bigr{)}$
$\displaystyle+\frac{(u-\kappa_{1}/\nu_{3})(u-\kappa_{1}/\nu_{4})}{q(\kappa_{1}/f-u)}(g\kappa_{2}\tilde{y}(u)-\tilde{y}(qu))=0,$

and the correspondence of the parameters is described by the action of
$\pi_{2}\pi_{1}\pi_{2}\pi_{1}(=\pi_{1}\pi_{2}\pi_{1}\pi_{2})$,

(2.16)
$\displaystyle(\nu_{1},\nu_{2},\nu_{3},\nu_{4},\nu_{5},\nu_{6},\nu_{7},\nu_{8};\kappa_{1},\kappa_{2};f,g)$
$\displaystyle\mapsto(\nu_{5},\nu_{6},\nu_{7},\nu_{8},\nu_{1},\nu_{2},\nu_{3},\nu_{4};\kappa_{1},\kappa_{2};\kappa_{1}/f,1/(\kappa_{2}g)).$

On $q$-$P(D^{(1)}_{5})$, a candidate of the time evolution by the symmetry of
$\widehat{W}(D^{(1)}_{5})$ is $(\pi_{2}\pi_{1}s_{2}s_{1}s_{0}s_{2})^{2}$,
which was essentially written in Sakai [3]. By a direct calculation, we have

(2.17)
$\displaystyle(\pi_{2}\pi_{1}s_{2}s_{1}s_{0}s_{2})^{2}(f)=\pi_{2}\pi_{1}s_{2}s_{1}s_{0}s_{2}(1/g)=\frac{q\nu_{3}\nu_{4}\nu_{7}\nu_{8}}{f\kappa_{1}}\frac{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}{(g-1/\nu_{1})(g-1/\nu_{2})},$

which does not coincide with

(2.18)
$\displaystyle\overline{f}=\frac{\nu_{3}\nu_{4}}{f}\frac{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}{(g-1/\nu_{1})(g-1/\nu_{2})}.$

Moreover we have
$(\pi_{2}\pi_{1}s_{2}s_{1}s_{0}s_{2})^{2}(\nu_{3})=q\nu_{3}\nu_{7}\nu_{8}/\kappa_{1}(\neq\nu_{3})$.
We can adjust the expressions as the following theorem.

######  Theorem 2.1.

(i) Let $\Xi$ be the transformation of the parameters defined by

(2.19)
$\displaystyle(\nu_{1},\nu_{2},\nu_{3},\nu_{4},\nu_{5},\nu_{6},\nu_{7},\nu_{8};\kappa_{1},\kappa_{2};f,g)$
$\displaystyle\mapsto\Bigl{(}\frac{\nu_{1}\nu_{5}\nu_{6}}{\kappa_{2}},\frac{\nu_{2}\nu_{5}\nu_{6}}{\kappa_{2}},\frac{\kappa_{1}}{q\nu_{4}},\frac{\kappa_{1}}{q\nu_{3}},\frac{\nu_{5}\nu_{1}\nu_{2}}{\kappa_{2}},\frac{\nu_{6}\nu_{1}\nu_{2}}{\kappa_{2}},\frac{\kappa_{1}}{q\nu_{8}},\frac{\kappa_{1}}{q\nu_{7}};$
$\displaystyle\qquad\qquad\qquad\qquad\frac{\kappa_{1}^{3}}{q^{2}\nu_{3}\nu_{4}\nu_{7}\nu_{8}},\frac{\nu_{1}\nu_{2}\nu_{5}\nu_{6}}{\kappa_{2}};\frac{f\kappa_{1}}{q\nu_{3}\nu_{4}},\frac{g\kappa_{2}}{\nu_{5}\nu_{6}}\Bigr{)}.$

Let $T$ be the transformation of the parameters defined by

(2.20) $T=\Xi\cdot(\pi_{2}\pi_{1}s_{2}s_{1}s_{0}s_{2})^{2}.$

Then

(2.21) $\displaystyle
T(f)=\frac{\nu_{3}\nu_{4}}{f}\frac{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}{(g-1/\nu_{1})(g-1/\nu_{2})},$
$\displaystyle
T(g)=\frac{1}{g\nu_{1}\nu_{2}}\frac{(T(f)-T(\kappa_{1}/\nu_{7}))(T(f)-T(\kappa_{1}/\nu_{8}))}{(T(f)-\nu_{3})(T(f)-\nu_{4})}.$

Namely the operator $T$ represents the time evolution of $q$-$P(D^{(1)}_{5})$
(see Eq. (1.1)). On the other parameters, we have

(2.22) $\displaystyle\displaystyle
T(\nu_{i})=\nu_{i}\;(i=1,2,\dots,8),\;T(\kappa_{1})=\kappa_{1}/q,\;T(\kappa_{2})=q\kappa_{2}.$

(ii) Let $G[s]$ and $D[c]$ be the operators in Eqs. (2.10) and (2.12). For any
elements $\zeta$ generated by
$\nu_{1},\nu_{2},\nu_{3},\nu_{4},\nu_{5}/\kappa_{2},\nu_{6}/\kappa_{2},\nu_{7}/\kappa_{1}$,
$\nu_{8}/\kappa_{1}$, $f$ and $g$, we have

(2.23)
$\Xi(\zeta)=G\Bigl{[}\frac{\kappa_{2}}{\nu_{5}\nu_{6}}\Bigr{]}D\Bigl{[}\frac{\kappa_{1}}{q\nu_{7}\nu_{8}}\Bigr{]}(\zeta).$

###### Proof.

Set $s=\pi_{2}\pi_{1}s_{2}s_{1}s_{0}s_{2}$. It follows from Eq. (2.1) that

(2.24) $\displaystyle
s(\nu_{1})=\nu_{7},\>s(\nu_{2})=\nu_{8},\>s(\nu_{3})=\kappa_{2}/\nu_{6},\>s(\nu_{4})=\kappa_{2}/\nu_{5},$
$\displaystyle
s(\nu_{5})=\nu_{3},\>s(\nu_{6})=\nu_{4},\>s(\nu_{7})=\kappa_{2}/\nu_{2},\>s(\nu_{8})=\kappa_{2}/\nu_{1},$
$\displaystyle
s(\kappa_{1})=\kappa_{2},\>s(\kappa_{2})=\kappa_{1}\kappa_{2}^{2}/(\nu_{1}\nu_{2}\nu_{5}\nu_{6})=q\nu_{3}\nu_{4}\nu_{7}\nu_{8}/\kappa_{1}.$

Then

(2.25) $\displaystyle
s^{2}(\nu_{1})=\kappa_{2}/\nu_{2},\>s^{2}(\nu_{2})=\kappa_{2}/\nu_{1},\>s^{2}(\nu_{3})=q\nu_{3}\nu_{7}\nu_{8}/\kappa_{1},\>s^{2}(\nu_{4})=q\nu_{4}\nu_{7}\nu_{8}/\kappa_{1},$
$\displaystyle
s^{2}(\nu_{5})=\kappa_{2}/\nu_{6},\>s^{2}(\nu_{6})=\kappa_{2}/\nu_{5},\>s^{2}(\nu_{7})=q\nu_{3}\nu_{4}\nu_{7}/\kappa_{1},\>s^{2}(\nu_{8})=q\nu_{3}\nu_{4}\nu_{8}/\kappa_{1},$
$\displaystyle
s^{2}(\kappa_{1})=q\nu_{3}\nu_{4}\nu_{7}\nu_{8}/\kappa_{1},\>s^{2}(\kappa_{2})=q\kappa_{2}^{3}/(\nu_{1}\nu_{2}\nu_{5}\nu_{6}).$

Let $\Xi$ be the operator defined in Eq. (2.19). We have

(2.26) $\Xi s^{2}(\nu_{i})=\nu_{i},\;(i=1,2,\dots,8),\;\Xi
s^{2}(\kappa_{1})=\kappa_{1}/q,\;\Xi s^{2}(\kappa_{2})=q\kappa_{2}.$

Therefore, we obtain Eq. (2.22).

It follows from Eq. (2.1) that $s(f)=1/g$ and

(2.27)
$s(g)=\frac{\kappa_{1}f}{q\nu_{3}\nu_{4}\nu_{7}\nu_{8}}\frac{(g-1/\nu_{1})(g-1/\nu_{2})}{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}.$

Then

(2.28)
$s^{2}(f)=\frac{1}{s(g)}=\frac{q\nu_{3}\nu_{4}\nu_{7}\nu_{8}}{\kappa_{1}f}\frac{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}{(g-1/\nu_{1})(g-1/\nu_{2})}.$

Since $\Xi(f)=f\kappa_{1}/(q\nu_{3}\nu_{4})$ and
$\Xi(g)=g\kappa_{2}/(\nu_{5}\nu_{6})$, we have

(2.29) $\Xi
s^{2}(f)=\Xi\Bigl{(}\frac{q\nu_{3}\nu_{4}\nu_{7}\nu_{8}}{f\kappa_{1}}\Bigr{)}\Xi\Bigl{(}\frac{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}{(g-1/\nu_{1})(g-1/\nu_{2})}\Bigr{)}=\frac{\nu_{3}\nu_{4}}{f}\frac{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}{(g-1/\nu_{1})(g-1/\nu_{2})}.$

Therefore we obtain the first equation of (2.21).

It follows from Eq. (2.27) that

(2.30) $\displaystyle
s^{2}(g)=s(f)s\Bigl{(}\frac{\kappa_{1}}{q\nu_{3}\nu_{4}\nu_{7}\nu_{8}}\Bigr{)}s\Bigl{(}\frac{(g-1/\nu_{1})(g-1/\nu_{2})}{(g-/\kappa_{2})(g-\nu_{6}/\kappa_{2})}\Bigr{)}$
$\displaystyle=\frac{1}{g}\frac{1}{s^{2}(\kappa_{2})}\frac{(1/s^{2}(f)-1/\nu_{7})(1/s^{2}(f)-1/\nu_{8})}{(1/s^{2}(f)-s^{2}(1/\nu_{4}))(1/s^{2}(f)-s^{2}(1/\nu_{3}))}.$

We apply the operation $\Xi$. It follows from $T=\Xi s^{2}$ and Eq. (2.26)
that

(2.31) $\displaystyle
T(g)=\frac{1}{\Xi(g)}\frac{1}{q\kappa_{2}}\frac{(1/T(f)-1/\Xi(\nu_{7}))(1/T(f)-1/\Xi(\nu_{8}))}{(1/T(f)-1/\nu_{4})(1/T(f)-1/\nu_{3})}$
$\displaystyle=\frac{\nu_{5}\nu_{6}}{gq\kappa_{2}^{2}}\frac{\nu_{3}\nu_{4}}{\Xi(\nu_{7})\Xi(\nu_{8})}\frac{(T(f)-\Xi(\nu_{8}))(T(f)-\Xi(\nu_{7}))}{(T(f)-\nu_{3})(T(f)-\nu_{4})}$

Then the second equation of (2.21) follows from
$\Xi(\nu_{7})=\kappa_{1}/(q\nu_{8})$, $\Xi(\nu_{8})=\kappa_{1}/(q\nu_{7})$ and
the relation
$\kappa_{1}^{2}\kappa_{2}^{2}=q\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}\nu_{6}\nu_{7}\nu_{8}$.
Therefore we obtain (i).

