# Hölder and Kurokawa meet Borwein–Dykshoorn and Adamchik

J.-P. Allouche
CNRS, IMJ-PRG, Sorbonne
4 Place Jussieu
F-75252 Paris Cedex 05, France
<EMAIL_ADDRESS>

###### Abstract

Following our discovery of a nice identity in a recent preprint of Hu and Kim,
we show a link between the Kurokawa multiple trigonometric functions and two
functions introduced respectively by Borwein-Dykshoorn and by Adamchik. In
particular several identities involving $\zeta(3)$, $\pi$ and the Catalan
constant $G$ that are proved in these three papers are related.

2010 Mathematics Subject Classification: 11M06; 33B15; 11M35; 33E20.

Key words: Kurokawa multiple sine; Borwein-Dykshoorn function; Adamchik
function; zeta values.

##  1 Introduction

There is a wealth of special functions arising from geometry and from
transcendence theory. The purpose of this paper is to provide identities
relating some of them. The beginning of the story here is a recent preprint of
Hu and Kim [11] that gives the nice identity

$\zeta(3)=\frac{4\pi^{2}}{21}\log\left(\frac{e^{\frac{4G}{\pi}}{\mathcal{C}}_{3}\left(\frac{1}{4}\right)^{16}}{\sqrt{2}}\right).$

where $G=\sum_{n\geq 0}\frac{(-1)^{n}}{(2n+1)^{2}}$ is the Catalan constant,
and ${\mathcal{C}}_{3}$ is the Kurokawa-Koyama triple cosine function (see
below).

The (slightly hidden) occurrences of $7\zeta(3)/4\pi^{2}$ and of $e^{G/2\pi}$
reminded us two (out of four) identities in a paper of Kachi and Tzermias
[12], namely:

$\lim_{n\to\infty}\prod_{k=1}^{2n+1}e^{-1/4}\left(1-\frac{1}{k+1}\right)^{\frac{k(k+1)}{2}(-1)^{k}}=\exp\left(\frac{7\zeta(3)}{4\pi^{2}}+\frac{1}{4}\right)$

and

$\lim_{n\to\infty}\prod_{k=1}^{2n+1}\left(1-\frac{2}{2k+1}\right)^{k(-1)^{k}}=\exp\left(\frac{2G}{\pi}+\frac{1}{2}\right).$

In that paper Kachi and Tzermias proved four identities, and they indicated
that they were not able to deduce them directly from the values of a function
introduced in 1993 by Borwein and Dykshoorn [4]. We provided in [2] a proof of
their identities, using the paper of Borwein and Dykshoorn and their function
$D$ for two of the identities, and a paper of Adamchik [1] and his function
$E$ for the remaining two. It was thus tempting to relate these functions $D$
and $E$ to the Kurokawa-Koyama triple cosine.

Furthermore, looking at the papers of Kurokawa, we found an expression of
$\zeta(3)$ resembling the identity given by Hu et Kim, namely (see [14, p.
62], also see [17, Theorem 2, p. 209])

$\zeta(3)=\frac{8\pi^{2}}{7}\log\left(\frac{2^{1/4}}{{\mathcal{S}}_{3}(1/2)}\right)$

where ${\mathcal{S}}_{3}$ is the triple sine function of Kurokawa (in
particular

${\mathcal{S}}_{3}(1/2)=e^{1/8}\prod_{n\geq
1}\left(\left(1-\frac{1}{4n^{2}}\right)^{n^{2}}e^{\frac{1}{4}}\right).$

It was then even more tempting to study these multiple trigonometric functions
and to (try to) find a link between all these results.

A possibly surprising fact is that the literature related to the functions of
Borwein-Dykshoorn and of Adamchik appears to be essentially disjoint from the
literature related to the Kurokawa multiple trigonometric functions, with the
notable exception of the book [28] where several papers of Kurokawa et al. are
cited in the references but do not seem to be exploited in the text. We
propose to enlarge the bridge between these two branches of the theory of
special functions.

In the sequel, we will recall the definitions of the multiple trigonometric
functions first introduced by Kurokawa in [14, 15], then the definitions of
the Borwein-Dykshoorn function $D$ given in [4], and of the Adamchik function
$E$ given in [1]. We will obtain close relations between these functions.
Furthermore we will show that an identity due to Holcombe [9] can also be
obtained using multiple trigonometric functions. We will also mention a link
between these functions and the generalized Euler constant function in [8]
(also see [30] and [27]). We will end with two questions, one of which
concerning an identity proved in [24], that we were not able to address with
multiple trigonometric functions.

