Voros Coefficients at the Origin and at the Infinity

Voros Coefficients at the Origin and at the Infinity
of the Generalized Hypergeometric Differential
Equations with a Large Parameter

Takashi AOKI a and Shofu UCHIDA b

T. Aoki and S. Uchida

a) Department of Mathematics, Kindai University, Higashi–Osaka 577-8502, Japan
<EMAIL_ADDRESS>
b) Graduate School of Science and Engineering, Kindai University,
b) Higashi–Osaka 577-8502, Japan<EMAIL_ADDRESS>
Received July 20, 2021, in final form December 30, 2021; Published online
January 03, 2022

Voros coefficients of the generalized hypergeometric differential equations
with a large parameter are defined and their explicit forms are given for the
origin and for the infinity. It is shown that they are Borel summable in some
specified regions in the space of parameters and their Borel sums in the
regions are given.

exact WKB analysis; Voros coefficients; generalized hypergeometric
differential equations

33C20; 34E20; 34M60

##  1 Introduction

The notion of the Voros coefficients is one of the keys in the exact WKB
analysis of differential equations with a large parameter. It has been
introduced mainly for second-order ordinary differential equations and used
effectively in the descriptions of the parametric Stokes phenomena, of
calculation of the monodromies and of the relations between Borel resummed WKB
solutions and classical special functions [4, 5, 6, 7, 8, 11, 17, 22, 25, 27,
28]. Recently, new insights into the relationship between cluster algebra and
Voros coefficients have been obtained in [19]. Furthermore, it is known that
Voros coefficient of the Gauss hypergeometric differential equation can be
described using the free energy in the topological recursion [18]. Besides,
the Voros coefficients have also appeared in the physics literature as the
spectral coordinates [15].

Our purpose is to define the Voros coefficients at the origin and at the
infinity for the generalized hypergeometric differential equation with a large
parameter and to give the explicit forms of them. Here the generalized
hypergeometric differential equation means the differential equation which
characterizes the generalized hypergeometric series (or function)
${}_{N}F_{N-1}$. It is well known [12] that this equation is of order $N$ and
obtained as a natural generalization of the Gauss hypergeometric differential
equation. The monodromy group of the generalized hypergeometric differential
equation plays a role in studying the solution of the equation. The
classification of the finite hypergeometric groups [23] and the differential
Galois group of the generalized hypergeometric equation have been obtained in
[10].

Basic notions and tools in the exact WKB analysis of higher-order or of
infinite-order differential equations with a large parameter are established
in [3]. Our discussions are based on the theory developed in this paper. For
example, WKB solutions of a higher-order differential equation are defined by
using the characteristic roots of the equation, which are branches of an
algebraic function. Once WKB solutions are introduced, one can define the
Voros coefficients in a similar manner to the second-order case. Fortunately,
our equation has ladder operators for parameters as in the case of the Gauss
hypergeometric differential equation. Therefore the basic idea of computation
of the explicit forms of the Voros coefficients is the same as the second-
order case [7, 8, 22, 27]. But in our case, we have to consider the integral
of the algebraic function. Although this fact causes some difficulties, some
of them can be overcomed by using an idea invented by Iwaki and Koike [17].
There is another difficulty, or complexity, coming from the number of
parameters. Our equation contains $(2N-1)$ parameters and we have to manage
some combinations of them. Thus our descriptions become somewhat complicated.

The plan of this article is as follows. In Section 2, we define WKB solutions
of the generalized hypergeometric differential equation with a large parameter
and investigate the local behavior of the solutions at the singularities of
the equation. In Section 3, we define the Voros coefficients at the origin and
at the infinity for our equation. The explicit forms of the Voros coefficients
are given (Theorem 3.9).

##  2 WKB solutions

###  2.1 The generalized hypergeometric differential equation
with a large parameter

Let $N$ be an integer greater than $1$, let $\bm{a}$ denote an $N$-tuple of
parameters $a_{i}\in\mathbb{C}$, $i=1,2,\dots,N$,) and let $\bm{b}$ denote an
$(N-1)$-tuple of parameters $b_{j}\in\mathbb{C}\setminus\\{0,-1,-2,\dots\\}$,
$j=1,2,\dots,N-1$. We set

$\displaystyle{}_{N}F_{N-1}\left(\begin{matrix}\bm{a}\\\
\bm{b}\end{matrix};x\right)=\sum_{k=0}^{\infty}\frac{(a_{1})_{k}(a_{2})_{k}\cdots(a_{N})_{k}}{(b_{1})_{k}(b_{2})_{k}\cdots(b_{N-1})_{k}k!}x^{k}$
(2.1)

and call this the generalized hypergeometric series [12]. Here $(\alpha)_{k}$
denotes the Pochhammer symbol $(={\Gamma(\alpha+k)}/{\Gamma(\alpha)})$. As is
well known, the radius of convergence of (2.1) equals $1$ and the right-hand
side of (2.1) defines a holomorphic function on the universal covering of
$\mathbb{C}\setminus\\{0,1\\}$, which is also denoted by ${}_{N}F_{N-1}$. This
series or function satisfy the following $N$-th order ordinary differential
equation:

$\displaystyle\left(\left(\prod_{j=1}^{N-1}(\vartheta_{x}+b_{j})\right)\partial_{x}-\left(\prod_{i=1}^{N}(\vartheta_{x}+a_{i})\right)\right)w=0.$
(2.2)

Here we set $\partial_{x}=\frac{{\rm d}}{{\rm d}x}$ and
$\vartheta_{x}=x\frac{{\rm d}}{{\rm d}x}$. We call (2.2) the generalized
hypergeometric differential equation. This equation has regular singular
points at $x=0,1,\infty$. We set $b_{N}=1$,
$\tilde{\bm{b}}=(b_{1},b_{2},\dots,b_{N-1},1)$ and $\bm{1}=(1,1,1,\dots,1)$.
We suppose that $a_{i}-a_{j},b_{i}-b_{j}\not\in\mathbb{Z}$, $1\leq i,j\leq N$,
$i\not=j$, and $\sum_{i=1}^{N}{(b_{i}-a_{i})}\not\in\mathbb{Z}$. Then there
exists a fundamental system of solutions around the singular point
$\big{\\{}w_{1}^{[\varrho]},\dots,w_{N}^{[\varrho]}\big{\\}}$ of (2.2). If
$\varrho=0,\infty$, they are given by

$\displaystyle w_{1}^{[0]}={}_{N}F_{N-1}\left(\begin{matrix}\bm{a}\\\
\bm{b}\end{matrix};x\right),$ $\displaystyle
w_{j+1}^{[0]}=x^{1-b_{j}}{}_{N}F_{N-1}\left(\begin{matrix}\bm{a}+(1-b_{j})\bm{1}\\\
\big{(}\tilde{\bm{b}}+(1-b_{j})\bm{1}\big{)}^{\lor}\end{matrix};x\right),$
$\displaystyle
w_{i}^{[\infty]}=(-x)^{-a_{i}}{}_{N}F_{N-1}\left(\begin{matrix}(1+a_{i})\bm{1}-\tilde{\bm{b}}\\\
((1-a_{i})\bm{1}+\bm{a})^{\lor}\end{matrix};\frac{1}{x}\right)$

for $j=1,\dots,N-1$, $i=1,\dots,N$, where $\ast^{\lor}$ indicates that the
entry $1+b_{j}-b_{j}$ or $1-a_{i}+a_{i}$ omitted in $\ast$ [24]. Connection
problem of (2.2) between $x=0$ and $x=\infty$ had already been studied well
[9, 21, 26].

We introduce a positive large parameter $\eta$ in $\bm{a}$ and $\bm{b}$ by
setting

$\displaystyle a_{i}={a_{i,0}}+a_{i,1}{\eta},\qquad i=1,2,\dots,N,$
$\displaystyle b_{j}={b_{j,0}}+b_{j,1}{\eta},\qquad j=1,2,\dots,N-1,$

where $a_{i,k},b_{j,k}\in\mathbb{C}$ for $k=0,1$. We consider the following
equation which contains the large parameter:

$\displaystyle{}_{N}P_{N-1}\psi=0,$ (2.3)

where we set

$\displaystyle{}_{N}P_{N-1}=\eta^{-N}\left(\left(\prod_{j=1}^{N-1}(\vartheta_{x}+b_{j})\right)\partial_{x}-\left(\prod_{i=1}^{N}(\vartheta_{x}+a_{i})\right)\right).$
(2.4)

We call (2.4) the generalized hypergeometric differential equation with the
large parameter. The singular points of the equation (2.3) (or (2.2)) are $0$,
$1$ and $\infty$.

The differential operator (2.4) is a WKB type operator defined on $\mathbb{C}$
in the sense of [3, Definition 2.1]. Let $\xi$ denote the dual variable of
$x$. Then $\xi$ is regarded as the symbol of the differential operator
$\partial_{x}$ (cf. [1, 2, 3]). We introduce a new variable $\zeta$ by setting
$\zeta=\eta^{-1}\xi$. Since the large parameter $\eta$ is regarded as the dual
variable of the variable $y$ of the Borel plane [20], $\eta$ indicates the
symbol of $\partial_{y}=\partial/\partial y$. Hence $\zeta$ designates the
symbol of the microdifferential operator $\partial_{y}^{-1}\partial_{x}$. Then
the total symbol $\sigma({}_{N}P_{N-1})(x,\zeta,\eta)$ in the sense of [3] can
be considered as a function $x$, $\zeta$ and $\eta^{-1}$. We write
$\sigma_{k}({}_{N}P_{N-1})(x,\zeta)$ the coefficient of $\eta^{-k}$ of
$\sigma({}_{N}P_{N-1})(x,\zeta,\eta)$. Then the total symbol can be expressed
in the form

$\displaystyle\sigma({}_{N}P_{N-1})(x,\zeta,\eta)=\sum_{k=0}^{N}\eta^{-k}\sigma_{k}({}_{N}P_{N-1})(x,\zeta).$
(2.5)

We call the leading term $\sigma_{0}({}_{N}P_{N-1})(x,\zeta)$ of (2.5) the
principal symbol of ${}_{N}P_{N-1}$. For an $m$-tuple
$\bm{c}=(c_{1},c_{2},\dots,c_{m})$ of parameters, we denote by
$\mathfrak{s}_{\ell}(\bm{c})$ the elementary symmetric polynomial of degree
$\ell$ of $c_{1},c_{2},\dots,c_{m}$. If $\ell>m$ or $\ell\leq-1$, we set
$\mathfrak{s}_{\ell}(\bm{c})=0$.

######  Lemma 2.1.

The total symbol $\sigma({}_{N}P_{N-1})(x,\zeta,\eta)$ is written in the form:

$\displaystyle\sigma({}_{N}P_{N-1})(x,\zeta,\eta)$
$\displaystyle\qquad{}=\sum_{k=1}^{N}\left(\sum_{j=k}^{N}\\!\left(\left\\{{j-1\atop
k-1}\right\\}\mathfrak{s}_{N-j}({\bm{b}})-\left\\{{j\atop
k}\right\\}\mathfrak{s}_{N-j}({\bm{a}})x\right)\\!\right)x^{k-1}\eta^{-N+k}\zeta^{k}-\eta^{-N}\mathfrak{s}_{N}({\bm{a}}).\\!\\!\\!\\!$
(2.6)

Here $\left\\{{j\atop k}\right\\}$ denotes the Stirling number of the second
kind [14, 24].