(ii) follows from confirming the relation
$\Xi(\nu_{5}/\kappa_{2})=1/\nu_{6}=G[\kappa_{2}/(\nu_{5}\nu_{6})](\nu_{5}/\kappa_{2})$
and so on. ∎

Theorem 2.1 asserts that the time evolution is essentially expressed as a
composition of Weyl group symmetry and the symmetry originated from a gauge
transformation and a dilation.

##  3\. The $q$-Painlevé equation of type $E^{(1)}_{6}$

The $q$-Painlevé equation of type $E^{(1)}_{6}$ ($q$-$P(E^{(1)}_{6})$) was
given as

(3.1)
$\displaystyle\frac{(fg-1)(\overline{f}g-1)}{f\overline{f}}=\frac{(g-1/\nu_{1})(g-1/\nu_{2})(g-1/\nu_{3})(g-1/\nu_{4})}{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})},$
$\displaystyle\frac{(fg-1)(f\underline{g}-1)}{g\underline{g}}=\frac{(f-\nu_{1})(f-\nu_{2})(f-\nu_{3})(f-\nu_{4})}{(f-\kappa_{1}/\nu_{7})(f-\kappa_{1}/\nu_{8})}.$

The equation $q$-$P(E^{(1)}_{6})$ is realized by the compatibility condition
for the Lax pair $L_{1}$ and $L_{2}$, where

(3.2) $\displaystyle L_{1}=$
$\displaystyle\frac{z(g\nu_{1}-1)(g\nu_{2}-1)(g\nu_{3}-1)(g\nu_{4}-1)}{g(fg-1)(gz-q)}-\frac{(g\kappa_{2}/\nu_{5}-1)(g\kappa_{2}/\nu_{6}-1){\kappa_{1}}^{2}}{qfg\nu_{7}\nu_{8}}$
$\displaystyle+\frac{(\nu_{1}-z/q)(\nu_{2}-z/q)(\nu_{3}-z/q)(\nu_{4}-z/q)}{f-z/q}\Bigl{\\{}\frac{g}{1-gz/q}-T_{z}^{-1}\Bigr{\\}}$
$\displaystyle+\frac{(\kappa_{1}/\nu_{7}-z)(\kappa_{1}/\nu_{8}-z)}{q(f-z)}\Bigl{\\{}\Bigl{(}\frac{1}{g}-z\Bigr{)}-T_{z}\Bigr{\\}}\,,$
$\displaystyle L_{2}=$
$\displaystyle\Bigl{(}1-\frac{f}{z}\Bigr{)}T+T_{z}-\Bigl{(}\frac{1}{g}-z\Bigr{)}\,.$

Then the linear $q$-differential equation $L_{1}y(z)=0$ is written as

(3.3)
$\displaystyle\Bigl{\\{}\frac{z(g\nu_{1}-1)(g\nu_{2}-1)(g\nu_{3}-1)(g\nu_{4}-1)}{g(fg-1)(gz-q)}-\frac{(g\kappa_{2}/\nu_{5}-1)(g\kappa_{2}/\nu_{6}-1){\kappa_{1}}^{2}}{qfg\nu_{7}\nu_{8}}\Bigr{\\}}y(z)$
$\displaystyle+\frac{(\nu_{1}-z/q)(\nu_{2}-z/q)(\nu_{3}-z/q)(\nu_{4}-z/q)}{f-z/q}\Bigl{(}\frac{g}{1-gz/q}y(z)-y(z/q)\Bigr{)}$
$\displaystyle+\frac{(\kappa_{1}/\nu_{7}-z)(\kappa_{1}/\nu_{8}-z)}{q(f-z)}\Bigl{(}\Bigl{(}\frac{1}{g}-z\Bigr{)}y(z)-y(qz)\Bigr{)}=0.$

Note that Eq. (3.3) was studied in [4] from the aspect of the initial-value
space of $q$-$P(E^{(1)}_{6})$, and it was observed that a variant of the
$q$-Heun equation appears.

The Weyl group symmetry of the $q$-Painlevé equation $q$-$P(E^{(1)}_{6})$ was
given by the following action of the operators $s_{0},\dots,s_{6}$, $\pi_{1}$
and $\pi_{2}$ for the parameters
$(\kappa_{1},\kappa_{2},\nu_{1},\dots,\nu_{8})$ and $(f,g)$ in [2].

(3.4) $\displaystyle s_{0}:$
$\displaystyle\>\nu_{7}\leftrightarrow\nu_{8},\quad
s_{1}:\>\nu_{5}\leftrightarrow\nu_{6},\quad
s_{3}:\>\nu_{1}\leftrightarrow\nu_{2},\quad
s_{4}:\>\nu_{2}\leftrightarrow\nu_{3},\quad
s_{5}:\>\nu_{3}\leftrightarrow\nu_{4},$ $\displaystyle s_{2}:$
$\displaystyle\>\nu_{1}\rightarrow\frac{\kappa_{2}}{\nu_{6}},\;\nu_{6}\rightarrow\frac{\kappa_{2}}{\nu_{1}},\;\kappa_{1}\rightarrow\frac{\kappa_{1}\kappa_{2}}{\nu_{1}\nu_{6}},\;f\rightarrow
f\,\frac{\kappa_{2}(\nu_{1}g-1)}{-(\kappa_{2}-\nu_{1}\nu_{6})fg+\nu_{1}\kappa_{2}g-\nu_{1}\nu_{6}},$
$\displaystyle s_{6}:$
$\displaystyle\>\nu_{1}\rightarrow\frac{\kappa_{1}}{\nu_{7}},\;\nu_{7}\rightarrow\frac{\kappa_{1}}{\nu_{1}},\;\kappa_{2}\rightarrow\frac{\kappa_{1}\kappa_{2}}{\nu_{1}\nu_{7}},\;g\rightarrow
g\,\frac{\nu_{7}(\nu_{1}-f)}{\kappa_{1}-\nu_{7}f+(\nu_{1}\nu_{7}-\kappa_{1})fg},$
$\displaystyle\pi_{1}:$ $\displaystyle\>q\rightarrow
1/q,\>\nu_{1}\rightarrow\nu_{2}/\kappa_{2},\>\nu_{2}\rightarrow\nu_{1}/\kappa_{2},\>\nu_{3}\rightarrow
1/\nu_{6},\>\nu_{4}\rightarrow 1/\nu_{5},\>\nu_{5}\rightarrow 1/\nu_{4},$
$\displaystyle\>\nu_{6}\rightarrow 1/\nu_{3},\>\nu_{7}\rightarrow
1/\nu_{7},\>\nu_{8}\rightarrow
1/\nu_{8},\>\kappa_{1}\rightarrow\nu_{1}\nu_{2}/(\kappa_{1}\kappa_{2}),\>\kappa_{2}\rightarrow
1/\kappa_{2},$
$\displaystyle\>f\rightarrow\frac{\nu_{1}\nu_{2}(1-fg)}{\kappa_{2}\\{\nu_{1}\nu_{2}g+f-(\nu_{1}+\nu_{2})fg\\}},\>g\rightarrow\kappa_{2}g,$
$\displaystyle\pi_{2}:$ $\displaystyle\>q\rightarrow 1/q,\>\nu_{1}\rightarrow
1/\nu_{1},\>\nu_{2}\rightarrow 1/\nu_{2},\>\nu_{3}\rightarrow
1/\nu_{3},\>\nu_{4}\rightarrow 1/\nu_{4},\>\nu_{5}\rightarrow 1/\nu_{8},$
$\displaystyle\>\nu_{6}\rightarrow 1/\nu_{7},\>\nu_{7}\rightarrow
1/\nu_{6},\>\nu_{8}\rightarrow 1/\nu_{5},\>\kappa_{1}\rightarrow
1/\kappa_{2},\>\kappa_{2}\rightarrow 1/\kappa_{1},\>f\leftrightarrow g.$

Then we can confirm that these operations satisfy the relations of the
extended Weyl group $\widetilde{W}(E^{(1)}_{6})$ whose Dynkin diagram is as
follows.
$1$$2$$3$$4$$5$$6$$0$ We investigate the action of the symmetry in Eq. (3.4)
on the equation $L_{1}y(z)=0$. It follows immediately that Eq. (3.3) is
invariant under the actions of $s_{0}$, $s_{1}$, $s_{3}$, $s_{4}$ and $s_{5}$.
We investigate the action of $s_{6}$ on Eq. (3.3). It follows from Eq. (3.4)
that $\nu_{1}$ and $\kappa_{1}/\nu_{7}$ are exchanged by the action of
$s_{6}$. Thus, we set

(3.5)
$y(z)=\tilde{y}(z)\frac{(q\nu_{1}/z;q)_{\infty}}{(q\kappa_{1}/(\nu_{7}z);q)_{\infty}}.$

Then it is shown as the case of $D^{(1)}_{5}$ that, if $y(z)$ is a solution of
Eq. (3.3), then $\tilde{y}(z)$ satisfies

(3.6)
$\displaystyle\Bigl{\\{}\frac{z(\tilde{g}\tilde{\nu}_{1}-1)(\tilde{g}\nu_{2}-1)(\tilde{g}\nu_{3}-1)(\tilde{g}\nu_{4}-1)}{\tilde{g}(f\tilde{g}-1)(\tilde{g}z-q)}-\frac{(\tilde{g}\tilde{\kappa}_{2}/\nu_{5}-1)(\tilde{g}\tilde{\kappa}_{2}/\nu_{6}-1){\kappa_{1}}^{2}}{qf\tilde{g}\tilde{\nu}_{7}\nu_{8}}\Bigr{\\}}\tilde{y}(z)$
$\displaystyle+\frac{(\tilde{\nu}_{1}-z/q)(\nu_{2}-z/q)(\nu_{3}-z/q)(\nu_{4}-z/q)}{f-z/q}\Bigl{(}\frac{\tilde{g}}{1-\tilde{g}z/q}\tilde{y}(z)-\tilde{y}(z/q)\Bigr{)}$
$\displaystyle+\frac{(\kappa_{1}/\tilde{\nu}_{7}-z)(\kappa_{1}/\nu_{8}-z)}{q(f-z)}\Bigl{(}\Bigl{(}\frac{1}{\tilde{g}}-z\Bigr{)}\tilde{y}(z)-\tilde{y}(qz)\Bigr{)}=0,$

where

(3.7)
$\displaystyle\tilde{g}=g\,\frac{\nu_{7}(\nu_{1}-f)}{\kappa_{1}-\nu_{7}f+(\nu_{1}\nu_{7}-\kappa_{1})fg},\;\tilde{\nu}_{1}=\frac{\kappa_{1}}{\nu_{7}},\;\tilde{\nu}_{7}=\frac{\kappa_{1}}{\nu_{1}},\tilde{\kappa}_{2}=\frac{\kappa_{1}\kappa_{2}}{\nu_{1}\nu_{7}}.$

Therefore, the gauge transformation defined in Eq. (3.5) induces the symmetry
$s_{6}$ of the $q$-Painlevé equation of type $E^{(1)}_{6}$.