##  2 Multiple trigonometric functions

In 1991–1992 (see [14, 15]) Kurokawa introduced the multiple sine functions
defined by ${\mathcal{S}}_{1}(z):=2\sin(\pi z)$, and for $r\geq 2$,

${\mathcal{S}}_{r}(z):=\exp\left(\frac{z^{r-1}}{r-1}\right)\prod_{n\geq
1}\left(P_{r}\left(\frac{z}{n}\right)P_{r}\left(-\frac{z}{n}\right)^{(-1)^{r-1}}\right)^{n^{r-1}}$

where

$P_{r}(z)=(1-z)\exp\left(z+\frac{z^{2}}{2}+\ldots+\frac{z^{r}}{r}\right).$

Since $P_{r}(z)=P_{r-1}(z)e^{z^{r}/r}$, so that
$e^{(z/n)^{r}}\cdot(e^{(-z/n)^{r}})^{(-1)^{r-1}}=1$, we can clearly write, for
$r\geq 2$,

${\mathcal{S}}_{r}(z):=\exp\left(\frac{z^{r-1}}{r-1}\right)\prod_{n\geq
1}\left(P_{r-1}\left(\frac{z}{n}\right)P_{r-1}\left(-\frac{z}{n}\right)^{(-1)^{r-1}}\right)^{n^{r-1}}.$

Note that in particular

${\mathcal{S}}_{2}(z)=e^{z}\prod_{n\geq
1}\left(\left(\frac{1-\frac{z}{n}}{1+\frac{z}{n}}\right)^{n}e^{2z}\right)$

is equal to the function $F$ studied by Hölder in [10, Eq. (4), p. 515]. Note
that ${\mathcal{S}}_{r}$ is equal to $1/\Lambda_{r}$ for $r>1$ where
$\Lambda_{r}$ is the function defined by Rovinskiĭ in 1991 (compare [14, (1)
p. 62] and [25, (5) p. 74]). Also see [22] and [26].

We are not going to give more details in this section about multiple
trigonometric functions, except for one of the motivations of Kurokawa, who
writes in [14]: as an application we report the calculation of the ”gamma
factors” of Selberg-Gangolli-Wakayama zeta functions of rank one locally
symmetric spaces, and for pointers to a limited number of references, e.g.,
[29] for an early approach of a related question, and the survey of Manin
[20].

##  3 The Borwein-Dykshoorn function

The Borwein-Dykshoorn function

$D(x)=\lim_{n\to\infty}\prod_{k=1}^{2n+1}\left(1+\frac{x}{k}\right)^{k(-1)^{k+1}}$

was introduced in [4] as a generalization of a result of Melzak [23] proving
that

$\lim_{n\to\infty}\prod_{k=1}^{2n+1}\left(1+\frac{2}{k}\right)^{k(-1)^{k+1}}=\frac{\pi
e}{2}\cdot$

This function can be extended to a meromorphic function on ${\mathbb{C}}$ with
poles at the negative even integers. We prove that this function is related to
${\mathcal{S}}_{2}$.

###### Theorem 1

The following equality holds:

$\frac{D(x)}{D(-x)}=e^{x}\
\frac{{\mathcal{S}}_{2}(x/2)^{4}}{{\mathcal{S}}_{2}(x)}\cdot$

Proof. Let $D_{n}(x)$ be defined by

$D_{n}(x):=\prod_{k=1}^{2n+1}\left(1+\frac{x}{k}\right)^{k(-1)^{k+1}}\cdot$

We can write

$D_{n}(x)=\prod_{j=1}^{n}\left(1+\frac{x}{2j}\right)^{-2j}\prod_{j=0}^{n}\left(1+\frac{x}{2j+1}\right)^{2j+1}=\prod_{j=1}^{n}\left(1+\frac{x}{2j}\right)^{-2j}\frac{\displaystyle\prod_{k=1}^{2n+1}\left(1+\dfrac{x}{k}\right)^{k}}{\displaystyle\prod_{k=1}^{n}\left(1+\dfrac{x}{2k}\right)^{2k}}\cdot$

Thus

$D_{n}(x)=\frac{\displaystyle\prod_{k=1}^{2n+1}\left(1+\dfrac{x}{k}\right)^{k}}{\displaystyle\prod_{j=1}^{n}\left(1+\dfrac{x}{2j}\right)^{4j}}$

which implies

$\frac{D_{n}(x)}{D_{n}(-x)}=\prod_{k=1}^{2n+1}\left(\frac{1+\dfrac{x}{k}}{1-\dfrac{x}{k}}\right)^{k}\left(\prod_{k=1}^{n}\left(\frac{1-\dfrac{x/2}{k}}{1+\dfrac{x/2}{k}}\right)^{k}\right)^{4}\cdot$

Multiplying the first product by $e^{-2x(2n+1)}$ and the second product by
$(e^{\frac{x}{2}}e^{nx})^{4}$ does not change the quantity
$D_{n}(x)D_{n}(-x)$, thus

$\frac{D_{n}(x)}{D_{n}(-x)}=\prod_{k=1}^{2n+1}\left(e^{-2x}\left(\frac{1+\dfrac{x}{k}}{1-\dfrac{x}{k}}\right)^{k}\right)\left(e^{\frac{x}{2}}\prod_{k=1}^{n}\left(e^{x}\left(\frac{1-\dfrac{x/2}{k}}{1+\dfrac{x/2}{k}}\right)^{k}\right)\right)^{4}$

which gives, when $n$ tends to infinity,

$\frac{D(x)}{D(-x)}=e^{x}\
\displaystyle\dfrac{{\mathcal{S}}_{2}(x/2)^{4}}{{\mathcal{S}}_{2}(x)}\cdot\ \
\ \ \ \Box$

A corollary of this result gives the value of an infinite product studied in
[12] which was proved again in [2] by using the Borwein–Dykshoorn function.