###### Proof.

By the definition of the Stirling number of the second kind, the $j$-th power
of the Euler operator $\vartheta_{x}$ becomes

$\displaystyle\vartheta_{x}^{j}=\sum_{k=0}^{j}\left\\{{j\atop
k}\right\\}x^{k}\partial_{x}^{k},\qquad j\geq 0.$ (2.7)

Expanding the right-hand side of (2.4) and using (2.7), we have

$\displaystyle\eta^{N}{}_{N}P_{N-1}$
$\displaystyle=\sum_{\begin{subarray}{c}0\leq j\leq N-1\\\ 0\leq k\leq
j\end{subarray}}\left\\{{j\atop
k}\right\\}\mathfrak{s}_{N-1-j}({\bm{b}})x^{k}\partial_{x}^{k+1}-\sum_{\begin{subarray}{c}0\leq
j\leq N\\\ 0\leq k\leq j\end{subarray}}\left\\{{j\atop
k}\right\\}\mathfrak{s}_{N-j}({\bm{a}})x^{k}\partial_{x}^{k}.$

Hence, we obtain (2.6). ∎

######  Remark 2.2.

The principal symbol $\sigma_{0}({}_{N}P_{N-1})(x,\zeta)$ is written in the
form:

$\displaystyle\sigma_{0}({}_{N}P_{N-1})(x,\zeta)$
$\displaystyle=\zeta\prod_{j=1}^{N-1}(x\zeta+b_{j,1})-\prod_{i=1}^{N}(x\zeta+a_{i,1})$
$\displaystyle=\sum_{k=1}^{N}\big{(}\mathfrak{s}_{N-k}({\bm{b}}_{1})-\mathfrak{s}_{N-k}({\bm{a}}_{1})x\big{)}x^{k-1}\zeta^{k}-\mathfrak{s}_{N}({\bm{a}}_{1}).$
(2.8)

Here we set ${\bm{a}}_{1}=(a_{1,1},a_{2,1},\dots,a_{N,1})$ and
${\bm{b}}_{1}=(b_{1,1},b_{2,1},\dots,b_{N-1,1})$.

###  2.2 Turning points and WKB solutions

A point $x_{\ast}\in\mathbb{C}$ is called a turning point of ${}_{N}P_{N-1}$
with the characteristic value $\zeta_{\ast}$ if

$\displaystyle\sigma_{0}({}_{N}P_{N-1})(x_{*},\zeta_{*})=\partial_{\zeta}\sigma_{0}({}_{N}P_{N-1})(x_{*},\zeta_{*})=0$

and $\sigma_{0}({}_{N}P_{N-1})(x,\zeta)$ does not vanish identically as a
function $\zeta$ (see [3, Definition 3.3]). The turning point $x_{\ast}$ with
the characteristic value $\zeta_{\ast}$ is said to be simple if

$\displaystyle\partial_{x}\sigma_{0}({}_{N}P_{N-1})(x_{*},\zeta_{*})\not=0,\qquad\partial_{\zeta}^{2}\sigma_{0}({}_{N}P_{N-1})(x_{*},\zeta_{*})\not=0$

(see [3, Definition 3.6]).

Let $\mathcal{P}({}_{N}P_{N-1})$ denote the set

$\displaystyle\\{(x_{\ast},\zeta_{\ast})\,|\,\sigma_{0}({}_{N}P_{N-1})(x_{*},\zeta_{*})=\partial_{\zeta}\sigma_{0}({}_{N}P_{N-1})(x_{*},\zeta_{*})=0\\}.$

This set can be regarded as a subset of $\mathbb{P}^{2}_{\mathbb{C}}$. Let
$\mathcal{P}_{\rm tp}({}_{N}P_{N-1})$ be the projection of
$\mathcal{P}({}_{N}P_{N-1})$ to the $x$-space. Note that the singular points
$0$, $1$, $\infty$ do not belong to $\mathcal{P}_{\rm tp}({}_{N}P_{N-1})$ for
generic ${\bm{a}}_{1}$ and ${\bm{b}}_{1}$. By the definition, the turning
points of (2.3) should satisfy the following equation:

$\displaystyle\operatorname{res}_{\zeta}\left(\sigma_{0}({}_{N}P_{N-1})(x,\zeta),\partial_{\zeta}\sigma_{0}({}_{N}P_{N-1})(x,\zeta)\right)=0.$
(2.9)

Here $\operatorname{res}_{\zeta}(F,G)$ denotes the resultant of $F$ and $G$
with respect to $\zeta$.

######  Lemma 2.3.

The resultant (2.9) is rewritten in the form

$\displaystyle\operatorname{res}_{\zeta}\big{(}\sigma_{0}({}_{N}P_{N-1})(x,\zeta),\partial_{\zeta}\sigma_{0}({}_{N}P_{N-1})(x,\zeta)\big{)}$
$\displaystyle\qquad{}=\frac{(-1)^{N-1}}{N^{N-2}}x^{(N-1)^{2}}(1-x)\operatorname{res}_{\zeta}\big{(}f(x,\zeta),g(x,\zeta)\big{)}.$
(2.10)

Here we set

$\displaystyle
f(x,\zeta)=\sum_{k=0}^{N-1}(k+1)(\mathfrak{s}_{N-k-1}({\bm{b}}_{1})-\mathfrak{s}_{N-k-1}({\bm{a}}_{1})x)\zeta^{k},$
(2.11) $\displaystyle
g(x,\zeta)=\sum_{k=1}^{N-1}(N-k)(\mathfrak{s}_{N-k}({\bm{b}}_{1})-\mathfrak{s}_{N-k}({\bm{a}}_{1})x)\zeta^{k}-N\mathfrak{s}_{N}({\bm{a}}_{1})x.$
(2.12)

###### Proof.

For the sake of simplicity, we show (2.10) for the case $N=3$. General case
can be proved in a similar way. For $N=3$, the principal symbol is written in
the form

$\displaystyle\sigma_{0}({}_{3}P_{2})(x,\zeta)=x^{2}q_{3}\zeta^{3}+xq_{2}\zeta^{2}+q_{1}\zeta+q_{0},$

Hence the left-hand side of (2.10) has the form

$\displaystyle\left|\begin{matrix}q_{3}x^{2}&q_{2}x&q_{1}&q_{0}&0\\\
0&q_{3}x^{2}&q_{2}x&q_{1}&q_{0}\\\ 3q_{3}x^{2}&2q_{2}x&q_{1}&0&0\\\
0&3q_{3}x^{2}&2q_{2}x&q_{1}&0\\\ 0&0&3q_{3}x^{2}&2q_{2}x&q_{1}\\\
\end{matrix}\right|.$ (2.13)

Firstly we eliminate the $(3,1)$-element by using the first row. Next we
eliminate the $(4,2)$-element by the second row. Then, expanding the
determinant by the first column, we have

$\displaystyle(-1)^{2}q_{3}x^{2}\left|\begin{matrix}q_{3}x^{2}&q_{2}x&q_{1}&q_{0}\\\
q_{2}x&2q_{1}&3q_{0}&0\\\ 0&q_{2}x&2q_{1}&3q_{0}\\\
0&3q_{3}x^{2}&2q_{2}x&q_{1}\\\ \end{matrix}\right|.$

We eliminate the $(1,4)$-element by using the third row and multiply the first
row by $3$. Then, factoring $x$ out from the first column, multiplying the
third and the fourth columns by $x$ and $x^{2}$, respectively, exchanging the
second row and the fourth row and finally, exchanging the third row and the
fourth row, we have

$\displaystyle\frac{(-1)^{2}}{3}q_{3}\left|\begin{matrix}3q_{3}x&2q_{2}x&q_{1}x&0\\\
0&3q_{3}x^{2}&2q_{2}x^{2}&q_{1}x^{2}\\\ q_{2}&2q_{1}&3q_{0}x&0\\\
0&q_{2}x&2q_{1}x&3q_{0}x^{2}\\\ \end{matrix}\right|.$

We factor $x$, $x^{2}$ and $x$ out from the first, the second and the fourth
row, respectively. Then we obtain the expression of (2.13) of the form

$\displaystyle\frac{(-1)^{2}}{3}q_{3}x^{4}\left|\begin{matrix}3q_{3}&2q_{2}&q_{1}&0\\\
0&3q_{3}&2q_{2}&q_{1}\\\ q_{2}&2q_{1}&3q_{0}x&0\\\ 0&q_{2}&2q_{1}&3q_{0}x\\\
\end{matrix}\right|.$

Thus we have

$\displaystyle\operatorname{res}_{\zeta}(\sigma_{0}({}_{3}P_{2})(x,\zeta),\partial_{\zeta}\sigma_{0}({}_{3}P_{2})(x,\zeta))=\frac{(-1)^{2}}{3}(1-x)x^{4}\operatorname{res}_{\zeta}\left(f(x,\zeta),g(x,\zeta)\right)$

with

$\displaystyle f(x,\zeta)=3q_{3}\zeta^{2}+2q_{2}\zeta+q_{1},$ $\displaystyle
g(x,\zeta)=q_{2}\zeta^{2}+2q_{1}\zeta+3q_{0}x.$

This proves Lemma 2.3 for $N=3$. ∎

We assume that the following conditions are satisfied.

######  Assumption 2.4.

  * (i)

All turning points of the equation (2.3) are simple.

  * (ii)

$\mathfrak{s}_{1}({\bm{a}}_{1})\not=\mathfrak{s}_{1}({\bm{b}}_{1})$.

  * (iii)

The leading coefficient of
$\operatorname{res}_{\zeta}\left(f(x,\zeta),g(x,\zeta)\right)$ with respect to
$x$ does not vanish.

  * (iv)

${\rm{Dis}}_{x}\left(\operatorname{res}_{\zeta}\left(f(x,\zeta),g(x,\zeta)\right)\right)\not=0.$
Here ${\rm{Dis}}_{x}(F)$ denotes the discriminant of $F$ with respect to $x$.

######  Lemma 2.5.

The number of elements in the set $\mathcal{P}({}_{N}P_{N-1})$ equals
$2(N-1)$.

###### Proof.

From Lemma 2.3, a point $(x,\zeta)$ belongs to $\mathcal{P}({}_{N}P_{N-1})$ if
and only if $(x,\zeta)$ satisfies $f(x,\zeta)=0$ and $g(x,\zeta)=0$. Here
$f(x,\zeta)$ and $g(x,\zeta)$ denote (2.11) and (2.12), respectively.
Eliminating $x$ from $f(x,\zeta)=0$ and $g(x,\zeta)=0$, we get the equation
$h(\zeta)=0$ for $\zeta$, where

$\displaystyle
h(\zeta)=\sum_{k=N}^{2(N-1)}\Bigg{(}\sum_{j=1}^{2N-k}j(N-k-j+1)\big{(}\mathfrak{s}_{j-1}({\bm{b}}_{1})\mathfrak{s}_{2N-k-j}({\bm{a}}_{1})-\mathfrak{s}_{j-1}({\bm{a}}_{1})\mathfrak{s}_{2N-k-j}({\bm{b}}_{1})\big{)}\Bigg{)}\zeta^{k}$
$\displaystyle\hphantom{h(\zeta)=}{}+\sum_{k=0}^{N-1}\Bigg{(}\sum_{j=0}^{k+1}N(k+j+1)\big{(}\mathfrak{s}_{N-k+j-1}({\bm{a}}_{1})\mathfrak{s}_{N-j}({\bm{b}}_{1})-\mathfrak{s}_{N-k+j-1}({\bm{b}}_{1})\mathfrak{s}_{N-j}({\bm{a}}_{1})\big{)}\Bigg{)}\zeta^{k}.$

We can see that the coefficient of the highest degree part of $h(\zeta)$ does
not vanish if
$\mathfrak{s}_{1}({\bm{a}}_{1})\not=\mathfrak{s}_{1}({\bm{b}}_{1})$. The
coefficient of $\zeta^{k}$ of $h(\zeta)$ is a homogeneous polynomial of degree
$2N-k-1$ of $\bm{a}_{1}$ and $\bm{b}_{1}$ which does not vanish identically.
Thus, we can show that there are distinct $2(N-1)$ roots of $h(\zeta)=0$ by
using Assumption 2.4. ∎

######  Remark 2.6.

If $N=3$,

$\displaystyle{\rm{Dis}}_{x}\left(\operatorname{res}_{\zeta}\left(f(x,\zeta),g(x,\zeta)\right)\right)=256\prod_{i=1,2,3}a_{i,1}\prod_{\begin{subarray}{c}i=1,2,3\\\
j=1,2\end{subarray}}(a_{i,1}-b_{j,1})h^{\prime}(\bm{a}_{1},\bm{b}_{1})^{3}$

holds. Here $h^{\prime}(\bm{a}_{1},\bm{b}_{1})$ is a homogeneous polynomial of
degree $9$ with respect to $\bm{a}_{1}$, $\bm{b}_{1}$. From
$h^{\prime}(\bm{a}_{1},\bm{b}_{1})=0$, we have the following conditions:

  * (i)

$b_{j,1}\neq 0$ for any $j$, $1\leq j\leq 2$.