We investigate the actions of the gauge transformation and dilation in Eq.
(3.4) on the equation $L_{1}y(z)=0$. Set $z=u/c$,
$y(z)=z^{d}\tilde{y}(u)=z^{d}\tilde{y}(cz)$. Then the function $\tilde{y}(u)$
satisfies

(3.8)
$\displaystyle\left\\{\frac{u(g\nu_{1}-1)(g\nu_{2}-1)(g\nu_{3}-1)(g\nu_{4}-1)}{q(g/c)(fg-1)(ug/(cq)-1)}-c^{2}\frac{(g\kappa_{2}/\nu_{5}-1)(g\kappa_{2}/\nu_{6}-1){\kappa_{1}}^{2}}{qfg\nu_{7}\nu_{8}}\right\\}\tilde{y}(u)$
$\displaystyle+\frac{(c\nu_{1}-u/q)(c\nu_{2}-u/q)(c\nu_{3}-u/q)(c\nu_{4}-u/q)}{cf-u/q}\left\\{\frac{g/c}{1-ug/(cq)}\tilde{y}(u)-\frac{\tilde{y}(u/q)}{cq^{d}}\right\\}$
$\displaystyle+\frac{(c\kappa_{1}/\nu_{7}-u)(c\kappa_{1}/\nu_{8}-u)}{q(cf-u)}\left\\{\left(\frac{c}{g}-u\right)\tilde{y}(u)-cq^{d}\tilde{y}(qu)\right\\}=0.$

If $cq^{d}=1$, then it is written in the form of Eq.(3.3), where the
parameters are changed as

(3.9)
$\displaystyle(\nu_{1},\nu_{2},\nu_{3},\nu_{4},\nu_{5}/\kappa_{2},\nu_{6}/\kappa_{2},\nu_{7}/\kappa_{1},\nu_{8}/\kappa_{1},f,g)$
$\displaystyle\mapsto(c\nu_{1},c\nu_{2},c\nu_{3},c\nu_{4},\nu_{5}/\kappa_{2}/c,\nu_{6}/\kappa_{2}/c,\nu_{7}/\kappa_{1}/c,\nu_{8}/\kappa_{1}/c,cf,g/c).$

We denote the transformation of the parameters in Eq. (3.9) by $S_{E_{6}}[c]$.

The time evolution of the $q$-Painlevé equation of type $E^{(1)}_{6}$ is
expressed as the following theorem.

######  Theorem 3.1.

(i) Let $\Xi$ be the transformation of the parameters defined by

(3.10)
$\displaystyle(\nu_{1},\nu_{2},\nu_{3},\nu_{4},\nu_{5},\nu_{6},\nu_{7},\nu_{8};\kappa_{1},\kappa_{2};f,g)$
$\displaystyle\mapsto\Bigl{(}\frac{\nu_{1}\kappa_{2}}{\nu_{5}\nu_{6}\kappa_{1}^{2}},\frac{\nu_{2}\kappa_{2}}{\nu_{5}\nu_{6}\kappa_{1}^{2}},\frac{\nu_{3}\kappa_{2}}{\nu_{5}\nu_{6}\kappa_{1}^{2}},\frac{\nu_{4}\kappa_{2}}{\nu_{5}\nu_{6}\kappa_{1}^{2}},\frac{q\nu_{5}\kappa_{1}}{\kappa_{2}},\frac{q\nu_{6}\kappa_{1}}{\kappa_{2}},\frac{\kappa_{1}}{q\nu_{8}},\frac{\kappa_{1}}{q\nu_{7}};$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\frac{\kappa_{2}}{q\nu_{5}\nu_{6}\nu_{7}\nu_{8}},\frac{\kappa_{2}}{q\nu_{5}\nu_{6}\kappa_{1}};\frac{f\kappa_{2}}{\nu_{5}\nu_{6}\kappa_{1}^{2}},\frac{g\nu_{5}\nu_{6}\kappa_{1}^{2}}{\kappa_{2}}\Bigr{)}.$

Let $T$ be the transformation of the parameters defined by

(3.11)
$T=\Xi\cdot(\pi_{1}\pi_{2}s_{4}s_{5}s_{3}s_{6}s_{4}s_{3}s_{0}s_{6})^{2}.$

Then

(3.12)
$\displaystyle\frac{(T(f)g-1)(fg-1)}{T(f)f}=\frac{(g-1/\nu_{1})(g-1/\nu_{2})(g-1/\nu_{3})(g-1/\nu_{4})}{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})},$
$\displaystyle\frac{(T(f)g-1)(T(f)T(g)-1)}{gT(g)}=\frac{(T(f)-\nu_{1})(T(f)-\nu_{2})(T(f)-\nu_{3})(T(f)-\nu_{4})}{(T(f)-T(\kappa_{1}/\nu_{7}))(T(f)-T(\kappa_{1}/\nu_{8}))}.$

Namely the operator $T$ represents the time evolution of $q$-$P(E^{(1)}_{6})$
(see Eq. (3.1)). On the other parameters, we have

(3.13) $\displaystyle\displaystyle
T(\nu_{i})=\nu_{i}\;(i=1,2,\dots,8),\;T(\kappa_{1})=\kappa_{1}/q,\;T(\kappa_{2})=q\kappa_{2}.$

(ii) For any element $\zeta$ generated by
$\nu_{1},\nu_{2},\nu_{3},\nu_{4},\nu_{5}/\kappa_{2},\nu_{6}/\kappa_{2},\nu_{7}/\kappa_{1}$,
$\nu_{8}/\kappa_{1}$, $f$ and $g$, we have

(3.14)
$\Xi(\zeta)=S_{E_{6}}\Bigl{[}\frac{\kappa_{2}}{\nu_{5}\nu_{6}\kappa_{1}^{2}}\Bigr{]}(\zeta).$

###### Proof.

Set $s=\pi_{1}\pi_{2}s_{4}s_{5}s_{3}s_{6}s_{4}s_{3}s_{0}s_{6}$. It follows
from Eq. (3.4) that

(3.15) $\displaystyle
s(\nu_{1})=\kappa_{2}/\nu_{4},\>s(\nu_{2})=\kappa_{2}/\nu_{3},\>s(\nu_{3})=\kappa_{2}/\nu_{2},\>s(\nu_{4})=\kappa_{2}/\nu_{1},$
$\displaystyle
s(\nu_{5})=\nu_{8},\>s(\nu_{6})=\nu_{7},\>s(\nu_{7})=\kappa_{2}/\nu_{5},\>s(\nu_{8})=\kappa_{2}/\nu_{6},$
$\displaystyle
s(\kappa_{1})=\kappa_{2},\>s(\kappa_{2})=\kappa_{1}\kappa_{2}^{3}/(\nu_{1}\nu_{2}\nu_{3}\nu_{4}\nu_{5}\nu_{6})=q\nu_{7}\nu_{8}\kappa_{2}/\kappa_{1}.$

Then

(3.16) $\displaystyle
s^{2}(\nu_{1})=q\nu_{1}\nu_{7}\nu_{8}/\kappa_{1},\>s^{2}(\nu_{2})=q\nu_{2}\nu_{7}\nu_{8}/\kappa_{1},\>s^{2}(\nu_{3})=q\nu_{3}\nu_{7}\nu_{8}/\kappa_{1},\>s^{2}(\nu_{4})=q\nu_{4}\nu_{7}\nu_{8}/\kappa_{1},$
$\displaystyle
s^{2}(\nu_{5})=\kappa_{2}/\nu_{6},\>s^{2}(\nu_{6})=\kappa_{2}/\nu_{5},\>s^{2}(\nu_{7})=q\nu_{7}\kappa_{2}/\kappa_{1},\>s^{2}(\nu_{8})=q\nu_{8}\kappa_{2}/\kappa_{1},$
$\displaystyle
s^{2}(\kappa_{1})=q\nu_{7}\nu_{8}\kappa_{2}/\kappa_{1},\>s^{2}(\kappa_{2})=q^{2}\nu_{7}\nu_{8}\kappa_{2}^{2}/(\nu_{5}\nu_{6}\kappa_{1}).$

Let $\Xi$ be the operator defined in Eq. (3.10). Then we have

(3.17) $\Xi s^{2}(\nu_{i})=\nu_{i},\;(i=1,2,\dots,8),\;\Xi
s^{2}(\kappa_{1})=\kappa_{1}/q,\;\Xi s^{2}(\kappa_{2})=q\kappa_{2}.$

Hence we obtain Eq. (3.13).

It follows from Eq. (3.4) that $s(f)=\kappa_{2}g$ and

(3.18)
$\frac{1}{\kappa_{2}s(g)}=g+\frac{f(g-1/\nu_{1})(g-1/\nu_{2})(g-1/\nu_{3})(g-1/\nu_{4})}{(1-fg)(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}.$

Then

(3.19)
$\frac{1}{s^{2}(f)}=\frac{1}{s(\kappa_{2})s(g)}=\frac{\kappa_{1}}{q\nu_{7}\nu_{8}}\Bigl{\\{}g+\frac{f(g-1/\nu_{1})(g-1/\nu_{2})(g-1/\nu_{3})(g-1/\nu_{4})}{(1-fg)(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}\Bigr{\\}}.$

It follows from Eq. (3.10) that

(3.20) $\displaystyle\frac{1}{T(f)}=\frac{1}{\Xi
s^{2}(f)}=\Xi\Bigl{(}\frac{\kappa_{1}}{q\nu_{7}\nu_{8}}\Bigr{)}\Xi\Big{\\{}g+\frac{f(g-1/\nu_{1})(g-1/\nu_{2})(g-1/\nu_{3})(g-1/\nu_{4})}{(1-fg)(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}\Bigr{\\}}$
$\displaystyle=\frac{\kappa_{2}}{\nu_{5}\nu_{6}\kappa_{1}^{2}}\frac{\nu_{5}\nu_{6}\kappa_{1}^{2}}{\kappa_{2}}\Big{\\{}g+\frac{f(g-1/\nu_{1})(g-1/\nu_{2})(g-1/\nu_{3})(g-1/\nu_{4})}{(1-fg)(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}\Bigr{\\}}.$

It is equivalent to

(3.21)
$\frac{(T(f)g-1)(fg-1)}{T(f)f}=\frac{(g-1/\nu_{1})(g-1/\nu_{2})(g-1/\nu_{3})(g-1/\nu_{4})}{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})},$

and we obtain the first equation of (3.12).

It follows from Eq. (3.18) that

(3.22)
$\displaystyle\frac{1}{s^{2}(g)}=s(\kappa_{2})s\Bigl{(}g+\frac{f(g-1/\nu_{1})(g-1/\nu_{2})(g-1/\nu_{3})(g-1/\nu_{4})}{(1-fg)(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})}\Bigr{)}$
$\displaystyle=s(\kappa_{2}g)+\frac{s(f)\prod_{j=1}^{4}(s(\kappa_{2}g)-s(\kappa_{2}/\nu_{j}))}{s(\kappa_{2})(1-s(f)s(g))(s(\kappa_{2}g)-s(\nu_{5}))(s(\kappa_{2}g)-s(\nu_{6}))}$
$\displaystyle=s^{2}(f)+\frac{\kappa_{2}g(s^{2}(f)-s^{2}(\nu_{4}))(s^{2}(f)-s^{2}(\nu_{3}))(s^{2}(f)-s^{2}(\nu_{2}))(s^{2}(f)-s^{2}(\nu_{1}))}{(s(\kappa_{2})-\kappa_{2}gs^{2}(f))(s^{2}(f)-\nu_{8})(s^{2}(f)-\nu_{7})}.$

We apply the operation $\Xi$. Then we have

(3.23)
$\displaystyle\frac{1}{T(g)}=T(f)+\frac{(T(f)-T(\nu_{4}))(T(f)-T(\nu_{3}))(T(f)-T(\nu_{2}))(T(f)-T(\nu_{1}))}{(\Xi(q\nu_{7}\nu_{8}\kappa_{2}/\kappa_{1})/\Xi(\kappa_{2}g)-T(f))(T(f)-\Xi(\nu_{8}))(T(f)-\Xi
s(\nu_{7}))}$
$\displaystyle=T(f)+\frac{g(T(f)-\nu_{1})(T(f)-\nu_{2})(T(f)-\nu_{3})(T(f)-\nu_{4})}{(1-T(f)g)(T(f)-\kappa_{1}/(q\nu_{7}))(T(f)-\kappa_{1}/(q\nu_{8}))}.$

Therefore we obtain the second equation of (3.12).