###### Corollary 1

The following identities hold

$\lim_{n\to\infty}\prod_{k=1}^{2n}\left(1-\frac{2}{2k+1}\right)^{k(-1)^{k}}=\
\exp\left(\frac{2G}{\pi}-\frac{1}{2}\right)$

and

$\lim_{n\to\infty}\prod_{k=1}^{2n+1}\left(1-\frac{2}{2k+1}\right)^{k(-1)^{k}}=\
\exp\left(\frac{2G}{\pi}+\frac{1}{2}\right)$

where $G=\displaystyle\sum_{n\geq 0}\frac{(-1)^{n}}{(2n+1)^{2}}$ is the
Catalan constant.

Proof. Since $\left(1-\dfrac{2}{4n+3}\right)^{-(2n+1)}$ tends to $e$ (take the
logarithm), it suffices to prove the second identity. We write

$\prod_{k=1}^{2n+1}\left(1-\frac{2}{2k+1}\right)^{k(-1)^{k}}=\ \
\prod_{k=1}^{2n+1}\left(\frac{2k+1}{2k-1}\right)^{k(-1)^{k+1}}=\ \
\frac{\displaystyle\prod_{k=1}^{2n+1}\left(1+\dfrac{1}{2k}\right)^{k(-1)^{k+1}}}{\displaystyle\prod_{k=1}^{2n+1}\left(1-\dfrac{1}{2k}\right)^{k(-1)^{k+1}}}\cdot$

Hence, using Theorem 1

$\lim_{n\to\infty}\prod_{k=1}^{2n+1}\left(1-\frac{2}{2k+1}\right)^{k(-1)^{k}}=\
\ \frac{D(1/2)}{D(-1/2)}\ =\
e^{1/2}\frac{{\mathcal{S}}_{2}(1/4)^{4}}{{\mathcal{S}}_{2}(1/2)}\cdot$

But we have ${\mathcal{S}}_{2}(1/2)=2^{1/2}$ and
${\mathcal{S}}_{2}(1/4)=2^{1/8}\exp\left(\frac{G}{2\pi}\right)$ (see [16, p.
852], hence

$\lim_{n\to\infty}\prod_{k=1}^{2n+1}\left(1-\frac{2}{2k+1}\right)^{k(-1)^{k}}=\
\ e^{1/2}\frac{{\mathcal{S}}_{2}(1/4)^{4}}{{\mathcal{S}}_{2}(1/2)}\ =\
\exp\left(\frac{2G}{\pi}+\frac{1}{2}\right)\cdot$

###### Remark 1

  * •

The function $D$ can also be written (see, e.g., [2])

$D(x)=e^{x}\prod_{k\geq
1}\left(e^{-x}\left(1+\dfrac{x}{k}\right)^{k}\right)^{(-1)^{k+1}}$

  * •

The left hand term of the equality in Theorem 1, say $f(x):=D(x)/D(-x)$, has
the property that $f(-x)=1/f(x)$. Hence this is the same for the right hand
term, but it is of course easy to see that the function ${\mathcal{S}}_{2}(x)$
itself satisfies the identity
${\mathcal{S}}_{2}(-x)=({\mathcal{S}}_{2}(x))^{-1}$ (see, e.g., [10, Eq.(5),
p. 516]).

  * •

Note that [4, Theorem, p. 204] gives an explicit (finite) formula in terms of
the function $\Gamma$ and of the generalized Gamma function $\Gamma_{1}$
defined in [3] and itself closely related to the Barnes function. There is
also a close formula for $D(x)$ in [1, Proposition 5, p. 284]. Also see [24,
Example 20, p. 139]. Finally, an expression of $D(x)$ in terms of the function
$\gamma_{\alpha}$ and its derivative is given in [2, p. 86], where
$\gamma_{\alpha}$ (see [8] and [27]; also see [30]) is defined by:

$\gamma_{\alpha}(z):=\sum_{n\geq
1}z^{n-1}\left(\alpha-n\log\left(1+\frac{\alpha}{n}\right)\right).$

##  4 The Adamchik function $E$

Adamchik defined in [1] a function $E$ by

$E(x):=\lim_{n\to\infty}\prod_{k=1}^{2N}\left(1-\dfrac{4x^{2}}{k^{2}}\right)^{k^{2}(-1)^{k+1}}.$

It is easy to see that $E(x)$ has also the following expression:

$E(x)=\prod_{k\geq
1}\left(e^{4x^{2}}\left(1-\dfrac{4x^{2}}{k^{2}}\right)^{k^{2}}\right)^{(-1)^{k+1}}.$

Recall that ${\mathcal{S}}_{3}(x)$ is defined by

${\mathcal{S}}_{3}(x)=e^{\frac{x^{2}}{2}}\prod_{n\geq
1}\left(e^{x^{2}}\left(1-\frac{x^{2}}{n^{2}}\right)^{n^{2}}\right).$

Also recall that the function ${\mathcal{C}}_{3}(x)$ is defined by (see, e.g.,
[18, p. 1])

${\mathcal{C}}_{3}(x)=\prod_{n\geq
1}\left(e^{x^{2}}\left(1-\dfrac{4x^{2}}{(2n-1)^{2}}\right)^{\frac{(2n-1)^{2}}{4}}\right)$

Now we state the following result.