  * (ii)

$b_{j,1}\neq b_{j^{\prime},1}$ for $j\not=j^{\prime}$, $1\leq j,j^{\prime}\leq
2$.

The leading coefficient of $\operatorname{res}_{\zeta}(f(x,\zeta),g(x,\zeta))$
with respect to $x$ does not vanish. Then we have

$\displaystyle\prod_{\begin{subarray}{c}i\neq
i^{\prime}\end{subarray}}(a_{i,1}-a_{i^{\prime},1})\neq 0.$

Outside the turning points, there are $N$ distinct roots of the algebraic
equation

$\sigma_{0}({}_{N}P_{N-1})(x,\zeta)=0$

in $\zeta$ of degree $N$. We call these roots the characteristic roots of
${}_{N}P_{N-1}$. We consider the Laurent expansion at the singular point of
each characteristic root. For the case of $\varrho=0$, we substitute
$\zeta=\sum_{k=m}^{\infty}c_{k}x^{k}$ for (2.8). Taking note of the degree of
the leading term of (2.8) with respect to $x$, we find $m=-1$ or $m=0$. If
$m=-1$, there are $N-1$ choices for the leading coefficient:

$\displaystyle c_{-1}=-b_{\ell,1},\qquad\ell=1,2,\dots,N-1.$

If $m=0$, we can see

$\displaystyle c_{0}=\frac{a_{1,1}a_{2,1}\cdots a_{N,1}}{b_{1,1}b_{2,1}\cdots
b_{N-1,1}}.$

The coefficients of higher-order terms $c_{m+\ell}$, $\ell>0$, are determined
recursively. Similarly, we can find the Laurent expansions of the
characteristic roots at $x=\infty$.

######  Definition 2.7.

We take local numbering of the characteristic roots of ${}_{N}P_{N-1}$ as
follows:

  * (i)

For $\varrho=0$, the roots are denoted by $\zeta_{m}^{(0)}$, $m=1,2,\dots,N$,
so that

$\displaystyle\zeta_{\ell}^{(0)}=-\frac{b_{\ell,1}}{x}+O(1),\qquad\ell=1,2,\dots,N-1,$
$\displaystyle\zeta_{N}^{(0)}=\frac{a_{1,1}a_{2,1}\cdots
a_{N,1}}{b_{1,1}\cdots b_{N-1,1}}+O(x)$

hold.

  * (ii)

For $\varrho=\infty$, the roots are denoted by $\zeta_{m}^{(\infty)}$,
$m=1,2,\dots,N$, so that

$\displaystyle\zeta_{m}^{(\infty)}=-\frac{a_{m,1}}{x}+O\left(\frac{1}{x^{2}}\right)$

hold.

Taking suitable branch cuts connecting simple turning points, we can regard
this numbering is defined globally. For a simple turning point $x_{\ast}$ and
a singular point $\varrho$, there are two numbers $j,k\in\\{1,2,\dots,N\\}$,
$j\not=k$ such that
$\zeta_{j}^{(\varrho)}(x_{\ast})=\zeta_{k}^{(\varrho)}(x_{\ast})$ holds. We
consider mainly the case where $\varrho=0$. Then we say that $x_{\ast}$ is a
simple turning point of type $(j,k)$ (see [16, Definition 1.2.1]).

The characteristic variety $\operatorname{Ch}({}_{N}P_{N-1})$ of
${}_{N}P_{N-1}$ is, by definition, an algebraic curve

$\displaystyle\operatorname{Ch}({}_{N}P_{N-1})=\\{(x,\zeta)\,|\,\sigma_{0}({}_{N}P_{N-1})(x,\zeta)=0\\}.$

This can be regarded as a compact Riemann surface $\Sigma$. There is a natural
projection

$\displaystyle\pi\colon\ \Sigma\rightarrow\mathbb{P}_{\mathbb{C}}^{1},$

which is an $N$-covering map.

######  Lemma 2.8.

The genus of $\Sigma$ equals $0$.

###### Proof.

By using the Riemann–Hurwitz formula [13, Section 17], we have the following
relation:

$\displaystyle
2-2g(\Sigma)=N\big{(}2-2g\big{(}\mathbb{P}^{1}_{\mathbb{C}}\big{)}\big{)}-\sum_{i=1}^{2(N-1)}(2-1).$

Here $g(X)$ denotes the genus of the compact Riemann surface $X$. Hence we get
$g(\Sigma)=0$. ∎

A WKB solution $\psi$ of (2.3) is a formal solution of the form

$\displaystyle\psi=\exp\left(\int S\,{\rm d}x\right),$ $\displaystyle S=\eta
S_{-1}+S_{0}+\eta^{-1}S_{1}+\cdots=\sum_{\ell=-1}^{\infty}\eta^{-\ell}S_{\ell}$

(see [3, Definition 3.2]). By inserting $\psi=\exp\big{(}\int S\,{\rm
d}x\big{)}$ into (2.3), we have the following nonlinear differential equation
for $S$:

$\displaystyle\operatorname{Ri}({}_{N}P_{N-1})(S)=0.$ (2.14)

Here we set

$\displaystyle\operatorname{Ri}({}_{N}P_{N-1})(S)=\exp\left(-\int S\,{\rm
d}x\right){}_{N}P_{N-1}\exp\left(\int S\,{\rm d}x\right).$

The leading term of the equation (2.14) with respect to $\eta^{-1}$ determines
$S_{-1}$. Therefore $S_{-1}$ should satisfy
$\sigma_{0}({}_{N}P_{N-1})(x,S_{-1})=0$. Hence we can take one of the
characteristic roots as $S_{-1}$. The higher-order terms $S_{j}$,
$j=0,1,2,\dots$, can be determined recursively and uniquely outside the
turning points if $S_{-1}$ is chosen. The following Lemma is obtained by the
same way as [13, Proposition 3.6]:

######  Lemma 2.9.

Let $S^{(\varrho,m)}$, $m=1,2,\dots,N$, be the formal solutions of (2.3) such
that the numbering are consistent with that of the leading terms given in
Definition 2.7. Then the formal solutions $S^{(\varrho,m)}$, $m=1,2,\dots,N$,
have the following local behaviors near $x=\varrho$, $\varrho=0,\infty$:

$\displaystyle
S^{(0,\ell)}=\frac{1-b_{\ell}}{x}+\frac{\prod\limits_{m=1}^{N}(1+a_{m}-b_{\ell})}{(b_{\ell}-2)\prod\limits_{\begin{subarray}{c}m=1\\\
m\not=\ell\end{subarray}}^{N-1}(b_{\ell}-b_{m}-1)}+O(x),\qquad\ell=1,2,\dots,N-1,$
(2.15) $\displaystyle S^{(0,N)}=\frac{a_{1}a_{2}\cdots a_{N}}{b_{1}b_{2}\cdots
b_{N-1}}+O(x),$ (2.16) $\displaystyle
S^{(\infty,\ell)}=-\frac{a_{\ell}}{x}-\frac{a_{\ell}\prod\limits_{m=1}^{N-1}(1+a_{\ell}-b_{m})}{\prod\limits_{\begin{subarray}{c}m=1\\\
m\not=\ell\end{subarray}}^{N}(1+a_{\ell}-a_{m})}\frac{1}{x^{2}}+O\left(\frac{1}{x^{3}}\right),\qquad\ell=1,2,\dots,N.$
(2.17)

Here we use the notation $O(x)$ in (2.15) and (2.16) in the sense that these
parts can be written as

$\displaystyle x\sum_{j=-1}^{\infty}\eta^{-j}f_{j}(x)$

with some holomorphic functions $f_{j}(x)$, $j=-1,0,1,\dots$, near $x=0$. The
notation $O\big{(}{1}/{x^{3}}\big{)}$ in (2.17) should be understood
similarly.

###  2.3 Factorization

We assume that there is a simple turning point $x_{\ast}$ of ${{}_{N}}P_{N-1}$
with the characteristic value $\zeta_{\ast}$. Then
$\sigma_{0}({}_{N}P_{N-1})(x,\zeta)$ is uniquely decomposed holomorphically in
a neighborhood of $(x_{\ast},\zeta_{\ast})$ in the form

$\displaystyle\sigma_{0}({}_{N}P_{N-1})(x,\zeta)=l(x,\zeta)r(x,\zeta),$

where $r(x,\zeta)$ is a Weierstrass polynomial of degree $2$ in $\zeta$ with
the center at $(x_{\ast},\zeta_{\ast})$ and
$l(x_{\ast},\zeta_{\ast})\allowbreak\not=0$. By the definition, $r(x,\zeta)$
has the form

$\displaystyle
r(x,\zeta)=(\zeta-\zeta_{*})^{2}+f_{1}(x)(\zeta-\zeta_{*})+f_{2}(x),$

where $f_{j}(x)$ vanishes at $x=x_{*}$ for $j=1,2$. We may assume
$\zeta_{j}^{(\varrho)}$ and $\zeta_{k}^{(\varrho)}$ are the roots of
$r(x,\zeta)=0$. We set

$\displaystyle{}_{N}{\tilde{P}}_{N-1}=\frac{1}{x^{N-1}(1-x)}{}_{N}{P}_{N-1}.$

Then $\sigma\big{(}{}_{N}{\tilde{P}}_{N-1}\big{)}(x,\zeta,\eta)$ becomes a
monic polynomial with respect to $\zeta$:

$\displaystyle\sigma\big{(}{}_{N}{\tilde{P}}_{N-1}\big{)}(x,\zeta,\eta)=\zeta^{N}+\sum_{k=0}^{N-1}\tilde{p}_{k}(x,\eta)\zeta^{k}.$

It follows from [3, Theorem 5.1] that there uniquely exist differential
operators $L$ and $R$ of WKB type near $x_{*}$ which satisfy

$\displaystyle{}_{N}{\tilde{P}}_{N-1}=LR$ (2.18)

and

  * (i)

The principal symbol $\sigma_{0}(R)(x,\zeta)$ of $R$ coincides with
$r(x,\zeta)$.

  * (ii)

For each $j>0$, the coefficient $\sigma_{j}(R)(x,\zeta)$ of $\eta^{-j}$ of the
symbol of $R$ is of degree at most one in $\zeta$.

  * (iii)

The principal symbol $\sigma_{0}(L)(x,\zeta)$ of $L$ does not vanish at
$(x_{*},\zeta_{*})$.