(ii) follows similarly to the case $D^{(1)}_{5}$. ∎

Note that the expression
$(\pi_{1}\pi_{2}s_{4}s_{5}s_{3}s_{6}s_{4}s_{3}s_{0}s_{6})^{2}$ in Eq. (3.11)
can be essentially found in Tsuda’s paper [6].

##  4\. The $q$-Painlevé equation of type $E^{(1)}_{7}$

The $q$-Painlevé equation of type $E^{(1)}_{7}$ ($q$-$P(E^{(1)}_{7})$) was
given as

(4.1)
$\displaystyle\frac{(fg-\kappa_{1}/\kappa_{2})(\overline{f}g-\kappa_{1}/(q\kappa_{2}))}{(fg-1)(\overline{f}g-1)}=\frac{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})(g-\nu_{7}/\kappa_{2})(g-\nu_{8}/\kappa_{2})}{(g-1/\nu_{1})(g-1/\nu_{2})(g-1/\nu_{3})(g-1/\nu_{4})},$
$\displaystyle\frac{(fg-\kappa_{1}/\kappa_{2})(f\underline{g}-q\kappa_{1}/\kappa_{2})}{(fg-1)(f\underline{g}-1)}=\frac{(f-\kappa_{1}/\nu_{5})(f-\kappa_{1}/\nu_{6})(f-\kappa_{1}/\nu_{7})(f-\kappa_{1}/\nu_{8})}{(f-\nu_{1})(f-\nu_{2})(f-\nu_{3})(f-\nu_{4})}.$

The equation $q$-$P(E^{(1)}_{7})$ is realized by the compatibility condition
for the Lax pair $L_{1}$ and $L_{2}$, where

(4.2) $\displaystyle
L_{1}=\frac{q(\kappa_{1}-\kappa_{2})(g\kappa_{2}-\nu_{5})(g\kappa_{2}-\nu_{6})(g\kappa_{2}-\nu_{7})(g\kappa_{2}-\nu_{8})}{g\kappa_{1}{\kappa_{2}}^{2}(fg\kappa_{2}-\kappa_{1})(g\kappa_{2}z-\kappa_{1})}$
$\displaystyle\quad-\frac{q(\kappa_{1}-\kappa_{2})(g\nu_{1}-1)(g\nu_{2}-1)(g\nu_{3}-1)(g\nu_{4}-1)}{g(fg-1)\kappa_{1}\nu_{1}\nu_{2}\nu_{3}\nu_{4}(gz-q)}$
$\displaystyle\quad+\frac{(q\nu_{1}-z)(q\nu_{2}-z)(q\nu_{3}-z)(q\nu_{4}-z)\\{(g\kappa_{2}z-\kappa_{1}q)-\kappa_{1}(gz-q)T_{z}^{-1}\\}}{\kappa_{1}q\nu_{1}\nu_{2}\nu_{3}\nu_{4}(fq-z)z^{2}(q-gz)}$
$\displaystyle\quad-\frac{q(\kappa_{1}-\nu_{5}z)(\kappa_{1}-\nu_{6}z)(\kappa_{1}-\nu_{7}z)(\kappa_{1}-\nu_{8}z)\\{\kappa_{1}(gz-1)-(g\kappa_{2}z-\kappa_{1})T_{z}\\}}{{\kappa_{1}}^{4}(f-z)z^{2}(g\kappa_{2}z-\kappa_{1})}\,,$
$\displaystyle L_{2}=(1-zg\kappa_{2}/\kappa_{1})T_{z}-(1-zg)+z(z-f)gT\,.$

Then the linear $q$-differential equation $L_{1}y(z)=0$ is written as

(4.3)
$\displaystyle\Bigl{\\{}\frac{q(\kappa_{1}-\kappa_{2})(g\kappa_{2}-\nu_{5})(g\kappa_{2}-\nu_{6})(g\kappa_{2}-\nu_{7})(g\kappa_{2}-\nu_{8})}{g\kappa_{1}{\kappa_{2}}^{2}(fg\kappa_{2}-\kappa_{1})(g\kappa_{2}z-\kappa_{1})}$
$\displaystyle-\frac{q(\kappa_{1}-\kappa_{2})(g\nu_{1}-1)(g\nu_{2}-1)(g\nu_{3}-1)(g\nu_{4}-1)}{g(fg-1)\kappa_{1}\nu_{1}\nu_{2}\nu_{3}\nu_{4}(gz-q)}\Bigr{\\}}y(z)$
$\displaystyle+\frac{(q\nu_{1}-z)(q\nu_{2}-z)(q\nu_{3}-z)(q\nu_{4}-z)}{q\nu_{1}\nu_{2}\nu_{3}\nu_{4}(fq-z)z^{2}}\Bigl{\\{}\frac{g\kappa_{2}z-\kappa_{1}q}{\kappa_{1}(q-gz)}y(z)+y(z/q)\Bigr{\\}}$
$\displaystyle+\frac{q(\kappa_{1}-\nu_{5}z)(\kappa_{1}-\nu_{6}z)(\kappa_{1}-\nu_{7}z)(\kappa_{1}-\nu_{8}z)}{{\kappa_{1}}^{4}(f-z)z^{2}}\Bigl{\\{}\frac{\kappa_{1}(1-gz)}{g\kappa_{2}z-\kappa_{1}}y(z)+y(qz)\Bigr{\\}}=0.$

Note that Eq. (4.3) was studied in [4] from the aspect of the initial-value
space of $q$-$P(E^{(1)}_{7})$, and it was observed that a variant of the
$q$-Heun equation appears.

The Weyl group symmetry of the $q$-Painlevé equation $q$-$P(E^{(1)}_{7})$ was
given by the following action of the operators $s_{0},\dots,s_{6}$, $\pi_{1}$
and $\pi_{2}$ for the parameters
$(\kappa_{1},\kappa_{2},\nu_{1},\dots,\nu_{8})$ and $(f,g)$ in [2].

(4.4) $\displaystyle s_{0}:$
$\displaystyle\>\kappa_{1}\leftrightarrow\kappa_{2},\>f\rightarrow
1/g,\>g\rightarrow 1/f,$ $\displaystyle s_{1}:$
$\displaystyle\>\nu_{3}\leftrightarrow\nu_{4},\quad
s_{2}:\>\nu_{2}\leftrightarrow\nu_{3},\quad
s_{3}:\nu_{1}\leftrightarrow\nu_{2},$ $\displaystyle s_{5}:$
$\displaystyle\>\nu_{5}\leftrightarrow\nu_{6},\quad
s_{6}:\>\nu_{6}\leftrightarrow\nu_{7},\quad
s_{7}:\>\nu_{7}\leftrightarrow\nu_{8}$ $\displaystyle s_{4}:$
$\displaystyle\>\nu_{1}\rightarrow\frac{\kappa_{2}}{\nu_{5}},\;\nu_{5}\rightarrow\frac{\kappa_{2}}{\nu_{1}},\;\kappa_{1}\rightarrow\frac{\kappa_{1}\kappa_{2}}{\nu_{1}\nu_{5}},$
$\displaystyle\>f\rightarrow\frac{-\kappa_{2}(\nu_{1}\nu_{5}-\kappa_{1})fg-\nu_{5}(\kappa_{1}-\kappa_{2})f+\kappa_{1}(\nu_{1}\nu_{5}-\kappa_{2})}{\nu_{5}\\{-(\nu_{1}\nu_{5}-\kappa_{2})fg+\nu_{1}(\kappa_{1}-\kappa_{2})g+(\nu_{1}\nu_{5}-\kappa_{1})\\}},$
$\displaystyle\pi:$ $\displaystyle\>q\rightarrow 1/q,\>\nu_{1}\rightarrow
1/\nu_{5},\>\nu_{2}\rightarrow 1/\nu_{6},\>\nu_{3}\rightarrow
1/\nu_{7},\>\nu_{4}\rightarrow 1/\nu_{8},\>\nu_{5}\rightarrow
1/\nu_{1},\>\nu_{6}\rightarrow 1/\nu_{2},$ $\displaystyle\>\nu_{7}\rightarrow
1/\nu_{3},\>\nu_{8}\rightarrow 1/\nu_{4},\>\kappa_{1}\rightarrow
1/\kappa_{1},\>\kappa_{2}\rightarrow 1/\kappa_{2},\>f\rightarrow
f/\kappa_{1},\>g\rightarrow\kappa_{2}g,$

Then we can confirm that these operations satisfy the relations of the
extended Weyl group $\widetilde{W}(E^{(1)}_{7})$ whose Dynkin diagram is as
follows.
$1$$2$$3$$4$$5$$6$$7$$0$

We investigate the action of the symmetry in Eq. (4.4) on the equation
$L_{1}y(z)=0$. It follows immediately that Eq. (4.3) is invariant under the
actions of $s_{1}$, $s_{2}$, $s_{3}$, $s_{5}$, $s_{6}$ and $s_{7}$. The action
of $s_{4}$ does not exchange $\nu_{1}$ and $\kappa_{1}/\nu_{5}$, which are
related with zeros of the coefficient of $y(z/q)$ and $y(qz)$, and the action
of $s_{0}s_{4}s_{0}$ exchanges them. More precisely, the action is written as

(4.5) $\displaystyle s_{0}s_{4}s_{0}:$
$\displaystyle\>\nu_{1}\rightarrow\frac{\kappa_{1}}{\nu_{5}},\;\nu_{5}\rightarrow\frac{\kappa_{1}}{\nu_{1}},\;\kappa_{2}\rightarrow\frac{\kappa_{1}\kappa_{2}}{\nu_{1}\nu_{5}},$
$\displaystyle\>g\rightarrow\frac{\nu_{5}\\{-(\nu_{1}\nu_{5}-\kappa_{1})+\nu_{1}(\kappa_{2}-\kappa_{1})g+(\nu_{1}\nu_{5}-\kappa_{2})fg\\}}{-\kappa_{1}(\nu_{1}\nu_{5}-\kappa_{2})-\nu_{5}(\kappa_{2}-\kappa_{1})f+\kappa_{2}(\nu_{1}\nu_{5}-\kappa_{1})fg}.$

Set

(4.6)
$y(z)=\tilde{y}(z)\frac{(q\nu_{1}/z;q)_{\infty}}{(q\kappa_{1}/(\nu_{5}z);q)_{\infty}}.$

It follows that, if $y(z)$ is a solution of Eq. (4.3), then $\tilde{y}(z)$
satisfies the equation as Eq. (4.3) whose parameters are changed by the action
of $s_{0}s_{4}s_{0}$.

We investigate the action of another gauge transformation in Eq. (4.4) on the
equation $L_{1}y(z)=0$. Set $z=u/c$ and $y(z)=\tilde{y}(u)$. If $y(z)$ is a
solution of Eq. (4.3), then the function $\tilde{y}(u)$ satisfies Eq. (4.3)
for the variable $u$, where the parameters are changed as

(4.7)
$\displaystyle(\nu_{1},\nu_{2},\nu_{3},\nu_{4},\nu_{5}/\kappa_{1},\nu_{6}/\kappa_{1},\nu_{7}/\kappa_{1},\nu_{8}/\kappa_{1},\kappa_{2}/\kappa_{1},f,g)$
$\displaystyle\mapsto(c\nu_{1},c\nu_{2},c\nu_{3},c\nu_{4},(\nu_{5}/\kappa_{1})/c,(\nu_{6}/\kappa_{1})/c,(\nu_{7}/\kappa_{1})/c,(\nu_{8}/\kappa_{1})/c,\kappa_{2}/\kappa_{1},cf,g/c).$

We denote the transformation of the parameters in Eq. (4.7) by $S_{E_{7}}[c]$.