###### Theorem 2

We have the following identity:

$E(x)=e^{2x^{2}}\dfrac{{\mathcal{S}}_{3}(2x)}{{\mathcal{S}}_{3}(x)^{8}}=e^{2x^{2}}\dfrac{{\mathcal{C}}_{3}(x)^{8}}{{\mathcal{S}}_{3}(2x)}\cdot$

Proof. We write

$E(x)=\prod_{k\geq
1}\left(e^{4x^{2}}\left(1-\dfrac{4x^{2}}{k^{2}}\right)^{k^{2}}\right)^{(-1)^{k+1}}=\
\ \frac{\displaystyle\prod_{\stackrel{{\scriptstyle\scriptstyle k\geq 1}}{{k\
\text{odd}}}}\left(e^{4x^{2}}\left(1-\dfrac{4x^{2}}{k^{2}}\right)^{k^{2}}\right)}{\displaystyle\prod_{k\geq
1}\left(e^{4x^{2}}\left(1-\dfrac{x^{2}}{k^{2}}\right)^{4k^{2}}\right)}\ \ =\ \
\frac{\displaystyle\prod_{k\geq
1}\left(e^{4x^{2}}\left(1-\dfrac{4x^{2}}{k^{2}}\right)^{k^{2}}\right)}{\displaystyle\prod_{k\geq
1}\left(e^{4x^{2}}\left(1-\dfrac{x^{2}}{k^{2}}\right)^{4k^{2}}\right)^{2}}\cdot$

Hence

$E(x)=\left({\mathcal{S}}_{3}(2x)e^{-2x^{2}}\right)\left({\mathcal{S}}_{3}(x)e^{-x^{2}/2}\right)^{-8}=e^{2x^{2}}\frac{{\mathcal{S}}_{3}(2x)}{{\mathcal{S}}_{3}(x)^{8}}\cdot$

Now (see, e.g., [18, Theorem 1.1, p. 123]),

${\mathcal{S}}_{3}(2x)={\mathcal{S}}_{3}(x)^{4}{\mathcal{C}}_{3}(x)^{4},$

which gives

$E(x)=e^{2x^{2}}\frac{{\mathcal{C}}_{3}(x)^{8}}{{\mathcal{S}}_{3}(2x)}\ \ \ \
\ \Box$

###### Remark 2

  * $*$

The first equality in Theorem 2 above and Equation (2.15) in [16, p. 850]
imply Corollary 1 in the paper of Adamchik [1, p. 286] where $E(x)$ is
expressed in terms of ${\text{L}i}_{2}(\pm e^{2i\pi x})$ and
${\text{L}i}_{3}(\pm e^{2i\pi x})$, up to replacing $i$ with $-i$, which does
not change the result since $E(x)$ is real (recall that
Li${}_{k}(x):=\sum_{n\geq 1}\frac{\sin nx}{n^{k}}$). It is also interesting to
compare these expressions and [31, Proposition 2 p. 618].

  * $*$

Note that [1, Proposition 6, p. 286] gives a close formula for $E$ in terms of
the Barnes function and of the generalized gamma function $\Gamma_{3}$.

We have as a corollary the following particular cases given in [1, p. 287].

###### Corollary 2

We have

$\ \ \
\lim_{N\to\infty}\prod_{k=1}^{2N}\left(1-\dfrac{1}{4k^{2}}\right)^{k^{2}(-1)^{k+1}}=\
\ \exp\left(\frac{1}{8}-\frac{2G}{\pi}+\frac{7\zeta(3)}{2\pi^{2}}\right)$

and

$\lim_{N\to\infty}\prod_{k=2}^{2N}\left(1-\dfrac{1}{k^{2}}\right)^{k^{2}(-1)^{k+1}}=\
\ \frac{\pi}{4}\exp\left(\frac{1}{2}+\frac{7\zeta(3)}{\pi^{2}}\right)$

Proof.

The left hand side of first equality is equal to $E(1/4)$ thus, using Theorem
2, to $e^{1/8}{\mathcal{C}}_{3}(1/4)^{8}{\mathcal{S}}_{3}(1/2)^{-1}$. But
${\mathcal{S}}_{3}(1/2)$ is given in [17, p. 206], where we find

$\zeta(3)=\frac{8\pi^{2}}{7}\log(2^{1/4}A^{-1}),\ \text{where}\
A:=e^{1/8}\prod_{n\geq
1}\left(e^{1/4}\left(1-\dfrac{1}{4n^{2}}\right)^{n^{2}}\right)$

which can be written

${\mathcal{S}}_{3}(1/2)=2^{1/4}\exp\left(-7\dfrac{\zeta(3)}{8\pi^{2}}\right).$

The value of ${\mathcal{C}}_{3}(1/4)$ is given in [18, p. 1], where we find

${\mathcal{C}}_{3}(1/4)=2^{1/32}\exp\left(\dfrac{21\zeta(3)}{64\pi^{2}}-\dfrac{G}{4\pi}\right).$