Thus constructed operator $R$ has the form

$\displaystyle
R=\big{(}\eta^{-1}\partial_{x}\big{)}^{2}+A(x,\eta)\eta^{-1}\partial_{x}+B(x,\eta)$

with

$\displaystyle A(x,\eta)=A_{0}(x)+A_{1}(x)\eta^{-1}+A_{2}(x)\eta^{-2}+\cdots,$
$\displaystyle B(x,\eta)=B_{0}(x)+B_{1}(x)\eta^{-1}+B_{2}(x)\eta^{-2}+\cdots$

and $L$ is an $(N-2)$-th order operator

$\displaystyle
L=\big{(}\eta^{-1}\partial_{x}\big{)}^{N-2}+\sum_{k=0}^{N-3}L_{k}(x,\eta)\big{(}\eta^{-1}\partial_{x}\big{)}^{k}$

with

$\displaystyle
L_{k}(x,\eta)=L_{k,0}(x)+L_{k,1}(x)\eta^{-1}+L_{k,2}(x)\eta^{-2}+\cdots.$

The operators $L$ and $R$ are constructed as follows. Relation (2.18) yields

$\displaystyle\tilde{p}_{2}=A+L_{0},\qquad\tilde{p}_{1}=B+AL_{0}+\eta^{-1}A^{\prime},\qquad\tilde{p}_{0}=BL_{0}+\eta^{-1}B^{\prime}$

for $N=3$ and

$\displaystyle\tilde{p}_{N-1}(x,\eta)=A+L_{N-3},$
$\displaystyle\tilde{p}_{k}(x,\eta)=\eta^{-N+k+2}\left(\binom{N-2}{k-1}\eta^{-1}A^{(N-k-1)}+\binom{N-2}{k}B^{(N-k-2)}\right)$
$\displaystyle\hphantom{\tilde{p}_{k}(x,\eta)=}{}+\sum_{j=k}^{N-3}\eta^{-j+k}\left(\binom{j}{k-1}\eta^{-1}A^{(j-k+1)}+\binom{j}{k}B^{(j-k)}\right)L_{j}$
$\displaystyle\hphantom{\tilde{p}_{k}(x,\eta)=}{}+L_{k-2}+AL_{k-1},\qquad
2\leq k\leq N-2,$
$\displaystyle\tilde{p}_{1}(x,\eta)=\eta^{-N+3}\big{(}\eta^{-1}A^{(N-2)}+(N-2)B^{(N-3)}\big{)}$
$\displaystyle\hphantom{\tilde{p}_{1}(x,\eta)=}{}+\sum_{j=1}^{N-3}\eta^{-j+1}\big{(}\eta^{-1}A^{(j)}+jB^{(j-1)}\big{)}L_{j}+AL_{0},$
$\displaystyle\tilde{p}_{0}(x,\eta)=\eta^{-N+2}B^{(N-2)}+\sum_{j=0}^{N-3}\eta^{-j}B^{(j)}L_{j}$

for $N\geq 4$. These relations determine $A$, $B$ and $L_{k}$, $0\leq k\leq
N-3$, if we choose the leading terms of them, which solve a system of
algebraic equations. For example, if $N=3$, we obtain $A$ by solving the
following equation:

$\displaystyle
A^{3}-2\tilde{p}_{2}A^{2}-\big{(}3\eta^{-1}A^{\prime}-\eta^{-1}\tilde{p}_{2}-\tilde{p}_{1}-\tilde{p}_{2}^{2}\big{)}A$
$\displaystyle\qquad{}+2\eta^{-1}\tilde{p}_{2}A^{\prime}+\eta^{-1}A^{\prime\prime}+\tilde{p}_{0}-\tilde{p}_{1}\tilde{p}_{2}-\eta^{-1}\tilde{p}_{1}^{\prime}=0.$

We consider a WKB solution

$\displaystyle\phi=\exp\left(\int T\,{\rm d}x\right),$ $\displaystyle T=\eta
T_{-1}+T_{0}+\eta^{-1}T_{1}+\cdots=\sum_{\ell=-1}^{\infty}\eta^{-\ell}T_{\ell}$

of $R\phi=0$. Here $T$ is determined by the Riccati equation associated with
this equation:

$\displaystyle\operatorname{Ri}(R)(T)=\eta^{-2}\big{(}T^{2}+T^{\prime}\big{)}+A\eta^{-1}T+B=0.$
(2.19)

Since we have (2.18), $\phi$ is a formal solution of (2.18). The following
lemma shows that we may regard it as a WKB solution of (2.18).

######  Lemma 2.10.

We have the following relation for $N\geq 3$:

$\displaystyle\operatorname{Ri}\big{(}{}_{N}{\tilde{P}}_{N-1}\big{)}(S)=\sum_{k=0}^{N-2}\frac{\eta^{-k}}{k!}\operatorname{Ri}\big{(}\textrm{{{\Large:}}}\partial^{k}_{\zeta}\sigma(L)(x,\zeta,\eta)\textrm{{\Large:}}\big{)}(S)\partial^{k}_{x}\operatorname{Ri}(R)(S).$

Here $\textrm{{{\Large:}}}\sigma(P)(x,\zeta,\eta)\textrm{{\Large:}}$
designates the differential operator defined by the total symbol
$\sigma(P)(x,\zeta,\eta)$ $($see [1, Definition 4.6] and [3]). Hence
$\operatorname{Ri}(R)(T)=0$ implies
$\operatorname{Ri}\big{(}{}_{N}{\tilde{P}}_{N-1}\big{)}(T)=0$.

######  Remark 2.11.

The differential operator
$\textrm{{{\Large:}}}\partial^{k}_{\zeta}\sigma(L)(x,\zeta,\eta)\textrm{{\Large:}}$
is recovered from the total symbol
$\partial^{k}_{\zeta}\sigma(L)(x,\zeta,\eta)$ by replacing $\zeta$ by
$\eta^{-1}\partial_{x}$ after moving all powers of $\zeta$ in each term in the
symbol to the rightmost part.

###### Proof.

Since we have (2.18),
$\operatorname{Ri}\big{(}{}_{N}\tilde{P}_{N-1}\big{)}(S)$ is computed in the
following form

$\displaystyle\operatorname{Ri}\big{(}{}_{N}\tilde{P}_{N-1}\big{)}(S)$
$\displaystyle={\rm e}^{-\int S\,{\rm d}x}LR{\rm e}^{\int S\,{\rm d}x}$
$\displaystyle={\rm e}^{-\int S\,{\rm d}x}L{\rm e}^{\int S\,{\rm d}x}{\rm
e}^{-\int S\,{\rm d}x}R{\rm e}^{\int S\,{\rm d}x}.$

By the definition of $\operatorname{Ri}(R)(S)$, this equals

${\rm e}^{-\int S\,{\rm d}x}L{\rm e}^{\int S\,{\rm
d}x}\operatorname{Ri}(R)(S),$

which can be written as

${\rm e}^{-\int S\,{\rm d}x}L\operatorname{Ri}(R)(S){\rm e}^{\int S\,{\rm
d}x}.$

The product of the differential operator $L$ of order $N-2$ and the
multiplying operator $\operatorname{Ri}(R)(S)$ has the symbol

$\sum_{k=0}^{N-2}\frac{\eta^{-k}}{k!}\partial_{\zeta}^{k}\sigma(L)(x,\zeta,\eta)\partial_{x}^{k}\operatorname{Ri}(R)(S).$

Since $\partial_{x}^{k}\operatorname{Ri}(R)(S)$ does not contain $\zeta$ for
each $k$, we can write

$\displaystyle L\operatorname{Ri}(R)(S)$
$\displaystyle=\textrm{{{\Large:}}}\sum_{k=0}^{N-2}\frac{\eta^{-k}}{k!}\partial_{\zeta}^{k}\sigma(L)(x,\zeta,\eta)\partial_{x}^{k}\operatorname{Ri}(R)(S)\textrm{{\Large:}}$
$\displaystyle=\sum_{k=0}^{N-2}\frac{\eta^{-k}}{k!}\partial_{x}^{k}\operatorname{Ri}(R)(S)\textrm{{{\Large:}}}\partial_{\zeta}^{k}\sigma(L)(x,\zeta,\eta)\textrm{{\Large:}}.$

Thus we have

${\rm e}^{-\int S\,{\rm d}x}L\operatorname{Ri}(R)(S){\rm e}^{\int S\,{\rm
d}x}=\sum_{k=0}^{N-2}\frac{\eta^{-k}}{k!}\operatorname{Ri}\big{(}\textrm{{{\Large:}}}\partial_{\zeta}^{k}\sigma(L)(x,\zeta,\eta)\textrm{{\Large:}}\big{)}(S)\partial_{x}^{k}\operatorname{Ri}(R)(S).$

This proves the lemma. ∎

###  2.4 Definition of Voros coefficients

Recall that the characteristic roots $\zeta_{j}^{(\varrho)}$ and
$\zeta_{k}^{(\varrho)}$ introduced in the preceding section satisfy the
quadratic equation

$\displaystyle\zeta^{2}+A_{0}\zeta+B_{0}=0.$

We may assume

$\displaystyle
S_{-1}^{(\varrho,j)}=\zeta_{j}=\frac{-A_{0}+\sqrt{A_{0}^{2}-4B_{0}}}{2},\qquad
S_{-1}^{(\varrho,k)}=\zeta_{k}=\frac{-A_{0}-\sqrt{A_{0}^{2}-4B_{0}}}{2}$

for suitable choice of the branch of the square root. We set

$\displaystyle S_{{\rm
even}}^{(\varrho,j,k)}=\frac{1}{2}\big{(}S^{(\varrho,j)}+S^{(\varrho,k)}\big{)},\qquad
S_{{\rm
odd}}^{(\varrho,j,k)}=\frac{1}{2}\big{(}S^{(\varrho,j)}-S^{(\varrho,k)}\big{)}.$
(2.20)

Then we have

$\displaystyle S_{{\rm even}}^{(\varrho,j,k)}+\frac{1}{2}\eta
A=-\frac{1}{2}\frac{\partial_{x}S_{{\rm odd}}^{(\varrho,j,k)}}{S_{{\rm
odd}}^{(\varrho,j,k)}}=-\frac{1}{2}\frac{d}{dx}\log S_{{\rm
odd}}^{(\varrho,j,k)}$

by using (2.19). Thus we can take the following normalization of integration:

$\displaystyle\int S_{{\rm even}}^{(\varrho,j,k)}\,{\rm d}x=-\frac{1}{2}\log
S_{{\rm odd}}^{(\varrho,j,k)}-\frac{1}{2}\eta\int A\,{\rm d}x.$

Here we choose a primitive function $\int A\,{\rm d}x$.

######  Definition 2.12.

The WKB solutions $\psi_{\pm}^{(\tau)}$ of (2.3) normalized at the turning
point $\tau$ of type $(j,k)$ are defined by

$\displaystyle\psi_{\pm}^{(\tau)}=\frac{1}{\sqrt{S_{{\rm
odd}}^{(\varrho,j,k)}}}\exp\left(-\frac{1}{2}\eta\int A\,{\rm
d}x\right)\exp\left(\pm\int_{\tau}^{x}S_{{\rm odd}}^{(\varrho,j,k)}\,{\rm
d}x\right).$

Here the integration of $S_{{\rm odd}}^{(\varrho,j,k)}$ from $\tau$ to $x$ is
understood as a half of the contour integral of it on the path starting from
$x$ on the $k$-th sheet, going around $\tau$ counterclockwise and back to $x$
on the $j$-th sheet.

We denote by $S^{(\varrho,j,k)}_{{\rm odd},\ell}$ the coefficient of
$\eta^{-\ell}$ of $S^{(\varrho,j,k)}_{{\rm odd}}$ and set
$S^{(\varrho,j,k)}_{{\rm odd},\leq 0}=\eta S^{(\varrho,j,k)}_{{\rm
odd},-1}+S^{(\varrho,j,k)}_{{\rm odd},0}$. It follows from Lemma 2.9 and
(2.20) that

$\displaystyle\underset{x=\varrho}{\operatorname{Res}}\,S_{\rm
odd}^{(\varrho,j,k)}\,{\rm
d}x=\underset{x=\varrho}{\operatorname{Res}}\,S_{{\rm odd},\leq
0}^{(\varrho,j,k)}\,{\rm d}x$

holds.