The time evolution of the $q$-Painlevé equation of type $E^{(1)}_{7}$ is
expressed as the following theorem.

######  Theorem 4.1.

(i) Let $\Xi$ be the transformation of the parameters defined by

(4.8)
$\displaystyle(\nu_{1},\nu_{2},\nu_{3},\nu_{4},\nu_{5},\nu_{6},\nu_{7},\nu_{8};\kappa_{1},\kappa_{2};f,g)$
$\displaystyle\mapsto\Bigl{(}\frac{\nu_{1}\kappa_{1}}{q\kappa_{2}},\frac{\nu_{2}\kappa_{1}}{q\kappa_{2}},\frac{\nu_{3}\kappa_{1}}{q\kappa_{2}},\frac{\nu_{4}\kappa_{1}}{q\kappa_{2}},\frac{\nu_{5}\kappa_{1}}{q\kappa_{2}},\frac{\nu_{6}\kappa_{1}}{q\kappa_{2}},\frac{\nu_{7}\kappa_{1}}{q\kappa_{2}},\frac{\nu_{8}\kappa_{1}}{q\kappa_{2}};\frac{\kappa_{1}^{3}}{q^{2}\kappa_{2}^{2}},\frac{\kappa_{1}^{2}}{q^{2}\kappa_{2}};\frac{f\kappa_{1}}{q\kappa_{2}},\frac{gq\kappa_{2}}{\kappa_{1}}\Bigr{)}.$

Let $T$ be the transformation of the parameters defined by

(4.9)
$T=\Xi\cdot(s_{4}s_{5}s_{1}s_{4}s_{6}s_{5}s_{1}s_{2}s_{4}s_{7}s_{6}s_{5}s_{1}s_{2}s_{3}s_{4}s_{0})^{2}.$

Then

(4.10)
$\displaystyle\frac{(T(f)g-\kappa_{1}/(q\kappa_{2}))(fg-\kappa_{1}/\kappa_{2})}{(T(f)g-1)(fg-1)}=\frac{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})(g-\nu_{7}/\kappa_{2})(g-\nu_{8}/\kappa_{2})}{(g-1/\nu_{1})(g-1/\nu_{2})(g-1/\nu_{3})(g-1/\nu_{4})},$
$\displaystyle\frac{(T(g)T(f)-\kappa_{1}/(q^{2}\kappa_{2}))(T(f)g-\kappa_{1}/(q\kappa_{2}))}{(T(g)T(f)-1)(T(f)g-1)}$
$\displaystyle=\frac{(T(f)-T(\kappa_{1}/\nu_{5}))(T(f)-T(\kappa_{1}/\nu_{6}))(T(f)-T(\kappa_{1}/\nu_{7}))(T(f)-T(\kappa_{1}/\nu_{8}))}{(T(f)-\nu_{1})(T(f)-\nu_{2})(T(f)-\nu_{3})(T(f)-\nu_{4})}.$

Namely the operator $T$ represents the time evolution of $q$-$P(E^{(1)}_{7})$
(see Eq. (4.1)). On the other parameters, we have

(4.11) $\displaystyle\displaystyle
T(\nu_{i})=\nu_{i}\;(i=1,2,\dots,8),\;T(\kappa_{1})=\kappa_{1}/q,\;T(\kappa_{2})=q\kappa_{2}.$

(ii) For any elements $\zeta$ generated by
$\nu_{1},\nu_{2},\nu_{3},\nu_{4},\nu_{5}/\kappa_{1},\nu_{6}/\kappa_{1},\nu_{7}/\kappa_{1}$,
$\nu_{8}/\kappa_{1}$, $\kappa_{2}/\kappa_{1}$, $f$ and $g$, we have

(4.12)
$\Xi(\zeta)=S_{E_{7}}\Bigl{[}\frac{\kappa_{1}}{q\kappa_{2}}\Bigr{]}(\zeta).$

###### Proof.

Set
$s=s_{4}s_{5}s_{1}s_{4}s_{6}s_{5}s_{1}s_{2}s_{4}s_{7}s_{6}s_{5}s_{1}s_{2}s_{3}s_{4}s_{0}$.
Then it follows from Eq. (4.4) that $s(\nu_{i})=\kappa_{2}/\nu_{9-i}$
$(i=1,\dots,8)$, $s(\kappa_{1})=\kappa_{2}$,
$s(\kappa_{2})=q\kappa_{2}^{2}/\kappa_{1}$, $s(f)=1/g$ and

(4.13)
$\frac{(s(g)/g-\kappa_{1}/(q\kappa_{2}))(fg-1)}{(s(g)/g-1)(fg-\kappa_{1}/\kappa_{2})}=\frac{\kappa_{1}/(q\kappa_{2})(g-1/\nu_{1})(g-1/\nu_{2})(g-1/\nu_{3})(g-1/\nu_{4})}{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})(g-\nu_{7}/\kappa_{2})(g-\nu_{8}/\kappa_{2})}.$

Therefore, $s^{2}(\nu_{j})=q\nu_{j}\kappa_{2}/\kappa_{1}$ $(j=1,\dots 8)$,
$s^{2}(\kappa_{1})=q\kappa_{2}^{2}/\kappa_{1}$ and
$s^{2}(\kappa_{2})=q^{3}\kappa_{2}^{3}/\kappa_{1}^{2}$.

Let $\Xi$ be the operator defined in Eq. (4.8). We have

(4.14) $\Xi s^{2}(\nu_{i})=\nu_{i},\;(i=1,2,\dots,8),\;\Xi
s^{2}(\kappa_{1})=\kappa_{1}/q,\;\Xi s^{2}(\kappa_{2})=q\kappa_{2}.$

Then we obtain Eq. (4.11). Note that
$\Xi(\kappa_{2}/\kappa_{1})=\kappa_{2}/\kappa_{1}$ and
$\Xi(\nu_{i}/\kappa_{2})=(q\kappa_{2}/\kappa_{1})(\nu_{i}/\kappa_{2})$
$(i=1,\dots,8)$.

It follows from Eq. (4.13) and $s^{2}(f)=1/s(g)$ that

(4.15)
$\frac{(q\kappa_{2}/\kappa_{1}-gs^{2}(f))(fg-1)}{(1-gs^{2}(f))(fg-\kappa_{1}/\kappa_{2})}=\frac{(g-1/\nu_{1})(g-1/\nu_{2})(g-1/\nu_{3})(g-1/\nu_{4})}{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})(g-\nu_{7}/\kappa_{2})(g-\nu_{8}/\kappa_{2})}.$

We apply the operation $\Xi$. It follows from Eq. (4.8) that

(4.16)
$\frac{(1-gT(f))(fg-1)}{(\kappa_{1}/(q\kappa_{2})-gT(f))(fg-\kappa_{1}/\kappa_{2})}=\frac{(g-1/\nu_{1})(g-1/\nu_{2})(g-1/\nu_{3})(g-1/\nu_{4})}{(g-\nu_{5}/\kappa_{2})(g-\nu_{6}/\kappa_{2})(g-\nu_{7}/\kappa_{2})(g-\nu_{8}/\kappa_{2})}.$

Therefore we obtain the first equation of (4.10).

We apply the operation $s$ to Eq. (4.13). It follows from $s(f)=1/g$ and
$s(g)=1/s^{2}(f)$ that

(4.17)
$\displaystyle\frac{(s^{2}(g)s^{2}(f)-\kappa_{1}/\kappa_{2}/q^{2})(gs^{2}(f)-1)}{(s^{2}(g)s^{2}(f)-1)(gs^{2}(f)-q\kappa_{2}/\kappa_{1})}$
$\displaystyle=\frac{(s^{2}(f)-\kappa_{2}/\nu_{5})(s^{2}(f)-\kappa_{2}/\nu_{6})(s^{2}(f)-\kappa_{2}/\nu_{7})(s^{2}(f)-\kappa_{2}/\nu_{8})}{(s^{2}(f)-s^{2}(\nu_{1}))(s^{2}(f)-s^{2}(\nu_{2}))(s^{2}(f)-s^{2}(\nu_{3}))(s^{2}(f)-s^{2}(\nu_{4}))}.$

By applying the operation $\Xi$, we obtain the second equation of (4.10). ∎

Note that the expression
$(s_{4}s_{5}s_{1}s_{4}s_{6}s_{5}s_{1}s_{2}s_{4}s_{7}s_{6}s_{5}s_{1}s_{2}s_{3}s_{4}s_{0})^{2}$
in Eq. (4.9) can be essentially found in [6].

## Acknowledgements

The author thanks Shoko Sasaki and Shun Takagi for discussions. He was
supported by JSPS KAKENHI Grant Number JP18K03378.

## References

  * [1] Jimbo M. and Sakai H., A $q$-Analog of the sixth Painlevé equation, Lett. Math. Phys., 38 (1996), 145–154
  * [2] Kajiwara K., Noumi M., and Yamada Y., Geometric aspects of Painlevé equations, J. Phys. A, 50 (2017), 073001.
  * [3] Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Commun. Math. Phys., 220 (2001), 165–229.
  * [4] Sasaki S., Takagi S. and Takemura K., $q$-Heun equation and initial-value space of $q$-Painlevé equation, arXiv:2110.13860.
  * [5] Sasaki S., Takagi S. and Takemura K., $q$-middle convolution and $q$-Painlevé equation, arXiv:2201.03960.
  * [6] Tsuda T., A geometric approach to tau-functions of difference Painlevé equations, Lett. Math. Phys., 85 (2008), 65–78.
  * [7] Yamada Y., A Lax formalism for the elliptic difference Painlevé equation, SIGMA, 5 (2009), paper 042.
  * [8] Yamada Y., Lax formalism for $q$-Painleve equations with affine Weyl group symmetry of type $E^{(1)}_{n}$, Int. Math. Res. Notices, 2011 (2011), 3823–3838.


Further author information: (Send correspondence to D.B.K.)
D.B.K.: E-mail<EMAIL_ADDRESS>Telephone: 1 801 581 5220

# Performance of the Upgraded VERITAS Stellar Intensity Interferometer (VSII)

David B. Kieda  Department of Physics and Astronomy, University of Utah, Salt
Lake City, UT 84112 USA  the VERITAS Collaboration
https://veritas.sao.arizona.edu

###### Abstract

The VERITAS Imaging Air Cherenkov Telescope array (IACT) was augmented in 2019
with high-speed focal plane electronics to create a new Stellar Intensity
Interferometry (SII) observational capability (VERITAS-SII, or VSII). VSII
operates during bright moon periods, providing high angular resolution
observations ( $<$ 1 mas) in the B photometric band using idle telescope time.
VSII has already demonstrated the ability to measure the diameters of two B
stars at 416 nm (Bet CMa and Eps Ori) with $<5$% accuracy using relatively
short (5 hours) exposures[1].

The VSII instrumentation was recently improved to increase instrumental
sensitivity and observational efficiency. This paper describes the upgraded
VSII instrumentation and documents the ongoing improvements in VSII
sensitivity. The report describes VSII’s progress in extending SII
measurements to dimmer magnitude stars and improving the VSII angular diameter
measurement resolution to better than 1%.