Here $G$ is the Catalan constant. Note that there seems to be a misprint in
[18], namely $G/4\pi$ there should be replaced with $-G/4\pi$ as above (also
see [11]). So we finally have

$E(1/4)=\exp\left(\dfrac{1}{8}-\dfrac{2G}{\pi}+\dfrac{7\zeta(3)}{2\pi^{2}}\right).$

##  5 Holcombe’s infinite product

In a 2013 paper [9] Holcombe proved that

$\pi=e^{3/2}\prod_{n\geq
2}\left(e\left(1-\frac{1}{n^{2}}\right)^{n^{2}}\right).$

This result can be deduced from the value of the derivative at $1$ of the
triple trigonometric function ${\mathcal{S}}_{3}$. Namely

${\mathcal{S}}_{3}(x)=e^{\frac{x^{2}}{2}}\prod_{n\geq
1}\left(e^{x^{2}}\left(1-\frac{x^{2}}{n^{2}}\right)^{n^{2}}\right)$

so that ${\mathcal{S}}_{3}(1)=0$ and

$\frac{{\mathcal{S}}_{3}(x)-{\mathcal{S}}_{3}(1)}{x-1}=-e^{\frac{3}{2}}(1+x)\prod_{n\geq
2}\left(e\left(1-\frac{1}{n^{2}}\right)^{n^{2}}\right).$

Letting $x$ tend to $1$, we thus have

${\mathcal{S}}_{3}^{\prime}(1)=-2e^{\frac{3}{2}}\prod_{n\geq
2}\left(e\left(1-\frac{1}{n^{2}}\right)^{n^{2}}\right).$

Using the value ${\mathcal{S}}_{3}^{\prime}(1)=-2\pi$ (see [18, Prop. 4.1, p.
125]) gives Holcombe’s result.

##  6 Miscellanea

Further links between the above mentioned functions and other special
functions can be found in the literature. We give only quick remarks in this
section.

Recall that the dilogarithm function, already alluded to in Remark 2, and the
Clausen function are respectively defined by

$\text{Li}_{2}(z):=\sum_{n\geq 1}\frac{z^{n}}{n^{2}}\ \ (\text{where}\ |z|\leq
1),\ \text{and}\ \ \text{Cl}_{2}(\theta)=\sum_{n\geq 1}\frac{\sin
n\theta}{n^{2}}\cdot$

The Clausen function Cl2 is related to the Barnes function $G$ (see, e.g.,
[28, Equation (2) p. 175]):

$\text{Cl}_{2}(x)=x\log\pi-x\log\left(\sin\left(\frac{x}{2}\right)\right)+2\pi\log\frac{G(1-\frac{x}{2\pi})}{G(1+\frac{x}{2\pi})}\cdot$

But we can find in [16] the relation

${\mathcal{S}}_{2}(z)=(2\sin\pi z)^{z}\exp\left(\frac{1}{2\pi}\sum_{n\geq
1}\frac{\sin(2\pi nz)}{n^{2}}\right)$

hence

${\mathcal{S}}_{2}(z)=(2\sin\pi
z)^{z}\exp\left(\frac{1}{2\pi}\text{Cl}_{2}(2\pi z)\right).$

Thus

$\hskip
85.35826pt\log\frac{G(1+t)}{G(1-t)}=t\log(2\pi)-\log{\mathcal{S}}_{2}(t).\hskip
85.35826pt(*)$

This identity can also be obtained directly from a relation for
${\mathcal{S}}_{2}(x)$ given in [16, Theorem 2.5, p. 847] and a relation given
in [5, (2.1), p. 94], namely

${\mathcal{S}}_{2}(x)=\exp\left(\int_{0}^{x}\pi t\cot(\pi t)\ dt\right)\ \
\text{and}\ \ \int_{0}^{x}\pi t\cot(\pi
t)dt=x\log(2\pi)+\log\frac{G(1-x)}{G(1+x)}\cdot$

Relation $(*)$ can be double checked with the values of
${\mathcal{S}}_{2}(1/2)=\sqrt{2}$ and
${\mathcal{S}}_{2}((1/4)=2^{1/8}e^{G/2\pi}$ given in [16, Examples 2.9, p.
852] and the values of $G(3/2)/G(1/2)=\Gamma(1/2)=\sqrt{\pi}$ and
$G(5/4)/G(3/4)=2^{1/8}\pi^{1/4}e^{-G/2\pi}$ given in [21, p. 271]. This
relation can also be used to give a closed form of values of
${\mathcal{S}}_{2}(t)$ using known values of $G(1+t)/G(1-t)$, e.g., for
$t=1/8$ or $t=1/12$ (see [6, Section 5] and the corrections and results in
[21, p. 272–273]).

###### Remark 3

In [5, p. 94] the authors write that the relation

$\int_{0}^{x}\pi t\cot(\pi t)dt=x\log(2\pi)+\log\frac{G(1-x)}{G(1+x)}\cdot$

is originally due to Kinkelin. Actually Kinkelin proves a similar result —see
the first equality after (26.) p. 135 in [13]— for a function which is
actually equal to $\Gamma(x)^{x-1}/G(x)$: as written in [7, p. 136] this
function has been comparatively neglected by researchers in favor of $G$.