######  Definition 2.13.

The WKB solutions $\psi_{\pm}^{(\varrho)}$ of (2.3) normalized at the singular
point $\varrho$ are defined by

$\displaystyle\psi_{\pm}^{(\varrho)}=\frac{1}{\sqrt{S^{(\varrho,j,k)}_{\rm
odd}}}\exp\left(-\frac{1}{2}\eta\int\\!A\,{\rm
d}x\right)\exp\left(\pm\int^{x}_{\varrho}\\!\left(S^{(\varrho,j,k)}_{\rm
odd}-S^{(\varrho,j,k)}_{{\rm odd},\leq 0}\right)\,{\rm
d}x\pm\int^{x}_{\tau}\\!S^{(\varrho,j,k)}_{{\rm odd},\leq 0}\,{\rm
d}x\right).$

######  Definition 2.14.

The Voros coefficient $V_{\varrho}^{(j,k)}$ at $x=\varrho$ of type $(j,k)$ is
defined by

$\displaystyle
V^{(j,k)}_{\varrho}=\int_{\varrho}^{\tau}\left(S^{(\varrho,j,k)}_{\rm{odd}}-S^{(\varrho,j,k)}_{\rm{odd},\leq
0}\right)\,{\rm d}x.$

The Voros coefficient relates two kinds of the normalization of WKB solutions
defined in the preceding subsection. We have the following formal relations:

$\displaystyle\psi_{\pm}^{(\varrho)}=\exp\big{(}V_{\varrho}^{(j,k)}\big{)}\psi_{\pm}^{(\tau)}.$

The Voros coefficient is rewritten as

$\displaystyle V^{(j,k)}_{\varrho}=\frac{1}{2}\int_{\gamma_{j,k}}(S-\eta
S_{-1}-S_{0})\,{\rm d}x.$

Here we consider that $S$ is defined on the Riemann surface $\Sigma$ of
$S_{-1}$ and $\gamma_{j,k}$ is a path on $\Sigma$ starting from the singular
point $\varrho$ on the $j$-th sheet, going to and detouring $\tau$
counterclockwise on the base space and back to the singular point $\varrho$ on
the $k$-th sheet.

\begin{overpic}[scale={0.50}]{path.pdf} \put(0.0,85.0){$k$-th sheet}
\put(65.0,0.0){$j$-th sheet} \put(45.0,55.0){$\tau$} \put(3.5,5.0){$\varrho$}
\put(80.0,50.0){$\gamma_{j,k}$} \end{overpic} Figure 1: The path
$\gamma_{j,k}$.

######  Remark 2.15.

For any $1\leq j,k,\ell\leq N$, the following relation holds from Lemma 2.8:

$\displaystyle
V^{(j,k)}_{\varrho}+V^{(k,\ell)}_{\varrho}=V^{(j,\ell)}_{\varrho}.$

Especially, $V^{(j,k)}_{\varrho}=-V^{(k,j)}_{\varrho}$ holds.

##  3 Explicit forms of the Voros coefficients

###  3.1 Ladder operators

######  Lemma 3.1.

Let $\mathcal{S}\left(\begin{smallmatrix}{\bm{a}}\\\
{\bm{b}}\end{smallmatrix}\right)$ be the linear space of all solutions of
${}_{N}P_{N-1}\psi=0$. The operators

$\displaystyle H(a_{i})=\vartheta_{x}+a_{i},\qquad i=1,2,\dots,N,$
$\displaystyle B(b_{j})=\vartheta_{x}+b_{j},\qquad j=1,2,\dots,N-1$

induce homomorphisms

$\displaystyle H(a_{i})\colon\ \mathcal{S}\left(\begin{matrix}{\bm{a}}\\\
{\bm{b}}\end{matrix}\right)\rightarrow\mathcal{S}\left(\begin{matrix}{\bm{a}}+{\bf{e}}_{i}\\\
{\bm{b}}\end{matrix}\right),\qquad i=1,2,\dots,N,$ $\displaystyle
B(b_{j})\colon\ \mathcal{S}\left(\begin{matrix}{\bm{a}}\\\
{\bm{b}}+{\bf{e}}_{j}\end{matrix}\right)\rightarrow\mathcal{S}\left(\begin{matrix}{\bm{a}}\\\
{\bm{b}}\end{matrix}\right),\qquad j=1,2,\dots,N-1,$ (3.1)

respectively. Here ${\bf e}_{m}$ denotes the $m$-th unit vector.

###### Proof.

We set

$\displaystyle{}_{N}\hat{P}_{N-1}={}_{N}P_{N-1}\big{|}_{a_{i,1}\mapsto
a_{i,1}+\eta^{-1}}$ (3.2)

for $1\leq i\leq N$. Then we have the following relation for $1\leq i\leq N$:

$\displaystyle H(a_{i})x{}_{N}P_{N-1}=x{}_{N}\hat{P}_{N-1}H(a_{i}).$ (3.3)

Hence $\psi\in\mathcal{S}\left(\begin{smallmatrix}{\bm{a}}\\\
{\bm{b}}\end{smallmatrix}\right)$ implies
$H(a_{i})\psi\in\mathcal{S}\left(\begin{smallmatrix}{\bm{a}}+{\bf e}_{i}\\\
{\bm{b}}\end{smallmatrix}\right)$. Similarly, (3.1) can be proved. ∎

######  Lemma 3.2.

Let $S$ be a formal solution of $\operatorname{Ri}({}_{N}P_{N-1})(S)=0$. Then

$\displaystyle\operatorname{Ri}\big{(}{}_{N}\hat{P}_{N-1}\big{)}\big{(}\partial_{x}\log(xS+a_{i})+S\big{)}=0,\qquad
i=1,2,\dots,N,$ (3.4)
$\displaystyle\operatorname{Ri}({}_{N}P_{N-1})\big{(}\partial_{x}\log\big{(}x\hat{S}+b_{j}\big{)}+\hat{S}\big{)}=0,\qquad
j=1,2,\dots,N-1$ (3.5)

hold. Here we set (3.2) and

$\displaystyle\hat{S}=S\big{|}_{b_{j,1}\mapsto b_{j,1}+\eta^{-1}}$ (3.6)

for $j=1,2,\dots,N-1$.

###### Proof.

We fix the index $i$, $1\leq i\leq N$. By the definition of
$\operatorname{Ri}(\ast)(\star)$, we have the following relations:

$\displaystyle\operatorname{Ri}\big{(}{}_{N}\hat{P}_{N-1}\big{)}\big{(}\partial_{x}\log(xS+a_{i})+S\big{)}=\frac{1}{xS+a_{i}}{\rm
e}^{-\int S\,{\rm d}x}{}_{N}\hat{P}_{N-1}(xS+a_{i}){\rm e}^{\int S\,{\rm
d}x},$ (3.7)
$\displaystyle\operatorname{Ri}\big{(}{}_{N}\hat{P}_{N-1}H(a_{i})\big{)}(S)={\rm
e}^{-\int S\,{\rm d}x}{}_{N}\hat{P}_{N-1}(xS+a_{i}){\rm e}^{\int S\,{\rm
d}x}.$ (3.8)

Combining (3.7) and (3.8), we have

$\displaystyle\operatorname{Ri}\big{(}{}_{N}\hat{P}_{N-1}\big{)}\big{(}\partial_{x}\log(xS+a_{i})+S\big{)}=\frac{1}{xS+a_{i}}\operatorname{Ri}\big{(}{}_{N}\hat{P}_{N-1}H(a_{i})\big{)}(S).$
(3.9)

From (3.3), we get

$\displaystyle\operatorname{Ri}(H(a_{i})x{}_{N}P_{N-1})(S)=\operatorname{Ri}\big{(}x{}_{N}\hat{P}_{N-1}H(a_{i})\big{)}(S).$
(3.10)

By using a similar argument employed in the proof of Lemma 2.10, the left-hand
side of (3.10) can be written in the form:

$\displaystyle\operatorname{Ri}(H(a_{i})x{}_{N}P_{N-1})(S)=\sum_{k\geq
0}\frac{\eta^{-k}}{k!}\operatorname{Ri}\big{(}\textrm{{{\Large:}}}\partial^{k}_{\zeta}(x\eta\zeta+a_{i})\textrm{{\Large:}}\big{)}(S)\partial^{k}_{x}\operatorname{Ri}(x{}_{N}P_{N-1})(S).$
(3.11)

Then the right-hand side of (3.11) is equal to

$\displaystyle(xS+a_{i})x\operatorname{Ri}({}_{N}P_{N-1})(S)+x\big{(}\operatorname{Ri}({}_{N}P_{N-1})(S)+x\partial_{x}\operatorname{Ri}({}_{N}P_{N-1})(S)\big{)}.$

Here we used the relation

$\displaystyle\operatorname{Ri}(x{}_{N}P_{N-1})(S)=x\operatorname{Ri}({}_{N}P_{N-1})(S).$

Since $\operatorname{Ri}({}_{N}P_{N-1})(S)=0$, the right-hand side of (3.11)
vanishes. Combining (3.9), (3.10) and (3.11), we obtain (3.4). The relation
(3.5) can be proved as well. ∎

######  Lemma 3.3.

The formal solution $S$ satisfies the following relations:

$\displaystyle\Delta_{a_{i,1}}S=\partial_{x}\log(xS+a_{i}),\qquad
i=1,2,\dots,N,$
$\displaystyle\Delta_{b_{j,1}}S=-\partial_{x}\log\big{(}x\hat{S}+b_{j}\big{)},\qquad
j=1,2,\dots,N-1,$

where we set (3.6) and

$\displaystyle\Delta_{\rho}S=S\big{|}_{\rho\,\mapsto\rho+\eta^{-1}}-S$

for $\rho=a_{1,1},a_{2,1},\dots,a_{N,1},b_{1,1},b_{2,1},\dots,b_{N-1,1}$.

######  Remark 3.4.

Note that we distinguish $\rho$ from $\varrho$.

######  Remark 3.5.

The difference operator $\Delta_{\rho}$ in $\rho$ by $\eta^{-1}$can be written
in the following form by using a formal differential operator of infinite
order:

$\displaystyle\Delta_{\rho}={\rm e}^{\eta^{-1}\partial_{\rho}}-1.$

###### Proof.

We denote $\hat{\psi}=H({a_{i}})\psi$, which belongs to
$\mathcal{S}\left(\begin{smallmatrix}{\bm{a}}+{\bf e}_{i}\\\
{\bm{b}}\end{smallmatrix}\right)$. Then

$\displaystyle\partial_{x}\log\hat{\psi}=\partial_{x}\log(xS+a_{i})+S$

holds. It follows from Lemma 3.2 that $\hat{S}$ satisfies the equation
obtained from $\operatorname{Ri}({}_{N}P_{N-1})(S)=0$ by replacing $a_{i}$ by
$a_{i}+1$. Hence we have

$\displaystyle\partial_{x}\log(xS+a_{i})+S=\frac{(\eta
S_{-1}+S_{0}+\cdots)+\partial_{x}(\eta S_{-1}+S_{0}+\cdots)}{x(\eta
S_{-1}+S_{0}+\cdots)+a_{i}}$
$\displaystyle\hphantom{\partial_{x}\log(xS+a_{i})+S=}{}+(\eta
S_{-1}+S_{0}+\cdots).$ (3.12)

The leading term with respect to $\eta^{-1}$ of the right-hand side of (3.12)
equals $S_{-1}$. We can choose the leading term of $\hat{S}$ as
$S_{-1}\big{|}_{a_{i,1}\mapsto a_{i,1}+\eta^{-1}}$. Thus, we conclude that
$\hat{S}=S\big{|}_{a_{i,1}\mapsto a_{i,1}+\eta^{-1}}$. Similarly, Lemma 3.3
can be proved for $b_{1,1},b_{2,1},\dots,b_{N-1,1}$. ∎

######  Lemma 3.6.

Let $(\rho,\rho_{0})$ denote one of
$(a_{1,1},a_{1,0}),\dots,(b_{N-1,1},b_{N-1,0})$. The Voros coefficient
$V_{\varrho}^{(j,k)}$, $j<k$, satisfies the following differential-difference
equations:

$\displaystyle\partial_{\rho}\Delta_{\rho}V_{\varrho}^{(j,k)}=f^{(j,k)}_{\rho}+\Delta_{\rho}g^{(j,k)}_{\rho}.$
(3.13)

Here $f^{(j,k)}_{\rho}$ are given as follows and here $g^{(j,k)}_{\rho}$ is a
linear function of $\eta$:

  * (i)

If $\varrho=0$,

$\displaystyle
2f_{a_{i,1}}^{(j,k)}=\frac{\eta}{1+a_{i}-b_{k}}-\frac{\eta}{1+a_{i}-b_{j}},\qquad
k\not=N,$ $\displaystyle
2f_{a_{i,1}}^{(j,N)}=\frac{\eta}{a_{i}}-\frac{\eta}{1+a_{i}-b_{j}},$
$\displaystyle
2f_{b_{m,1}}^{(j,k)}=\frac{\eta}{1+b_{m}-b_{j}}-\frac{\eta}{1+b_{m}-b_{k}},\qquad
k\not=N,m\not=j,k,$ $\displaystyle
2f_{b_{j,1}}^{(j,k)}=-\frac{\eta}{1+b_{j}-b_{k}}-\frac{\eta}{b_{j}-1}-\sum_{\begin{subarray}{c}\ell=1\\\
\ell\not=j\end{subarray}}^{N-1}\frac{\eta}{b_{j}-b_{\ell}}-\sum_{\ell=1}^{N}\frac{\eta}{a_{\ell}-b_{j}},\qquad
k\not=N,$ $\displaystyle
2f_{b_{k,1}}^{(j,k)}=\frac{\eta}{1+b_{k}-b_{j}}+\frac{\eta}{b_{k}-1}+\sum_{\begin{subarray}{c}\ell=1\\\
\ell\not=k\end{subarray}}^{N-1}\frac{\eta}{b_{k}-b_{\ell}}+\sum_{\ell=1}^{N}\frac{\eta}{a_{\ell}-b_{k}},\qquad
k\not=N,$ $\displaystyle
2f_{b_{m,1}}^{(j,N)}=\frac{\eta}{1+b_{m}-b_{j}}-\frac{\eta}{b_{m}},\qquad
m\not=j,$ $\displaystyle
2f_{b_{j,1}}^{(j,N)}=\sum_{\ell=1}^{N}\frac{\eta}{b_{j}-a_{\ell}}-\frac{\eta}{b_{j}}-\frac{\eta}{b_{j}-1}-\sum_{\begin{subarray}{c}\ell=1\\\
\ell\not=j\end{subarray}}^{N-1}\frac{\eta}{b_{j}-b_{\ell}},$

$i=1,2,\dots,N$ and $m=1,2,\dots,N-1$.