###### keywords:

Intensity Interferometry, Imaging Atmospheric Cherenkov Telescope Arrays,
Stellar Diameter Measurements

##  1 THE VERITAS STELLAR INTENSITY INTERFEROMETER (VSII)

###  1.1 Overview

The Stellar Intensity interferometry (SII) technique measures correlated
fluctuations in light intensity between spatially separated telescopes. The
VERITAS Stellar Intensity Interferometer (VSII) is currently the world’s most
sensitive SII astronomical observatory. VSII is implemented through an
instrumentation augmentation of the existing VERITAS gamma-ray observatory
[2]. VSII operates for 5-10 days/month during bright moon periods when the
intensity of scattered moonlight severely limits VERITAS gamma-ray
observations. This paper describes the VSII instrumentation, including recent
upgrades and improvements in the VSII observatory. This paper also updates the
comprehensive VSII survey of northern sky bright stars [3] that is currently
in progress. This survey has performed nearly 500 hours of observation on 48
different astronomical targets since December 2019.

###  1.2 VERITAS Observatory

The VSII Observatory uses the optical telescopes of the VERITAS gamma-ray
observatory. The VERITAS Observatory [4] consists of an array of 4 IACTs
located at an altitude of 1268 m.a.s.l at the F.L. Whipple Observatory (FLWO)
near Amado, AZ. Each IACT contains an f/1.0 12 m-diameter segmented Davies-
Cotton reflector employing 345 hexagonal facets, resulting in 110 $m^{2}$ of
light-collecting area. The lateral separation between nearest-neighbor IACT is
approximately 80-120 m (Figure 1). The Davies-Cotton reflector design creates
approximately a 4 nanosecond spread to photons arriving at the telescope’s
focal plane. The combination of large telescope primary mirror area, fast
optics, and 100+ meter telescope baselines enables VSII to provide sub-
milliarcsecond angular resolution at short optical wavelengths (B/V bands)[5]

Figure 1: Baseline separations of the VERITAS array of 12-m diameter Gamma-ray
telescopes (Amado, AZ USA) . The VERITAS Stellar Intensity Interferometer uses
the VERITAS telescopes during bright moonlight periods when Very High Energy
(VHE) gamma-ray observations are not performed.

###  1.3 VSII Focal Plane Instrumentation

####  1.3.1 Removable Focal Plane Plate

Each VERITAS telescope is instrumented with a removable VSII Focal Plane Plate
(FPP) that is securely mounted in front of the 499-pixel VERITAS camera
(Figure 2). The removable FPPs can be easily installed or removed in about 15
minutes per telescope by a single individual. The FPPs are designed to be
securely mounted with no modifications or changes to the VERITAS cameras. The
FPP instrumentation employs a collapsible 45∘ mirror that reflects the
starlight from the 12-m Davies-Cotton primary reflector onto an SII focal
plane diaphragm that is oriented perpendicular to the telescope optical axis.
The collapsible mirror allows the VERITAS camera shutter to be closed during
the daytime or bad weather. This capability allows the SII FPPs to remain
mounted on the VERITAS focal plane for the entire 5-10 day duration of the SII
observing period.

####  1.3.2 Optical Instrumentation

The SII focal plane employs a Semrock FF01-420/5-2 narrowband optical filter
(420 nm/5 nm width) [6] located in a diaphragm mount that is located directly
in front of a fast light sensor (Hamamatsu super-bialkali R10560
photomultiplier tube (PMT)). The diaphragm mount includes a central aperture
for the Semrock filter. A flat oval-shaped white reflective screen enables the
observer to center the light from the primary mirror onto the Semrock filter.
The SII FPP instrumentation plate scale is automatically matched to the
VERITAS focal plane point spread function (PSF) as it uses the same size PMTs
used for the VERITAS gamma-ray camera.

Figure 2: VSII removable Focal Plane Plate (FPP). The FPP contains a
collapsible mirror, HV supply, Photomultiplier tube, RF shielded electronics
box, and a narrowband optical filer. An FPP is mounted in front of each
VERITAS camera and is secured in place by hexagonal nuts tightened on three
threaded rods.

####  1.3.3 Narrowband Optical Filter Effective Bandwidth

The center bandpass wavelength of a narrowband optical interference filter
will shift to shorter wavelengths for light arriving at inclined angles. The
Semrock FF01-420 filter coating exhibits a high refractive index ($n=2.38$)
that reduces the wavelength shift for non-normal incident photons. A
simulation was used to calculate the shift in optical bandpass wavelength for
the VERITAS f/1.0 mirror optics by weighting the angular distribution of
incoming photons by the relevant mirror area at each angle of incidence. The
resulting incidence angle distribution was then convolved with the standard
formula describing the shift in center bandpass wavelength with incidence
angle[6]. The resulting calculation indicates an effective bandwidth of
$\delta\lambda\approx 13\ nm$ about a $\lambda_{0}\approx 416\ nm$ center
wavelength, with reduced light transmission compared to normal incidence.

###  1.4 Signal Transport

####  1.4.1 Battery Powered High Voltage Supply

The PMT is powered by a custom battery-powered High Voltage (HV) Power
supply[7]. The HV supply battery uses two high-capacity (4400 ma-hr) TENERGY
Li-Ion batteries permanently mounted in the VERITAS camera. The battery pack
is recharged during the daytime. The battery pack has sufficient capacity to
provide HV for several nights of SII observations when fully charged. The HV
setting is remotely controlled through a pulse width modulated (PWM) signal
supplied through a fiber optic interface. The fiber optic PWM signal is
generated in the electronics trailer of each VERITAS telescope by an Ethernet-
controlled Arduino Yun connected to a fiber optic transmitter.

Previously, the HV control system worked well in the lab, but occasionally
would fail to control the HV module level adequately in the field, often
requiring the observer to manually clean the fiber optic end and re-seat the
fiber optic cable into the transmitter/receiver. This failure mode was
eventually linked to the longer fiber optic cable being used on the telescopes
than in the lab, so the LED driver light signal was marginally above the
optoelectronic receiver’s threshold. In summer 2021, the LED fiber drivers
were upgraded to an ultra-bright version (Industrial Fiber Optics IF-E96E)
that has twice the light output of the original LED fiber drivers. After
upgrading the LED driver, the HV control system has been operated for an
entire year without encountering additional failures.

####  1.4.2 Preamplifier and Signal Cables

The output of the PMT is amplified by a high-speed (200 MHz bandwidth)
transimpedance ($2\times 10^{4}$ V/A) preamplifier (FEMTO HCA-200M-20K-C). The
distance between the PMT output and the input to the FEMTO preamplifier is
kept as short as possible ($<$2 cm) to minimize the input capacitance load on
the preamplifier input. This configuration maximizes the preamplifier
bandwidth and decreases electronic noise. The output from the FEMTO
preamplifier drives a 45 m long double-shielded RG-223 coaxial cable. The
cable is routed along the VERITAS telescope quadrapod arms and the optical
support structure, and terminates at the SII data acquisition system, located
in the telescope electronics trailer next to each VERITAS telescope. Special
care is made to ensure that each SII focal plane instrumentation component is
electrically isolated from any telescope ground to minimize ground-loop pickup
of local RF noise.

###  1.5 Electronic Readout

####  1.5.1 Digitization

A standalone National Instruments (NI) data acquisition system, using a PXI
Express (PXIe) crate with a high-speed backplane (4 GB/sec), is deployed at
each VERITAS telescope to provide local data recording (Figure 3). Each SII
data acquisition crate holds a high-speed PXIe controller (PXIe-8133) and a
high-speed interface (PXIe-8262) connected to a high-speed (700 MB/sec) RAID
disk (NI-8265). Each PMT signal is continuously digitized at 250 MHz by an NI
FlexRio Module (NI-5761, 12-bit resolution, DC-coupled). The 250 MHz
digitization rate provides a sampling period that matches the 4 nsec time
dispersion of the VERITAS optics. The digitized data from each PMT is
truncated to 8 bits and merged into a single continuous data stream by a
Virtex-5 FPGA processor. The data stream is transported across the high-speed
PXIe backplane and recorded to the RAID disk. The VSII PXIe-based data
acquisition uses commercially available modules, and the VSII data acquisition
is controlled using LabView software.

Figure 3: Schematics drawing of the 2019-era VSII electronics system. Each
telescope contains a standalone VSII data acquisition system that operates
independently. The timing of the DAQ clocks between telescopes is synchronized
using a commercially available White Rabbit module from Seven Solutions. A
separate fiber optic system is used to simultaneously trigger the start of the
observing run at each telescope.

####  1.5.2 RAID Storage

VSII was initially configured to write continuously streaming data from the
VSII observations to NI -8265 RAID storage systems. At each telescope, the
8265 RAID system, consisting of 12 independent 1 TB disks configured in a
RAID0 configuration, supports continuous 700 MB/sec data streaming from the NI
5761 DAQ system digitizer. In the 2020-2021 observing season, the 1 TB drives
were replaced by larger (2 TB) disks in each RAID system, raising the overall
high-speed storage capability of each RAID drive to 24 TB. In Spring 2021, all
VSII crate controllers were upgraded to include high-speed USB 3.0 ports
(PXIe-8135). This change allowed the mounting of additional low-cost RAID
drive capacity (80 TB/telescope). These disks are used for archival storage,
as their continuous write speed ( 250 MB/sec) is too slow to support reliable
250 MHz data streaming from the NI-5761 digitizers.

In Fall 2022, two crate controllers were upgraded to next-generation version
(PXIe-8861) controllers with dual Thunderbolt-3 ports. The Thunderbolt 3 ports
have sufficient bandwidth (3 GB/sec) to allow the streaming of the SII DAQ
data directly onto very large capacity (80 TB) RAID disks. This capability
provides a low-cost expansion of available disk space for nightly
observations, and sufficient disk space to run for 10+ nights without needing
to process data. As of July 2022, each SII DAQ system has more than 150 TB of
high-speed data storage available for observations. At a sampling rate of 250
MHz (8-bit resolution), a one hour SII observation uses 865 GB of disk space.

####  1.5.3 Synchronization

The intensity signals recorded between separated telescopes must be
synchronized within a small fraction of the DAQ sampling period. If the DAQ
clocks are not 100% synchronized, the effective bandwidth of the measurement
is degraded, resulting in a lower Signal-to-Noise Ratio (SNR). The distributed
VSII data acquisition system incorporates synchronization of each telescope’s
DAQ FPGA clock using a commercially available Seven Solutions White Rabbit
clock synchronization system[8]. The White Rabbit timing system uses single-
mode fiber (SMF) optic links to distribute a centrally generated 10 MHz clock
to each SII DAQ system telescope with a $<200\ psec$ RMS precision.

###  1.6 Two-telescope Correlations

####  1.6.1 High-Speed Data Transport

An independent SII 10 GBase-T fiber optic data transfer network is used to
provide high-speed network access to all four SII RAID data disks. This 10
Gb/sec bandwidth allows each VERITAS-SII DAQ system to cross-correlate with
every other SII telescope data stream. A 5/125 SMF carries the VSII high-speed
datastream from each SII DAQ system to a central NetGear M4300-12X10 GBase-T
managed Ethernet switch. The NetGear switch isolates the SII-related data
traffic from the primary VERITAS observatory network. An independent 1 Gb/sec
network interface is also established to each crate controller and HV control
system in each telescope to allow access to these devices without competing
for bandwidth with the main SII data stream.