This discussion opens the way to other links with several other special
functions (e.g., the dilogarithm already mentioned, the inverse tangent
integral Ti${}_{2}(z):=\sum_{j\geq 1}(-1)^{j+1}y^{j}/j^{2}$ and the Legendre
chi-function $\chi_{2}(z):=\sum_{k\geq 0}y^{2k+1}/(2k+1)^{2}$, both defined
for $|z|\leq 1$) but we will not go further in this direction: the reader can
consult in particular [6, 21] and the references therein, with a special
mention for the book of Lewin [19].

##  7 Conclusion

We end this paper with two questions.

First, we note that combining the expression of $D(x)/D(-x)$ given above in
terms of ${\mathcal{S}}_{2}(x/2)$ and ${\mathcal{S}}_{2}(x)$ and the
expression of $D(x)$ in terms of the Hessami Pilehrood-Sondow-Hadjicostas
function given in [2], one can obtain a relation between this function and
${\mathcal{S}}_{2}(x)$. Is it possible to obtain a relation with the higher
multiple trigonometric functions ${\mathcal{S}}_{r}(x)$?

Second, an interesting identity is given in [24, p. 125] where it is proved by
unusual methods:

$\lim_{n\to\infty}\left[e^{\frac{n}{4}(4n+1)}n^{-\frac{1}{8}-n(n+1)}(2\pi)^{-\frac{n}{2}}\prod_{k=1}^{2n}\Gamma\left(1+\dfrac{k}{2}\right)^{k(-1)^{k}}\right]=2^{\frac{1}{12}}\exp\left(\dfrac{5}{24}-\dfrac{3}{2}\zeta^{\prime}(-1)-\dfrac{7\zeta(3)}{16\pi^{2}}\right)$

where the right hand constant can also be written

$(2e)^{1/12}A^{3/2}\exp\left(-\frac{7\zeta(3)}{16\pi^{2}}\right)$

with $A:=\exp(\frac{1}{12}-\zeta^{\prime}(-1))$ is the Glaisher-Kinkelin
constant. We did not succeed in finding a relation between this identity with
multiple trigonometric functions. Of course the quantity $7\zeta(3)/4\pi^{2}$
occurs, but, after all, this is nothing but $-7\zeta^{\prime}(-2)$ which looks
somehow more mundane. Is it possible to find a proof of this identity that
uses multiple trigonometric functions?

Acknowledgments We would like to thank S. Hu and M.-S. Kim for discussions
about their preprint [11]. We would also like to thank Olivier Ramaré and
Sanoli Gun, in particular for their help in obtaining some of the references.