  * (ii)

If $\varrho=\infty$,

$\displaystyle
2f_{a_{i,1}}^{(j,k)}=\frac{\eta}{a_{i}-a_{k}}-\frac{\eta}{a_{i}-a_{j}},\qquad
i\not=j,k,$ $\displaystyle
2f_{a_{j,1}}^{(j,k)}=-\frac{\eta}{a_{k}-a_{j}}-\frac{\eta}{a_{j}}+\sum_{\begin{subarray}{c}\ell=1\\\
\ell\not=j\end{subarray}}^{N}\frac{\eta}{1+a_{j}-a_{\ell}}-\sum_{\ell=1}^{N-1}\frac{\eta}{1+a_{j}-b_{\ell}},$
$\displaystyle
2f_{a_{k,1}}^{(j,k)}=\frac{\eta}{a_{j}-a_{k}}+\frac{\eta}{a_{k}}-\sum_{\begin{subarray}{c}\ell=1\\\
\ell\not=k\end{subarray}}^{N}\frac{\eta}{1+a_{k}-a_{\ell}}+\sum_{\ell=1}^{N-1}\frac{\eta}{1+a_{k}-b_{\ell}},$
$\displaystyle
2f_{b_{m,1}}^{(j,k)}=\frac{\eta}{b_{m}-a_{j}}-\frac{\eta}{b_{m}-a_{k}},$

$i=1,2,\dots,N$ and $m=1,2,\dots,N-1$.

The system of differential-difference equations (3.13) satisfy the
compatibility conditions:

$\displaystyle\partial_{\tau}\Delta_{\tau}\partial_{\rho}\Delta_{\rho}V_{\varrho}^{(j,k)}=\partial_{\rho}\Delta_{\rho}\partial_{\tau}\Delta_{\tau}V_{\varrho}^{(j,k)},\qquad\forall\rho,\tau\in\\{a_{1,1},\dots,a_{N,1},b_{1,1},\dots,b_{N-1,1}\\}.$

###### Proof.

We consider the integral

$\displaystyle\int_{{\tilde{\gamma}}_{j,k}}\Delta_{\rho}S\,{\rm d}x$

for $\rho\in\\{a_{1,1},\dots,a_{N,1},b_{1,1},\dots,b_{N-1,1}\\}$. Here the
path $\tilde{\gamma}_{j,k}$ of integration is taken as follows: Let $x$ be a
point near the origin and $x^{(m)}$ the point on the $m$-th sheet of Riemann
surface $\Sigma$ such that $\pi(x^{(m)})=x$. We take a path on $\Sigma$
starting from $x^{(j)}$, going to and detouring $\tau$ counterclockwise on the
base space and back to $x^{(k)}$ and denote this by $\tilde{\gamma}_{j,k}$.

\begin{overpic}[scale={0.50}]{path.pdf} \put(0.0,85.0){$k$-th sheet}
\put(65.0,0.0){$j$-th sheet} \put(45.0,55.0){$\tau$} \put(3.5,5.0){$x$}
\put(80.0,50.0){$\tilde{\gamma}_{j,k}$} \end{overpic} Figure 2: The path
$\tilde{\gamma}_{j,k}$.

It follows from Lemma 2.9 that this integral can be written in the form

$\displaystyle c^{(j,k)}_{-1}\log x+c_{0}^{(j,k)}+O(x)$

with some constant series $c^{(j,k)}_{-1}$ and $c^{(j,k)}_{0}$ of $\eta^{-1}$.
Here we use the notation $O(x)$ in the same sense as in Lemma 2.9. Let
$d_{m}^{(j,k)}$ denote the coefficient of $\eta^{-m}$ of $c_{0}^{(j,k)}$:

$\displaystyle c_{0}^{(j,k)}=\sum_{m=0}^{\infty}d_{m}^{(j,k)}\eta^{-m}.$
(3.14)

On the other hand, by using Lemma 3.3, we have

$\displaystyle\int_{{\tilde{\gamma}}_{j,k}}\Delta_{\rho}S\,{\rm
d}x=\begin{cases}\displaystyle\log\left(\frac{xS^{(0,k)}+a_{i}}{xS^{(0,j)}+a_{i}}\right),&\rho=a_{i,1},\
1\leq i\leq N,\\\
\displaystyle-\log\left(\frac{x\hat{S}^{(0,k)}+b_{m}}{x\hat{S}^{(0,j)}+b_{m}}\right),&\rho=b_{m,1},\
1\leq m\leq N-1.\end{cases}$ (3.15)

Expanding the right-hand side of (3.15) with respect to $x$, we obtain the
coefficient $c_{0}^{(j,k)}$. Moreover, Lemma 2.9 implies that $c_{0}^{(j,k)}$
is written as the logarithm of a product of several linear functions in
$\rho$, whose coefficients of $\rho$ are $-1$ or $1$.

Let $F_{\ell}^{(m)}$ be a primitive function of $S^{(0,m)}_{\ell}$ with
respect to $x$ and let $\int^{x}{\rm d}x$ denote the termwise integration in
$x$ of the Laurent expansion of the integrand at $x=0$, i.e.,

$\displaystyle\int^{x}\left(\frac{p_{-1}}{x}+p_{0}+p_{1}x+\cdots\right){\rm
d}x=p_{-1}\log x+p_{0}x+\frac{1}{2}p_{1}x^{2}+\cdots.$

We set

$\displaystyle
r^{(j,k)}_{\ell}=F^{(k)}_{\ell}(x)-F^{(j)}_{\ell}(x)-\left(\int^{x}S^{(0,k)}_{\ell}\,{\rm
d}x-\int^{x}S^{(0,j)}_{\ell}\,{\rm d}x\right).$ (3.16)

Note that $r^{(j,k)}_{\ell}$ does not depend on $x$ and $\eta$. We let apply
$\Delta_{\rho}$ on both sides of (3.16). Then we have

$\displaystyle\int_{\tilde{\gamma}_{j,k}}\Delta_{\rho}S_{\ell}\,{\rm
d}x=\int^{x}\Delta_{\rho}S^{(0,k)}_{\ell}\,{\rm
d}x-\int^{x}\Delta_{\rho}S^{(0,j)}_{\ell}\,{\rm
d}x+\Delta_{\rho}r_{\ell}^{(j,k)}.$ (3.17)

Since $\Delta_{\rho}={\rm e}^{\eta^{-1}\partial_{\rho}}-1$, the right most
term of (3.17) can be written in the form

$\displaystyle\Delta_{\rho}r_{\ell}^{(j,k)}=\left(\eta^{-1}\partial_{\rho}+\frac{1}{2}\eta^{-2}\partial_{\rho}^{2}+\cdots\right)r_{\ell}^{(j,k)}.$

Multiplying both members of (3.17) by $\eta^{-\ell}$ and summing them up in
$\ell$, we have

$\displaystyle\int_{\tilde{\gamma}_{j,k}}\Delta_{\rho}S\,{\rm
d}x=\sum_{\ell=-1}^{\infty}\eta^{-\ell}\left(\int^{x}\Delta_{\rho}S^{(0,k)}_{\ell}\,{\rm
d}x-\int^{x}\Delta_{\rho}S^{(0,j)}_{\ell}\,{\rm
d}x+\Delta_{\rho}r_{\ell}^{(j,k)}\right).$ (3.18)

The constant term of the right-hand side of (3.18) at $x=0$ is equal to

$\displaystyle\eta\left(\eta^{-1}\partial_{\rho}+\frac{1}{2}\eta^{-2}\partial_{\rho}^{2}+\cdots\right)r_{-1}^{(j,k)}+\left(\eta^{-1}\partial_{\rho}+\frac{1}{2}\eta^{-2}\partial_{\rho}^{2}+\cdots\right)r_{0}^{(j,k)}+\cdots$
$\displaystyle\qquad{}=\partial_{\rho}r_{-1}^{(j,k)}+\left(\frac{1}{2}\partial_{\rho}^{2}r_{-1}^{(j,k)}+\partial_{\rho}r_{0}^{(j,k)}\right)\eta^{-1}+\cdots.$
(3.19)

Comparing the coefficients of $\eta^{-m}$, $m=0,1$, of (3.14) and the right-
hand side of (3.19), we have

$\displaystyle d_{0}^{(j,k)}=\partial_{\rho}r^{(j,k)}_{-1},$ $\displaystyle
d_{1}^{(j,k)}=\frac{1}{2}\partial_{\rho}^{2}r_{-1}^{(j,k)}+\partial_{\rho}r_{0}^{(j,k)}.$
(3.20)

By the definition of $V^{(j,k)}_{0}$, we have

$\displaystyle\Delta_{\rho}V^{(j,k)}_{0}=\frac{1}{2}\lim_{x\to
0}\left(\int_{\tilde{\gamma}_{j,k}}\Delta_{\rho}S\,{\rm
d}x-\eta\int_{\tilde{\gamma}_{j,k}}\Delta_{\rho}S_{-1}\,{\rm
d}x-\int_{\tilde{\gamma}_{j,k}}\Delta_{\rho}S_{0}\,{\rm d}x\right).$ (3.21)

By the definition of $c_{0}^{(j,k)}$ and $r^{(j,k)}_{\ell}$, $\ell=-1,0$, the
right-hand side of (3.21) is written in the form:

$\displaystyle\Delta_{\rho}V^{(j,k)}_{0}=\frac{1}{2}\left(c_{0}^{(j,k)}-\eta\Delta_{\rho}r^{(j,k)}_{-1}-\Delta_{\rho}r^{(j,k)}_{0}\right).$
(3.22)

Differentiating (3.22) with respect to $\rho$ and using (3.20), we have

$\displaystyle\partial_{\rho}\Delta_{\rho}V^{(j,k)}_{0}=\frac{1}{2}\left(\partial_{\rho}c_{0}^{(j,k)}-\Delta_{\rho}\left(\eta
d_{0}^{(j,k)}+d_{1}^{(j,k)}-\frac{1}{2}\partial_{\rho}d_{0}^{(j,k)}\right)\right).$

As we remarked above, $c_{0}^{(j,k)}$ is written as the logarithm of a product
of several linear functions in $\rho$. Setting

$g_{\rho}^{(j,k)}=-\frac{1}{2}\left(\eta
d_{0}^{(j,k)}+d_{1}^{(j,k)}-\frac{1}{2}\partial_{\rho}d_{0}^{(j,k)}\right),$

we have the lemma for $\varrho=0$. Similarly, we get the differential-
difference equations for Voros coefficients at the infinity. ∎