####  1.6.2 FPGA-based correlators

Cross-correlation of the data streams between individual telescopes is
performed using a pipelined algorithm implemented on a National Instruments
FPGA card mounted in each VSII DAQ system. The correlation system searches the
directory structure of the previous night’s SII observations, matches
individual telescope observation files into pairs for correlation, and then
stages a batch process to perform all identified correlation pairs. A 1-hour
SII observation generates 865 GB of data per telescope, and the FPGA
correlator requires approximately 1.5 hours to complete the correlation of a
single telescope pair on the 1 hour observation. The VSII data transfer
network has more than sufficient bandwidth to accommodate the simultaneous
operation of multiple correlators on two independent DAQ systems.

The main limitation in running multiple correlators in parallel is providing
single-user access to the RAID disks. If two correlators are trying to access
the same RAID disk on one telescope, the RAID will spend a substantial amount
of time accessing different files on the RAID disk. This will cause the
correlation time to be significantly lengthened for both correlators. Nightly
VSII operations have developed a more practical approach of using two RAID
drives per telescope, alternating between writing to the two independent disks
on sequential observing nights. A single telescope can handle providing data
to two independent correlator processes if the data is on separate disks, so
in practice, up to four correlators may be run in parallel.

####  1.6.3 Processed File Size Reduction

Each two-telescope correlation file is considerably smaller than the raw data
file. A one-hour, two-telescope cross-correlation file is approximately 33 MB,
compared to the combined 1.7 TB size of the two raw data files. Once all two-
telescope cross-correlations are completed, the correlation files are uploaded
to the University of Utah Center for High Performance Computing for archiving,
distribution and analysis. The majority of the raw data is discarded once the
cross-correlations are completed. Secondary raw data files containing
electronic calibrations and measurements of sky brightness are substantially
smaller in size. Consequently, it is possible to permanently retain them for
correcting systematic effects for the final analysis.

####  1.6.4 Performance

For a one-hour observation using two independent correlators, approximately
4.5 hours of VSII DAQ system processing is required to process all six
baseline pairs. It has become routine practice to process a fraction of the
files the next day to assess the data quality from the previous night and
process the remainder after the completion of the monthly SII observing
period. In general, an entire week of nightly SII observations requires 7-10
days of computation to complete the correlations between every telescope pair.

###  1.7 RF Noise Reduction

The photomultiplier tube (PMT) is encased in a tight-fitting brass tube to
reduce RF pickup into the PMT dynodes. However, FFT analysis of data from each
SII PMT in December 2020 revealed the presence of both persistent and
transient RF noise (Figure 4, upper panel). Prominent RF pickup specific to
the FLWO observatory site is observed at a measured frequency of 79 MHz. This
noise appears to be an alias of higher frequency RF noise undersampled at the
250 MHz digitization rate. Additional RF shielding was added to the FPPs in
January 2021 to reduce RF pickup. The PMT and the preamplifier were fully
encased in an RF-shielded enclosure (Focal Plane Module, or FPM). The FPM
reduced the RF pickup in each SII channel by a factor of 8 or more (Figure 4,
lower panel). In addition, inline low-pass (140 MHz) anti-aliasing filters
(Mini-Circuits SLP-150+) were installed on all VSII data acquisition systems
in Summer 2021, and subsequent all observations have used the filters to
reduce the RF pickup.

After making these improvements, the residual RF noise has been eliminated
from any of the telescope traces, and the noise does not appear as a visible
RF signal in any of the signal traces. There is still a residual level,
however, which causes occasional ‘noise-assisted’ flipping of the LSB of the
digitizer in the datastream. Although this RF effect is not visible in the raw
datastream, it manifests as a low-level, persistent 79 MHz signal in the
cross-correlations between telescopes. The exceptional power of the cross-
correlation analysis allows these random, correlated bit flips to be visible
in the two-telescope correlations. Subsequent computational processing on the
correlated data can effectively remove this residual component. In addition,
we are installing (Summer 2022) triply shielded RG-223 signal cables; these
have demonstrated additional RF pickup attenuation by an additional 50% (or
more).

Figure 4: Effect of RF shielding of Telescope 3 Focal plane electronics. Left
photo: RF shielded electronics enclosure for FPP electronics. Right Plots: The
noise spectrum is calculated using a standard Fast Fourier Transform (FFT) on
raw data from Telescope 3 during an observing run. Horizontal Axis: Frequency
(MHz). Vertical axis: FFT power spectrum (arbitrary units). Upper plot: Noise
spectrum before RF shielding. Bottom Plot: Noise spectrum after RF shielding.

###  1.8 Zero Baseline Focal Plane Plate

In June 2021, a specialized ”zero baseline (ZB) beamsplitter” FPP was
constructed and deployed on VERITAS Telescope 1. The ZB FPP adds a non-
polarizing 50-50 beamsplitter after the narrowband filter to the optical beam
path. Two independent FPMs mounted perpendicular to each other on the FPP read
out the split optical beam. Special care is taken to ensure the FPMs do not
share a common ground connection on the FPP to eliminate ground-loop noise
pickup. Each FPM uses independent batteries, HV supplies, signal cables, and
DAQ systems in the trailer. The beamsplitter was constructed using
commercially available PVC tubing combined with a commercial 25 x 35.36 x 1mm
VIS, Elliptical Plate Beamsplitter (Edmund #48-915). Successful commissioning
of the ZB FPP was performed using SII observations of several stars in
Summer/Fall 2021.

Figure 5: VSII Beamsplitter performance. Upper photo: VSII beamsplitter plate.
The two PMTs (A, B) and their respective HV supplies are labeled. Bottom plot:
Observed Two-Telescope Zero baseline correlation observed on Alp Lyr (Vega)
October 16, 2021. The 30-minute observation was taken in modest moonlight
conditions. Horizontal Axis: Time lag between T0 and T1 observation. Vertical
axis: correlation $(g2-1)\times 10^{6}$.

The ZB FPP performance was evaluated in Fall 2021 by observing the bright A0V
star Vega (Figure 5). One of the challenges of using the ZB FPP is that the
light at each ZB PMT is reduced by a factor of 2 due to the beamsplitter, plus
additional optical losses in the beamsplitter PVC pipe. The reduced photon
flux at the beamsplitter PMTs also affects the correlation of Telescope 1 with
the other telescopes in the VSII array, significantly reducing their
sensitivity. To improve photon collection efficiency in the beamsplitter tube,
in November 2021, a front-surface mirror-polished aluminum 6061 foil (McMaster
#1655T1) was inserted into the PVC beamsplitter tube. The ZB FPP beamsplitter
was used in observation in November-December 2021, and analysis of the
observational data is currently underway.

####  1.8.1 Summary of Hardware Configuration Changes

Figure 6 provides a summary of the hardware changes that have been made to the
VSII instrumentation from the beginning of the Northern Sky SII survey
(December 2019) until the time of this manuscript (July 2022).

Figure 6: Timeline for various hardware improvements in the VSII observatory.
VSII would record 15-20 hours of observations per month at the start of the
survey period (December 2019). Presently, VSII system can record more tha 60
hours of data during full moon periods when VERITAS does not perform VHE
gamma-ray observations. Recent VSII system improvements have created the
ability to perform multi-hour observations routinely.

##  2 VSII Observing

###  2.1 Overview

VSII has been in regular 4-telescope operations on a lunar-monthly (full-
moon/rear-full-moon period) cadence since December 2019, recording almost 500
hours of observations to date. VSII was not in operation from March 2020
through September 2020 due to the COVID-related shutdown of the Whipple
Observatory. The observing sequence has evolved to include nightly planning of
observing constraints imposed by moon location and other considerations (e.g.,
stellar magnitude, estimated stellar diameter, source elevation, weather,
etc.)[2].

###  2.2 Planning of Nightly Observations

The suitability of a target for any given night of observation depends upon
several factors. These factors include source-specific characteristics
(stellar classification and magnitude, stellar diameter,
single/binary/multiple star configuration), observatory characteristics
(observatory latitude/longitude, telescope diameters, telescope separations,
electronic noise, and optical efficiencies), and astronomical considerations
(Source RA/Dec, number of hours observable on a given night, seasonal
nighttime visibility of the source). VSII uses the public domain ASIIP
software package [9, 10] to list and rank the suitability of potential sources
for any given observation night.

ASIIP simulations are performed for a specified SII observing week during the
year, and potential observation targets (drawn from several catalogs,
including the JMMC catalog [11]) are ranked according to the estimated error
in the determination of the stellar diameter. For example, larger diameter
stars ($>1\ mas$) may not be resolved by the large VERITAS baselines, and the
source radius can only be effectively constrained by low-elevations
observations on a given night. In contrast, smaller stars ($<0.5\ mas$) may be
unresolved if the observations are constrained to low elevations on a given
night. At the beginning of the observing night, the simulated visibility
curves for each target (e.g., Figure 1 ) are reviewed, and an observing
sequence of 2-5 stellar observations is generated. Then, the observing plan is
sequenced to provide an appropriate set of observation hours and baselines for
each target to result in a quality measurement of each visibility curve.

###  2.3 Observing Sequence

Preparations for Nightly SII observations begin about 1 hour before twilight.
The observer opens the shutter to each VSII camera, and deploys a collapsible
mirror to a 45-degree position to reflect the starlight onto a photomultiplier
tube (PMT) mounted transversely to the optical axis. The PMT High Voltage (HV)
supply battery is re-connected after daytime charging, and the HV supply is
powered up. The Data Acquisition (DAQ) system for each telescope is then
started, and short (10-30 second duration) calibration runs (with HV = 0V) are
performed with the telescopes parked in their stow positions. The pedestals of
the data traces are recorded and used for removing the DC offset in each
electronics channel. About 30 minutes before the start of the observation, a
low HV level is turned on to verify each PMT is operational and that the
telescopes are ready for beginning observations. The telescopes are then
slewed to the first target of the night, and a VERITAS focal-plane CCD camera
for each VERITAS telescope is used to micro-adjust each telescope’s pointing,
centering the point spread function of the starlight onto the narrowband
filter. Finally, the HV in each telescope is adjusted to provide a nominal DC
current in each PMT (typically 10 microamps for an $m_{V}=2$ star).

Once the PMT currents are equalized, the telescopes are slewed $0.5^{\circ}$
off from the star’s sky position, and a 1-minute OFF run is taken to measure
the night sky background near the star’s sky field. The telescope is then
slewed to point at the target star, and a 30-180 minute continuous observation
is performed (an ON run). Each observation run is triggered at every telescope
by a central trigger pulse distributed to each DAQ system via a fiber optic
system. As the VERITAS telescopes track the target over 4-6 hours, the
projected distance between the telescope combinations continuously changes
with the movement of the star position in elevation and azimuth. The
measurement of the correlated fluctuations between two telescopes
(interferometric visibility $|V(r)|^{2}$) is therefore sampled over a range of
projected distances $r$ during a single night’s observations.

During the ON run, the starlight’s point spread function (PSF) on the SII
focal plane narrowband filter is tracked. If the PSF moves off the narrowband
filter window, manual telescope tracking adjustments are performed to recenter
the starlight on the light sensor. At the end of the ON run, another OFF run
is taken with a $0.5^{\circ}$ offset. The ON-OFF observing sequence is
repeated throughout the night, as necessary. Moving to a new observing target
requires slewing to the desired target, making the manual tracking adjustments
to center the PSF onto the optical filter, setting the HV to equalize the PMT
gains, and then performing the nominal OFF-ON-OFF run sequence.