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Virginia, Charlottesville, USA
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A. Ledovskoy, C. Neu, B. Tannenwald, Y. Wang, E. Wolfe, F. Xia Wayne State
University, Detroit, USA
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- Madison, Madison, WI, USA
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Savin, V. Sharma, W.H. Smith, D. Teague, S. Trembath-reichert †: Deceased
1: Also at Vienna University of Technology, Vienna, Austria
2: Also at IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France
3: Also at Universidade Estadual de Campinas, Campinas, Brazil
4: Also at Federal University of Rio Grande do Sul, Porto Alegre, Brazil
5: Also at UFMS, Nova Andradina, Brazil
6: Also at Universidade Federal de Pelotas, Pelotas, Brazil
7: Also at Université Libre de Bruxelles, Bruxelles, Belgium
8: Also at University of Chinese Academy of Sciences, Beijing, China
9: Also at Institute for Theoretical and Experimental Physics named by A.I.
Alikhanov of NRC ‘Kurchatov Institute’, Moscow, Russia
10: Also at Joint Institute for Nuclear Research, Dubna, Russia
11: Also at Cairo University, Cairo, Egypt
12: Now at British University in Egypt, Cairo, Egypt
13: Also at Purdue University, West Lafayette, USA
14: Also at Université de Haute Alsace, Mulhouse, France
15: Also at Tbilisi State University, Tbilisi, Georgia
16: Also at Erzincan Binali Yildirim University, Erzincan, Turkey
17: Also at CERN, European Organization for Nuclear Research, Geneva,
Switzerland
18: Also at RWTH Aachen University, III. Physikalisches Institut A, Aachen,
Germany
19: Also at University of Hamburg, Hamburg, Germany
20: Also at Brandenburg University of Technology, Cottbus, Germany
21: Also at Institute of Physics, University of Debrecen, Debrecen, Hungary,
Debrecen, Hungary
22: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary
23: Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös
Loránd University, Budapest, Hungary, Budapest, Hungary
24: Also at IIT Bhubaneswar, Bhubaneswar, India, Bhubaneswar, India
25: Also at Institute of Physics, Bhubaneswar, India
26: Also at Shoolini University, Solan, India
27: Also at University of Hyderabad, Hyderabad, India
28: Also at University of Visva-Bharati, Santiniketan, India
29: Also at Isfahan University of Technology, Isfahan, Iran
30: Now at INFN Sezione di Bari a, Università di Bari b, Politecnico di Bari
c, Bari, Italy
31: Also at Italian National Agency for New Technologies, Energy and
Sustainable Economic Development, Bologna, Italy
32: Also at Centro Siciliano di Fisica Nucleare e di Struttura Della Materia,
Catania, Italy
33: Also at Scuola Normale e Sezione dell’INFN, Pisa, Italy
34: Also at Riga Technical University, Riga, Latvia, Riga, Latvia
35: Also at Malaysian Nuclear Agency, MOSTI, Kajang, Malaysia
36: Also at Consejo Nacional de Ciencia y Tecnología, Mexico City, Mexico
37: Also at Warsaw University of Technology, Institute of Electronic Systems,
Warsaw, Poland
38: Also at Institute for Nuclear Research, Moscow, Russia
39: Now at National Research Nuclear University ’Moscow Engineering Physics
Institute’ (MEPhI), Moscow, Russia
40: Also at St. Petersburg State Polytechnical University, St. Petersburg,
Russia
41: Also at University of Florida, Gainesville, USA
42: Also at Imperial College, London, United Kingdom
43: Also at P.N. Lebedev Physical Institute, Moscow, Russia
44: Also at California Institute of Technology, Pasadena, USA
45: Also at Budker Institute of Nuclear Physics, Novosibirsk, Russia
46: Also at Faculty of Physics, University of Belgrade, Belgrade, Serbia
47: Also at Università degli Studi di Siena, Siena, Italy
48: Also at INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy,
Pavia, Italy
49: Also at National and Kapodistrian University of Athens, Athens, Greece
50: Also at Universität Zürich, Zurich, Switzerland
51: Also at Stefan Meyer Institute for Subatomic Physics, Vienna, Austria,
Vienna, Austria
52: Also at Burdur Mehmet Akif Ersoy University, BURDUR, Turkey
53: Also at Şırnak University, Sirnak, Turkey
54: Also at Department of Physics, Tsinghua University, Beijing, China,
Beijing, China
55: Also at Beykent University, Istanbul, Turkey, Istanbul, Turkey
56: Also at Istanbul Aydin University, Application and Research Center for
Advanced Studies (App. & Res. Cent. for Advanced Studies), Istanbul, Turkey
57: Also at Mersin University, Mersin, Turkey
58: Also at Piri Reis University, Istanbul, Turkey
59: Also at Gaziosmanpasa University, Tokat, Turkey
60: Also at Ozyegin University, Istanbul, Turkey
61: Also at Izmir Institute of Technology, Izmir, Turkey
62: Also at Marmara University, Istanbul, Turkey
63: Also at Kafkas University, Kars, Turkey
64: Also at Istanbul Bilgi University, Istanbul, Turkey
65: Also at Hacettepe University, Ankara, Turkey
66: Also at Adiyaman University, Adiyaman, Turkey
67: Also at Vrije Universiteit Brussel, Brussel, Belgium
68: Also at School of Physics and Astronomy, University of Southampton,
Southampton, United Kingdom
69: Also at IPPP Durham University, Durham, United Kingdom
70: Also at Monash University, Faculty of Science, Clayton, Australia
71: Also at Bethel University, St. Paul, Minneapolis, USA, St. Paul, USA
72: Also at Karamanoğlu Mehmetbey University, Karaman, Turkey
73: Also at Bingol University, Bingol, Turkey
74: Also at Georgian Technical University, Tbilisi, Georgia
75: Also at Sinop University, Sinop, Turkey
76: Also at Mimar Sinan University, Istanbul, Istanbul, Turkey
77: Also at Texas A&M University at Qatar, Doha, Qatar
78: Also at Kyungpook National University, Daegu, Korea, Daegu, Korea


for ${\rm PPP}[m,\nu]$–a.a. $\omega$. Let us fix also $\gamma\in(0,1)$. If $X$
is a random variable with law $\nu$, then $\gamma X$ has law $\nu_{\gamma}$
because of the special form of $\nu_{1,\alpha}^{+}$ (see Definition 9.1).
Hence the marked SPP given by $\tilde{\omega}:=\\{(\gamma x,\gamma
E_{x})\,:\,x\in\widehat{\omega}\\}$ with $\omega$ sampled according to ${\rm
PPP}[m,\nu]$ has law ${\rm PPP}[\gamma^{-d}m,\nu_{\gamma}]$. On the other
hand, when $\mathcal{G}[1,1](\omega)$ percolates, also
$\mathcal{G}[\gamma,1](\tilde{\omega})$ percolates since the map
$(x,E_{x})\mapsto(\gamma x,\gamma E_{x})$ induces a vertex bijection between
the two graphs and since, given $x\not=y$ in $\widehat{\omega}$, $\\{x,y\\}$
is an edge of of $\mathcal{G}[1,1](\omega)$ if and only if it holds $|\gamma
x-\gamma y|+|\gamma E_{x}|+|\gamma E_{y}|+|\gamma E_{x}-\gamma
E_{y}|\leq\gamma$. Hence, by the above observations,
$\mathcal{G}[\gamma,1](\omega)$ percolates a.s. when $\omega$ is sampled
according to ${\rm PPP}[\gamma^{-d}m,\nu_{\gamma}]$. In conclusion, given
$m^{\prime}>0$, $\mathcal{G}[\gamma,1](\omega)$ with $\omega$ sampled by ${\rm
PPP}[m^{\prime},\nu_{\gamma}]$ percolates a.s. if
$\gamma^{d}m^{\prime}>\lambda_{c}^{+}(\alpha)$.