######  Remark 3.7.

If $N=3$, we have

$\displaystyle
2\partial_{a_{i,1}}\Delta_{a_{i,1}}V_{0}^{(1,2)}=-\frac{\eta}{a_{i}-b_{1}+1}+\frac{\eta}{a_{i}-b_{2}+1}$
$\displaystyle\qquad{}+\Delta_{a_{i,1}}\left(-\eta\log\left(\frac{a_{i,1}-b_{2,1}}{a_{i,1}-b_{1,1}}\right)+\frac{2a_{i,0}-2b_{1,0}+1}{2a_{i,1}-2b_{1,1}}-\frac{2a_{i,0}-2b_{2,0}+1}{2a_{i,1}-2b_{2,1}}\right),$
$\displaystyle
2\partial_{a_{i,1}}\Delta_{a_{i,1}}V_{0}^{(j,3)}=\frac{\eta}{a_{i}-b_{j}+1}+\frac{\eta}{a_{i}}$
$\displaystyle\qquad{}+\Delta_{a_{i,1}}\left(-\eta\log\left(\frac{a_{i,1}}{a_{i,1}-b_{j,1}}\right)-\frac{b_{j,0}-1}{a_{i,1}-b_{j,1}}+\frac{\left(1-2a_{i,0}\right)b_{j,1}}{2a_{i,1}(a_{i,1}-b_{j,1})}\right),$
$\displaystyle 2\partial_{b_{m,1}}\Delta_{b_{m,1}}V_{0}^{(1,2)}$
$\displaystyle\qquad{}=(-1)^{m-1}\left(\sum_{i=1}^{3}\frac{\eta}{b_{m}-a_{i}}-\sum_{\ell\in\\{1,2\\}\setminus\\{m\\}}\left(\frac{\eta}{b_{m}-1}-\frac{\eta}{b_{m}-b_{\ell}}-\frac{\eta}{b_{m}-b_{\ell}+1}\right)\right)$
$\displaystyle\qquad\quad{}+\Delta_{b_{m,1}}(-1)^{m-1}\Bigg{(}{-}\eta\log\left(\frac{\left(a_{1,1}-b_{m,1}\right)\left(a_{2,1}-b_{m,1}\right)\left(a_{3,1}-b_{m,1}\right)}{b_{m,1}\left(b_{1,1}-b_{2,1}\right){}^{2}}\right)$
$\displaystyle\qquad\quad{}-\frac{1}{2}\left(\sum_{i=1}^{3}\frac{2(a_{i,0}-b_{m,0})+1}{a_{i,1}-b_{m,1}}+\frac{3-2b_{m,0}}{b_{m,1}}-\frac{4\left(b_{1,0}-b_{2,0}\right)}{b_{1,1}-b_{2,1}}\right)\Bigg{)},$
$\displaystyle
2\partial_{b_{m,1}}\Delta_{b_{m,1}}V_{0}^{(j,3)}=\frac{\eta}{b_{m}-b_{j}+1}-\frac{\eta}{b_{m}}$
$\displaystyle\qquad{}+\Delta_{b_{m,1}}(-1)^{m-1}\left(-\eta\log\left(1-\frac{b_{j,1}}{b_{m,1}}\right)-\frac{b_{j,0}-1}{b_{j,1}-b_{m,1}}+\frac{(2b_{m,0}-1)b_{j,1}}{2b_{m,1}\left(b_{j,1}-b_{m,1}\right)}\right),$
$\displaystyle
2\partial_{b_{m,1}}\Delta_{b_{m,1}}V_{0}^{(m,3)}=\sum_{i=1}^{3}\frac{\eta}{b_{m}-a_{i}}-\frac{\eta}{b_{m}-1}-\frac{\eta}{b_{m}}-\frac{\eta}{b_{m}-b_{j}}$
$\displaystyle\qquad{}+\Delta_{b_{m,1}}\Bigg{(}{-}\eta\log\left(\frac{\left(a_{1,1}-b_{m,1}\right)\left(a_{2,1}-b_{m,1}\right)\left(a_{3,1}-b_{m,1}\right)}{b_{m,1}^{2}\left(b_{m,1}-b_{j,1}\right)}\right)$
$\displaystyle\qquad{}-\frac{1}{2}\left(\sum_{i=1}^{3}\frac{2(a_{i,0}-b_{m,0})+1}{a_{i,1}-b_{m,1}}+\frac{4(1-b_{m,0})}{b_{m,1}}+\frac{2(b_{j,0}-b_{m,0})+1}{b_{m,1}-b_{j,1}}\right)\Bigg{)}$

for $i=1,2,3$, $m=1,2$ and $j\in\\{1,2\\}\setminus\\{m\\}$.

The system of differential-difference equations given in Lemma 3.6
characterizes $V_{\varrho}^{(j,k)}$, for we have the following lemma:

######  Lemma 3.8.

Let ${W}=\sum_{n=-1}^{\infty}w_{n}\eta^{-n}$ be a formal solution of the
differential-difference equation

$\displaystyle\partial_{\rho}\Delta_{\rho}W=0$ (3.23)

for $\rho=a_{1,1},\dots,a_{N,1},b_{1,1},\dots,b_{N-1,1}$ and suppose that
$w_{n}$ is a homogeneous function of degree $(-n)$ with respect to
$a_{1,1},\dots,a_{N,1},b_{1,1},\dots,b_{N-1,1}$ and $w_{-1}=w_{0}=0$. Then, we
have

$\displaystyle W=0.$

###### Proof.

By the definition of $\Delta_{\rho}$, we have

$\displaystyle\partial_{\rho}\Delta_{\rho}W=\sum_{\ell=0}^{\infty}\left(\sum_{n=-1}^{\ell-1}\frac{1}{(\ell-n)!}\partial_{\rho}^{\ell-n+1}w_{n}\right)\eta^{-\ell}$

for all $\rho\in\\{a_{1,1},\dots,a_{N,1},b_{1,1},\dots,b_{N-1,1}\\}$.
Comparing the coefficients for both sides of (3.23) with respect to the powers
of $\eta$, we have

$\displaystyle\sum_{n=-1}^{\ell-1}\frac{1}{(\ell-n)!}\partial_{\rho}^{\ell-n+1}w_{n}=0,\qquad
n=0,1,2,\dots.$

Since $w_{-1}=w_{0}=0$, we have $\partial_{\rho}^{2}w_{n}=0$ for $n=1,2,\dots$
recursively. By the assumption, $w_{n}$ is a homogeneous function of degree
$(-n)$ with respect to $a_{1,1},\dots,a_{N,1},b_{1,1},\dots,b_{N-1,1}$.
Therefore we obtain

$\displaystyle w_{n}=0,\qquad n=1,2,\dots.$ ∎

###  3.2 Voros coefficients

We have the explicit forms of the Voros coefficients $V_{\varrho}^{(j,k)}$ at
$\varrho$, $\varrho=0,\infty$.

######  Theorem 3.9.

Let $\varrho$ be $0$ or $\infty$. If there exists a simple turning point of
type $(j,k)$, $j<k$, of (2.3), then the Voros coefficients
$V_{\varrho}^{(j,k)}$ are written in the form:

$\displaystyle\frac{1}{2}\sum_{\ell=2}^{\infty}\frac{(-1)^{\ell+1}\eta^{1-\ell}}{\ell(\ell-1)}V^{(j,k)}_{\varrho,\ell},$

where

$\displaystyle
V^{(j,k)}_{0,\ell}=\frac{B_{\ell}(b_{j,0}-1)}{b_{j,1}^{\ell-1}}-\frac{B_{\ell}(b_{k,0}-1)}{b_{k,1}^{\ell-1}}+\sum_{i=1}^{N}\frac{B_{\ell}(b_{k,0}-a_{i,0})}{(b_{k,1}-a_{i,1})^{\ell-1}}$
$\displaystyle\hphantom{V^{(j,k)}_{0,\ell}=}{}-\sum_{i=1}^{N}\frac{B_{\ell}(b_{j,0}-a_{i,0})}{(b_{j,1}-a_{i,1})^{\ell-1}}+\sum_{\begin{subarray}{c}m=1\\\
m\not=j\end{subarray}}^{N-1}\frac{B_{\ell}(b_{j,0}-b_{m,0})}{(b_{j,1}-b_{m,1})^{\ell-1}}-\sum_{\begin{subarray}{c}m=1\\\
m\not=k\end{subarray}}^{N-1}\frac{B_{\ell}(b_{k,0}-b_{m,0})}{(b_{k,1}-b_{m,1})^{\ell-1}},\qquad
k\not=N,$ $\displaystyle
V^{(j,N)}_{0,\ell}=\frac{B_{\ell}(b_{j,0}-1)}{b_{j,1}^{\ell-1}}-\sum_{i=1}^{N}\frac{B_{\ell}(a_{i,0})}{a_{i,1}^{\ell-1}}-\sum_{i=1}^{N}\frac{B_{\ell}(b_{j,0}-a_{i,0})}{(b_{j,1}-a_{i,1})^{\ell-1}}$
$\displaystyle\hphantom{V^{(j,N)}_{0,\ell}=}{}+\sum_{\begin{subarray}{c}m=1\\\
m\not=j\end{subarray}}^{N-1}\frac{B_{\ell}(b_{j,0}-b_{m,0})}{(b_{j,1}-b_{m,1})^{\ell-1}}+\sum_{m=1}^{N-1}\frac{B_{\ell}(b_{m,0})}{b_{m,1}^{\ell-1}},$
$\displaystyle
V^{(j,k)}_{\infty,\ell}=\sum_{m=1}^{N-1}\left(\frac{B_{\ell}(b_{m,0}-a_{k,0})}{(b_{m,1}-a_{k,1})^{\ell-1}}-\frac{B_{\ell}(b_{m,0}-a_{j,0})}{(b_{m,1}-a_{j,1})^{\ell-1}}\right)$
$\displaystyle\hphantom{V^{(j,k)}_{\infty,\ell}=}{}+\left(\frac{B_{\ell}(a_{j,0})}{a_{j,1}^{\ell-1}}+\sum_{\begin{subarray}{c}m=1\\\
m\not=j\end{subarray}}^{N-1}\frac{B_{\ell}(a_{i,0}-a_{j,0})}{(a_{i,1}-a_{j,1})^{\ell-1}}\right)-\left(\frac{B_{\ell}(a_{k,0})}{a_{k,1}^{\ell-1}}+\sum_{\begin{subarray}{c}m=1\\\
m\not=k\end{subarray}}^{N-1}\frac{B_{\ell}(a_{i,0}-a_{k,0})}{(a_{i,1}-a_{k,1})^{\ell-1}}\right).$

Here $B_{\ell}(t)$ denotes the $\ell$-th Bernoulli polynomial defined by

$\displaystyle\frac{x{\rm e}^{xt}}{{\rm
e}^{x}-1}=\sum_{\ell=0}^{\infty}\frac{B_{\ell}(t)}{\ell!}x^{\ell}.$

###### Proof.

To obtain the explicit forms of $V_{\varrho}^{(j,k)}$, we will solve (3.13).
We may write $f_{\rho}^{(j,k)}$ in the form

$\displaystyle
f_{\rho}^{(j,k)}=\frac{1}{2}\sum_{m}\frac{1}{u^{(j,k)}_{\rho,m}+v^{(j,k)}_{\rho,m}\eta^{-1}},$
(3.24)

where $u^{(j,k)}_{\rho,m}$ and $v^{(j,k)}_{\rho,m}$ being linear functions of
$\rho$ and $\rho_{0}$, respectively, which are independent of $\eta$. The
explicit forms of $u^{(j,k)}_{\rho,m}$ and $v^{(j,k)}_{\rho,m}$ are given by
Lemma 3.6. The coefficient of $\rho$ of $u^{(j,k)}_{\rho,m}$ is equal to $1$
or $-1$, which denotes $\epsilon^{(j,k)}_{\rho,m}$ $(=\pm 1)$. The right-hand
side of (3.24) is obtained from $u^{(j,k)}_{\rho,m}$ by the shift:

$\displaystyle\frac{1}{2}\sum_{m}\frac{1}{u^{(j,k)}_{\rho,m}+v^{(j,k)}_{\rho,m}\eta^{-1}}=\frac{1}{2}\sum_{m}\frac{1}{u^{(j,k)}_{\rho,m}}\bigg{|}_{\rho\mapsto\rho+\epsilon^{(j,k)}_{\rho,m}v^{(j,k)}_{\rho,m}\eta^{-1}}.$

Thus (3.13) is written in the following form:

$\displaystyle\partial_{\rho}\Delta_{\rho}V_{\varrho}^{(j,k)}=\frac{1}{2}\sum_{m}{\rm
e}^{\epsilon^{(j,k)}_{\rho,m}v_{\rho,m}^{(j,k)}\eta^{-1}\partial_{\rho}}\frac{1}{u_{\rho,m}^{(j,k)}}+\Delta_{\rho}g^{(j,k)}_{\rho}.$

Hence (3.13) can be solved by using a formal differential operator of infinite
order:

$\displaystyle V_{\varrho}^{(j,k)}$
$\displaystyle=\frac{1}{2}\sum_{m}\frac{{\rm
e}^{\epsilon^{(j,k)}_{\rho,m}v_{\rho,m}^{(j,k)}\eta^{-1}\partial_{\rho}}}{\big{(}{\rm
e}^{\eta^{-1}\partial_{\rho}}-1\big{)}}\partial_{\rho}^{-1}\frac{1}{u_{\rho,m}^{(j,k)}}+\partial_{\rho}^{-1}g^{(j,k)}_{\rho}$
$\displaystyle=\frac{1}{2}\sum_{\ell=0}^{\infty}\sum_{m}\frac{B_{\ell}\big{(}\epsilon^{(j,k)}_{\rho,m}v_{\rho,m}^{(j,k)}\big{)}}{\ell!}\eta^{1-\ell}\partial_{\rho}^{\ell-2}\frac{1}{u_{\rho,m}^{(j,k)}}+\partial_{\rho}^{-1}g^{(j,k)}_{\rho}.$

Using Lemma 3.8, we have

$\displaystyle
V_{\varrho}^{(j,k)}=\frac{1}{2}\sum_{\ell=2}^{\infty}\sum_{m}\frac{B_{\ell}\big{(}\epsilon^{(j,k)}_{\rho,m}v_{\rho,m}^{(j,k)}\big{)}}{\ell!}\eta^{1-\ell}\partial_{\rho}^{\ell-2}\frac{1}{u_{\rho,m}^{(j,k)}}+C_{\rho}(\check{\rho}),$

where $C_{\rho}(\check{\rho})$ is an arbitrary function of
$\\{a_{1,1},\dots,a_{N,1},b_{1,1},\dots,b_{N-1,1}\\}\setminus\\{\rho\\}$. We
repeat the above discussion for every
$\rho\in\\{a_{1,1},\dots,a_{N,1},b_{1,1},\dots,b_{N-1,1}\\}$. Then we can
determine $C_{\rho}(\check{\rho})$ by adjustment. Thus we have the theorem. ∎

######  Remark 3.10.

If $N=3$, we have

$\displaystyle V^{(1,2)}_{0,\ell}=\sum_{\begin{subarray}{c}i=1,2,3\\\
j=1,2\end{subarray}}(-1)^{j}\frac{B_{\ell}(b_{j,0}-a_{i,0})}{(b_{j,1}-a_{i,1})^{\ell-1}}$
$\displaystyle\hphantom{V^{(1,2)}_{0,\ell}=}{}+\frac{B_{\ell}(b_{1,0}-1)}{b_{1,1}^{\ell-1}}-\frac{B_{\ell}(b_{2,0}-1)}{b_{2,1}^{\ell-1}}+\frac{B_{\ell}(b_{1,0}-b_{2,0}+1)+B_{\ell}(b_{1,0}-b_{2,0})}{(b_{1,1}-b_{2,1})^{\ell-1}},$
$\displaystyle
V^{(1,3)}_{0,\ell}=-\sum_{i=1,2,3}\frac{B_{\ell}(b_{1,0}-a_{i,0})}{(b_{1,1}-a_{i,1})^{\ell-1}}-\sum_{i=1,2,3}\frac{B_{\ell}(a_{i,0})}{a_{i,1}^{\ell-1}}$
$\displaystyle\hphantom{V^{(1,3)}_{0,\ell}=}{}+\frac{B_{\ell}(b_{1,0})+B_{\ell}(b_{1,0}-1)}{b_{1,1}^{\ell-1}}+\frac{B_{\ell}(b_{2,0})}{b_{2,1}^{\ell-1}}+\frac{B_{\ell}(b_{1,0}-b_{2,0})}{(b_{1,1}-b_{2,1})^{\ell-1}},$
$\displaystyle
V^{(2,3)}_{0,\ell}=-\sum_{i=1,2,3}\frac{B_{\ell}(b_{2,0}-a_{i,0})}{(b_{2,1}-a_{i,1})^{\ell-1}}-\sum_{i=1,2,3}\frac{B_{\ell}(a_{i,0})}{a_{i,1}^{\ell-1}}$
$\displaystyle\hphantom{V^{(2,3)}_{0,\ell}=}{}+\frac{B_{\ell}(b_{2,0})+B_{\ell}(b_{2,0}-1)}{b_{2,1}^{\ell-1}}+\frac{B_{\ell}(b_{1,0})}{b_{1,1}^{\ell-1}}+\frac{B_{\ell}(b_{2,0}-b_{1,0})}{(b_{2,1}-b_{1,1})^{\ell-1}},$
$\displaystyle
V^{(j,k)}_{\infty,\ell}=\sum_{m=1,2}\left(\frac{B_{\ell}(b_{m,0}-a_{k,0})}{(b_{m,1}-a_{k,1})^{\ell-1}}-\frac{B_{\ell}(b_{m,0}-a_{j,0})}{(b_{m,1}-a_{j,1})^{\ell-1}}\right)$
$\displaystyle\hphantom{V^{(j,k)}_{\infty,\ell}=}{}+\left(\frac{B_{\ell}(a_{j,0})}{a_{j,1}^{\ell-1}}+\sum_{\begin{subarray}{c}i=1\\\
i\not=j\end{subarray}}^{3}\frac{B_{\ell}(a_{i,0}-a_{j,0})}{(a_{i,1}-a_{j,1})^{\ell-1}}\right)-\left(\frac{B_{\ell}(a_{k,0})}{a_{k,1}^{\ell-1}}+\sum_{\begin{subarray}{c}i=1\\\
i\not=k\end{subarray}}^{3}\frac{B_{\ell}(a_{i,0}-a_{k,0})}{(a_{i,1}-a_{k,1})^{\ell-1}}\right).$

We consider the Borel summability of $V_{\varrho}^{(j,k)}$. We set

$\displaystyle\tilde{V}(\kappa,\kappa_{0})=\frac{1}{2}\sum_{\ell=2}^{\infty}\frac{(-1)^{\ell+1}\eta^{1-\ell}}{\ell(\ell-1)}\frac{B_{\ell}(\kappa_{0})}{\kappa^{\ell-1}}.$

As we saw in the preceding theorem, $V_{\varrho}^{(j,k)}$ are expressed as a
finite sum of the formal series of the form $\tilde{V}(\kappa,\kappa_{0})$ for
suitable choice of $\kappa$, $\kappa_{0}$. Hence it is sufficient to consider
the Borel summability of $\tilde{V}(\kappa,\kappa_{0})$. Its summability is
well known: the formal series $\tilde{V}(\kappa,\kappa_{0})$ of $\eta^{-1}$ is
Borel summable if $\operatorname{Re}(\kappa)\neq 0$ and the Borel sum of
$\tilde{V}(\kappa,\kappa_{0})$ equals

$\displaystyle\frac{1}{2}\log\left(\frac{\sqrt{2\pi}(\kappa\eta)^{\kappa_{0}+\kappa\eta-\frac{1}{2}}}{{\rm
e}^{\kappa\eta}\Gamma(\kappa_{0}+\kappa\eta)}\right),\qquad\operatorname{Re}(\kappa)>0,$
$\displaystyle\frac{1}{2}\log\left(\frac{(-\kappa\eta)^{\kappa_{0}+\kappa\eta-\frac{1}{2}}\Gamma\left(1-(\kappa_{0}+\kappa\eta)\right)}{\sqrt{2\pi}{\rm
e}^{\kappa\eta}}\right),\qquad\operatorname{Re}(\kappa)<0,$

(see [6, Lemma 4.6]). This expression is obtained by the Binet formula [12].
Thus, we have

######  Corollary 3.11.

Let $\varrho$ be $0$ or $\infty$. Also let

$\displaystyle
D(\kappa)=\\{(a_{1,1},\dots,a_{N,1},b_{1,1},\dots,b_{N-1,1})\,|\,\operatorname{Re}(\kappa)\not=0\\},$

where $\kappa$ is a linear homogeneous function of
$a_{1,1},\dots,a_{N,1},b_{1,1},\dots,b_{N-1,1}$. The Voros coefficient
$V_{\varrho}^{(j,k)}$ of type $(j,k)$, $j<k$, is Borel summable if
$(a_{1,1},\dots,a_{N,1},b_{1,1},\dots,b_{N-1,1})$ belongs to
$D^{(j,k)}_{\varrho}$. Here we set

$\displaystyle D^{(j,k)}_{0}=D(b_{j,1})\cap
D(b_{k,1})\cap\bigcap_{i=1}^{N}{D(b_{j,1}-a_{i,1})}\cap\bigcap_{\begin{subarray}{c}m=1\\\
m\not=j\end{subarray}}^{N-1}{D(b_{j,1}-b_{m,1})}$
$\displaystyle\hphantom{D^{(j,k)}_{0}=}{}\cap\bigcap_{i=1}^{N}{D(b_{k,1}-a_{i,1})}\cap\bigcap_{\begin{subarray}{c}m=1\\\
m\not=k\end{subarray}}^{N-1}{D(b_{k,1}-b_{m,1})},\qquad k\not=N,$
$\displaystyle
D^{(j,N)}_{0}=\bigcap_{i=1}^{N}{D(a_{i,1})}\cap\bigcap_{m=1}^{N-1}{D(b_{m,1})}\cap\bigcap_{i=1}^{N}{D(b_{j,1}-a_{i,1})}\cap\bigcap_{\begin{subarray}{c}m=1\\\
m\not=j\end{subarray}}^{N-1}{D(b_{j,1}-b_{m,1})},$ $\displaystyle
D^{(j,k)}_{\infty}=D(a_{j,1})\cap
D(a_{k,1})\cap\bigcap_{m=1}^{N-1}{D(b_{m,1}-a_{k,1})}\cap\bigcap_{m=1}^{N-1}{D(b_{m,1}-a_{j,1})}$
$\displaystyle\hphantom{D^{(j,k)}_{\infty}=}{}\cap\bigcap_{\begin{subarray}{c}i=1\\\
i\not=j\end{subarray}}^{N}{D(a_{i,1}-a_{j,1})}\cap\bigcap_{\begin{subarray}{c}i=1\\\
i\not=k\end{subarray}}^{N}{D(a_{i,1}-a_{k,1})}.$

##  4 Concluding remarks and future problems

We have obtained explicit forms of the Voros coefficients at the origin and at
the infinity (Theorem 3.9). To give an explicit form of the Voros coefficient
at $x=1$ is our future problem. The difficulty of the problem comes from the
multiplicity of the characteristic exponents at $x=1$.

As in the case of the Gauss hypergeometric differential equation [8] and of
its confluent families [22, 27], we may get the formulas describing the
parametric Stokes phenomena for WKB solutions of (2.3) by using Corollary 3.11
under the assumption that their formal solutions are Borel summable in some
region. The Borel summability of WKB solutions of our equation is also a
future problem.

### Acknowledgements

The first author is supported by JSPS KAKENHI Grant No. 18K03385. The authors
would like to thank the referees for their careful readings and detailed
comments on this article.

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