At the end of the observing night, the HV is turned off and the telescopes are
brought to their stow positions. The HV battery is re-connected to the battery
charger, and the collapsible mirror is folded down. The camera shutter is then
closed to protect the focal plane instrumentation during the daytime.

###  2.4 Moon Angle Constraints

The narrowband Semrock interference filter (5 $nm$ bandpass, normal incidence)
[6] substantially reduces the intensity of background light impinging on the
VSII photomultiplier tubes (PMTs). This allows VSII observation to occur at
all moon phases, including the full moon. In practice, the moonlight restricts
the observability of specific stars on a given night. The effects include:

  1. 1.

Moonlight shining directly on the focal plane. Direct moonlight on the focal
plane makes it difficult to view the location of the starlight PSF on the
interference filter aperture using the focal plane CCD camera. The direct
moonlight makes it challenging to visualize the tracking adjustments necessary
to keep the starlight PSF centered on the narrowband filter. To first order,
this issue rules out any observations when the moon angle to the target source
is $>90^{\circ}$. In practice, the VSII focal plane is slightly recessed into
the VERITAS camera body, providing additional baffling against the moonlight.
This baffling allows targets to be observed with moon angles $<95^{\circ}$.

  2. 2.

Moonlight is scattered by the atmosphere through both Rayleigh and Mie
Scattering. The scattering functions are peaked in the forward direction,
creating a halo of scattered moonlight around the moon. SII observations are
challenging when the target star is less than $30^{\circ}$ away from the
moon’s sky position. Moon angles less than $30^{\circ}$ also make it
challenging to make tracking adjustments.

  3. 3.

Atmospheric conditions can cause additional constraints on the observability
of targets. Patchy clouds in the night sky can strongly scatter moonlight onto
the focal plane at unanticipated angles, making it difficult to perform
reliable tracking or estimate night sky background biases in the visibility
curves. Low clouds on the horizon can restrict observation to higher
elevations (longer baselines), making it difficult to sample the central peak
of the visibility curve for stars with diameters greater than 1 mas.
Alternatively, atmospheric conditions that restrict a given night’s
observations to low elevations (short baselines) will result in smaller stars
being unresolved.

  4. 4.

Observing stars at low elevation will noticeably dim their magnitude due to
the increased atmospheric depth. If moon angle constraints force observations
to occur only at low elevation angles, visible dimmer magnitude targets (
$m_{V}>2.5$ ) may not be feasible.

  5. 5.

Dimmer stars ($m_{V}>3.0$ ) are difficult to track under most moonlight
conditions, regardless of the moon angle. Therefore, the nightly observation
plan must schedule time between astronomical twilight and moonrise/moonset to
perform such observations.

  6. 6.

The VERITAS Telescopes are located adjacent to other telescopes (e.g., the
prototype CTA Schwartzchild-Couder telescope [12]) and other buildings and
equipment located at the FLWO basecamp. For moon elevations near rise or set
(elevation $<30^{\circ}$), the moonlight may offer serendipitous scatter from
the parked SCT telescope primary or from other objects in the basecamp. The
reflected moonlight can project an upward moonbeam into the night sky which
occasionally crosses the path of the focal plane of a VSII telescope. This
effect causes unanticipated illumination of the VSII focal plane. The
scattering of moonlight off other telescopes or equipment is difficult to
predict ahead of time. If it is observed, the observers assess whether it
compromises the ability to perform reliable observations, If so, the observing
schedule is changed to select a different star.

  7. 7.

In addition, substantial sky background (compared to the target starlight)
will bias the visibility curves, resulting in systematic errors in stellar
radius measurement. The night sky background light correction is most
straightforward when the background light is less intense, and the background
does not change significantly across short angular distances across the night
sky.

Figure 7: HR-diagram type summary of VSII survey Observations
(12/2020-6/2022). Individual stellar observations are plotted as a function of
stellar classification and visual magnitude. Each circle represents a set of
VSII observations. The area of the circle is proportional to the cumulative
number of VSII observation hours. Green circles indicates single stars. Blue
circles indicates binary or multi-star systems.

###  2.5 VSII Observing Priorities

At the beginning of the observing week, the first target observed is chosen to
be a bright SII reference (previously observed) star ($m_{V}<2.2$) with $0.6\
mas<stellar\ radius<0.85\ mas$ to verify the basic operational status of VSII.
The star is chosen for an appropriate moon angle ($30^{\circ}<moon\
angle<95^{\circ}$) and a strong visibility signal with a 1 to 1.5-hour
exposure.

Priority for subsequent observations is evaluated using the following
priorities:

  1. 1.

$30^{\circ}<moon\ angle<95^{\circ}$

  2. 2.

Observation time $>1$ hour

  3. 3.

$0.4\ mas<stellar\ diameter<1.2\ mas$

  4. 4.

Quality of ASIIP constraint on the stellar diameter

  5. 5.

Prefer $m_{V}<3$

  6. 6.

Prefer O, then B, then A stars

  7. 7.

Previously unobserved targets have a priority

  8. 8.

Underexposed targets have a priority

  9. 9.

Short period orbital binaries (e.g., Spica) establish a multi-day priority to
map out a visibility curve at different phases of the orbit.

  10. 10.

Unusual stellar characteristics (e.g., Cepheid, fast rotators, etc.) gain
priority over ‘vanilla’-type singular stars.

##  3 VSII Observations 2019-2022

From December 1, 2019 through June 1, 2022, VSII has performed 496 hours of
observations on 48 different targets. These targets include 26 single stars
and 22 binary/multiple star systems. Figure 7 provides a ”Hertzprung-Russel”
(HR) type summary of the VSII observations. VSII observations have been
performed on stellar classifications ranging from O9 through A8, with an
emphasis on stellar classifications between O9 and B3. The stellar magnitudes
observed range between $+0.026<m_{V}<+4.2$. Several targets have received
exposures approaching 50 hours.

###  3.1 Analysis of VSII Observations

Raw data from individual source observations are processed on-site into a
”correllelogram” using the pipelined two-telescope cross-correlation algorithm
that the FGPAs host in the VSII Data Acquisition crates. Once the
correllelogram for each telescope pair is computed, each correllelogram is
analyzed to extract the magnitude of the visibility at the specific telescope
separation. Figure 4 illustrates the analysis steps of each two-telescope
correllelogram. This procedure is complicated by the presence of different
levels of RF 79 MHz noise in the raw data stream.

The first VSII observation took place before introducing the focal plane RF
shielding and the anti-aliasing filters in 2020-2021. These observations had
significantly larger RF noise levels than the photocurrent signal stream. The
raw correllelogram is analyzed by fitting a noise model that includes a
dominant 79 MHz component and several side frequencies. The weighting of each
frequency component is iteratively adjusted to match the raw correllelogram
data until the values converge on a satisfactory fit (Figure 8, upper panel).
Next, the residual between the noise model and the raw data is calculated, and
the residual data is corrected for the changing optical path delay during the
observation using the known projected distance (OPD) difference between the
two telescopes to the target (Figure 8, middle panel). Finally a Gaussian fit
is used to extract the peak of the visibility curve at the expected time lag
between telescopes (Figure 8, bottom panel).

Subsequent VSII observations incorporated RF shielded focal planes
(Spring-2020) and 140 MHz anti-aliasing filters (Summer 2021). These
observations had RF pickup noise which is a small fraction of the photon
stream signal. For these observations, an FFT is applied to the correllelogram
data to identify the RF noise frequencies, and a combination of a Fourier
Series model and machine learning tools is be applied to eliminate the RF
noise contribution. Figure 9 demonstrates the reduction of RF noise in more
recent VSII observations. The left heatmap shows the correlation signal on the
T3-T4 correllelogram measured on Eta Uma in December 2021. The g2 coherence
signal is seen as the bright yellow vertical line , and the RF noise
contamination is seen as the striped patterns in the background fluctuations
to the left and the right of the g2 signal. The plot on the right shows the
same data after RF noise subtraction; the correllelogram is free from any
noticeable level of RF noise. Identifiable g2 peaks have been identified for
VSII observations of stars as dim as $m_{V}=+3.75$.

###  3.2 Visibility Curves and Measurement of Stellar Diameters

A visibility curve is calculated using the above-measured visibility peaks and
known telescope separations for every telescope pair in the observation. The
final visibility curve must be corrected for the presence of night sky
background light. After this correction, a suitable stellar diameter model is
fitted to the visibility curve to accurately measure the stellar diameter [1,
5] at an effective wavelength $\lambda=416\ nm$. We perform our preliminary
analysis fits using a Uniform Disk (UD) model to the visibility
$|\gamma(d)|^{2}$ as a function of the projected telescope separation $d$ :

$|\gamma(d)|^{2}=g^{(2)}(\tau=0,d)-1$ $|\gamma(d)|=2\frac{J_{1}(\pi\theta
d/\lambda)}{\pi\theta d/\lambda}$

.

Two preliminary visibility curves obtained by recent VSII observations are
shown in Figure 10. With the present UD analysis, VSII routinely provided
measurements of stellar diameters with 3-4% resolution for bright stars
($m_{V}=1.5-2$) with short (2-5 hour) exposures. Improvements in the VSII
telescope hardware, analysis, and calibration are projected to improve the
resolution in the future to 1% or better. These improvements are the main
focus of VSII analysis during 2022-2023.

Figure 8: Analysis of a raw two-telescope correllelogram for the extraction of
the visibility at a specific telescope separation. This observation of Zet Oph
took place on March 31, 2021. The correllelogram measures the correlation
between fluctuations in light intensities recorded by VSII telescopes 3 and 4
as a function of the time lag between the photon arrival at each telescope.
Upper panel: Fitting the raw correllelogram to remove RF noise pickup. Middle
panel: Residual correlation signal before and after correction for optical
path delay. Bottom panel: Fitting the Optical Path Delay (OPD) corrected
correllelogram to a Gaussian model to extract the magnitude of the two-
telescope correlation (visibility). Horizontal axis : time-lag between
telescope data streams (nsec). Vertical axis: scaled interferometric
visibility (scaling of units depends upon each panel)  Figure 9: 2D heatmap of
a single observing run correllelogram before and after RF noise subtraction.
The heatmap illustrates the time evolution of the T3-T4 correllelogram
(horizontal axis, the color scale is the g2 correlation magnitude) as a
function of time from the start of the observing run. Yellow vertical line:
evolution of the g2 peak through the observing run. Left panel: Raw
correllelogram data, including RF noise. The RF noise pickup is seen as the
correlated vertical ripples in the background (blue) region. Right panel:
Residual correlation signal after applying RF noise rejection. The residual
noise fluctuations in the background region are Gaussian distributed; this
noise can be used to estimate the limiting magnitude of the VSII observatory.
The observations were taken on December 20, 2021 on the star Eta Uma.  Figure
10: VSII observations of visibility curves for two bright B stars. The
analysis is preliminary as they have been uncorrected for the night sky
background. Uniform Disk models have been fitted to the visibility curves
demonstrating the ability to measure the UD diameter to an accuracy of 3-4%
with short (2-5 hour) observations.

##  4 Acknowledgements

Both observations and analysis of data for the VERITAS Observatory are
supported by grants from the US Department of Energy Office of Science, the US
National Science Foundation, the Smithsonian Institution, and NSERC in Canada.
The authors gratefully acknowledge support under NSF Grant #AST 1806262 for
the fabrication and commissioning of the VERITAS-SII instrumentation. We
acknowledge the excellent work of the technical support staff at the Fred
Lawrence Whipple Observatory and the collaborating institutions in the
construction and operation of the instrument.

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