Let us fix $\lambda_{*}\in(\lambda_{c}^{+}(\alpha),\lambda)$. By Lemma 9.3, if
$\omega$ is sampled by ${\rm PPP}[\lambda_{*},\nu]$, then
$\omega_{\gamma}:=\\{(x,E_{x})\,:\,x\in\widehat{\omega},\,|E_{x}|\leq\gamma\\}$
has law ${\rm PPP}[\nu(\gamma)\lambda_{*},\nu_{\gamma}]$. In particular, since
$\mathcal{G}[\gamma,1](\omega)$ contains
$\mathcal{G}[\gamma,1](\omega_{\gamma})$ and by the previous conclusion, to
prove that $\mathcal{G}[\gamma,1](\omega)$ a.s. percolates with $\omega$
sampled by ${\rm PPP}[\lambda_{*},\nu]$ it is enough to have
$\gamma^{d}\nu(\gamma)\lambda_{*}>\lambda_{c}^{+}(\alpha)$. As
$\lambda_{*}>\lambda_{c}^{+}(\alpha)$ and $\nu(\gamma)=\gamma^{\alpha+1}$, it
is enough to take $\gamma$ enough close to $1$. This proves the validity of
(A2).

##  Appendix C Failure of the FKG inequality in the MA random resistor network

The results on LR crossings of [17] are derived under the assumption that the
FKG inequality holds for the random graphs treated there. In the specific case
of the MA random resistor network, the most general form of the FKG inequality
would be the following one. First we call an event $A\in\mathcal{B}(\Omega)$
_increasing_ if, given $\omega\in\Omega$, it holds $\omega\in A$ whenever
$\mathcal{G}[\zeta,\beta](\omega^{\prime})\subset\mathcal{G}[\zeta,\beta](\omega)$
for some $\omega^{\prime}\in A$. The above inclusion means that the vertex set
and edge set of $\mathcal{G}[\zeta,\beta](\omega^{\prime})$ are included
rispectively in the vertex set and edge set of
$\mathcal{G}[\zeta,\beta](\omega)$.

We claim that the above general FKG does not always hold when the energy marks
in the MA resistor network appear with different sign. To explain the reason
let us consider first the following toy model. Consider the points $0=(0,0)$,
$e_{1}=(1,0)$, $e_{2}=(0,1)$ in ${\mathbb{Z}}^{2}$. We take independent random
variables $E_{0},E_{e_{1}},E_{e_{2}}$ uniformly distributed on $[-1,1]$. We
consider the random graph $G$ with vertex set $V=\\{0,e_{1},e_{2}\\}$ and
edges given by the pairs $\\{x,y\\}$ with $x\not=y$ in $V$ such that
$|x-y|+|E_{x}|+|E_{y}|+|E_{x}-E_{y}|\leq 4$. Let us consider the decreasing
(with respect to the partial order given by inclusion of graphs) events $A,B$
defined as $A:=\big{\\{}\text{neither }\\{0,e_{1}\\}\text{ nor
}\\{e_{1},e_{2}\\}\text{ is an edge}\big{\\}}$ and
$B:=\big{\\{}\\{0,e_{2}\\}\text{ is not an edge}\big{\\}}$. We claim that both
$A$ and $B$ have positive probability to take place but $A\cap B$ is
impossible, hence $A$ and $B$ cannot be positively correlated (thus violating
the FKG inequality). Let us prove our claim. We observe that
$w(a,b):=|a|+|b|+|a-b|$ equals $2\max\\{|a|,|b|\\}$ if $ab\geq 0$ and it
equals $2|a-b|$ if $ab\leq 0$. Hence if $w(a,b)>2$ with $a,b\in[-1,1]$, then
$a,b$ must have different signs. The event $A$ corresponds to the requirement
that $w(E_{0},E_{e_{1}})>3$ and $w(E_{e_{1}},E_{e_{2}})>4-\sqrt{2}>2$. In
particular, the event $A$ has positive probability and implies that
$E_{0},E_{e_{1}}$ have different signs and that $E_{e_{1}},E_{e_{2}}$ have
different signs. Similarly, the event $B$ has positive probability and implies
that $E_{0}$, $E_{e_{2}}$ have different signs. The constraint on the signs
makes the event $A\cap B$ impossible.

Considering the MA random resistor network
$\mathcal{G}[\zeta,\beta]=\mathcal{G}[4,1]$ and by playing with the above toy
example, one can easily show that the FKG inequality fails when the energy
marks uniformly distributed on $[-1,1]$.

Acknowledgments: I thank Hlafo Alfie Mimun for useful simulations concerning
the violation of the FKG inequality in the MA resistor network.

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