Theory and Modern Applications

# On the asymptotic expansion of the q-dilogarithm

## Abstract

In this work, we study some asymptotic expansion of the q-dilogarithm at $$q=1$$ and $$q=0$$ by using the Mellin transform and an adequate decomposition allowed by the Lerch functional equation.

## 1 Introduction

Euler’s dilogarithm is defined by 

$$\mathit{Li}_{2}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{2}}, \quad |z|< 1.$$
(1.1)

In , Kirillov defines the following q-analog of the dilogarithm $$\mathit{Li}_{2}(z)$$:

$$\mathit{Li}_{2}(z;q)=\sum_{n=1}^{\infty}\frac{z^{n}}{n(1-q^{n})}, \quad |z|< 1, 0 < q < 1,$$
(1.2)

and he observes the following remarkable formula (, Section 2.5, Lemma 8):

$$\sum_{n=0}^{\infty}\frac{z^{n}}{(q,q)_{n}}=\exp \bigl(\mathit{Li}_{2}(z,q)\bigr), \quad |z|< 1, |q|< 1,$$
(1.3)

where

$$(q,q)_{0}=1, \qquad (q,q)_{n}=\prod _{k=0}^{n-1}\bigl(1-q^{k}\bigr), \quad n=1, 2,\ldots .$$
(1.4)

It seems a precise formulation of (1.3) going back to Ramanujan (see , Chapter 27, Entry 6) is given an asymptotic series for $$\mathit{Li}_{2}(z; q)$$ and Hardy and Littlewood  proved that for $$|q|=1$$, the identity holds inside the radius of convergence of either series.

Let $$\omega=e^{zx+2\mathrm{i}\theta}$$ with $$\operatorname{Re} (z)>1$$, $$x>0$$, and $$0<\theta<1$$. The main result of this work is the following complete asymptotic expansion of the q-dilogarithm function $$\mathit{Li}_{2}(\omega;e^{-x})$$ at $${x\rightarrow0}$$:

\begin{aligned} \mathit{Li}_{2}\bigl(\omega,e^{-x}\bigr) \sim& \mathit{Ci}_{2}( \theta) \frac{1}{x}+\biggl(\frac{1}{2}-z\biggr)\mathit{Ci}_{1}( \theta)+\sum_{n=1} ^{\infty}\frac{(-1)^{n+1}}{(n+1)(n+1)!} \\ &{}\times B_{n+1} (z)B_{n+1}\bigl(1,e^{2\mathrm{i}\pi\theta} \bigr)x^{n} \quad \text{as } x\rightarrow0 \end{aligned}
(1.5)

and

$$\mathit{Li}_{2}\bigl(\omega,e^{-x}\bigr) \sim\frac{4\gamma}{\pi}B_{2}( \theta)\frac{\mathrm{i}}{x}+4\sum_{n=1}^{\infty} \mathrm{i}^{n}\frac{\psi^{(n-1)}(z)B_{n+1} (\theta)}{(n+1)!}\biggl(\frac{2\pi}{x} \biggr)^{n},\quad x\rightarrow\infty.$$
(1.6)

In Section 2.5, Corollary 10 of , Kirillov and Ueno and Nishizawa derived the asymptotic expansion (1.5) by using the Euler-Maclaurin summation formula; see also , an integral representation of Barnes type for the q-dilogarithm. Second, we use the Lerch functional equation to decompose the integrand and to apply the Cauchy theorem.

## 2 q-Dilogarithm

The polylogarithm is defined in the unit disk by the absolutely convergent series 

$$\mathit{Li}_{s}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}},\quad |z|< 1.$$
(2.1)

Several functional identities satisfied by the polylogarithm are available in the literature (see ). For $$n=2, \ldots$$ , the function $$\mathit{Li}_{n}(z)$$ can also be represented as

$$\mathit{Li}_{n}(z)= \int_{0}^{z}\frac{\mathit{Li}_{n-1}(t)}{t}\, dt, \quad n\in \mathbb {N}, \qquad \mathit{Li}_{1}(z)=-\log(1-z)= \int_{0}^{z}\frac{dt}{1-t},$$
(2.2)

which is valid for all z in the cut plane $$\mathbb{C}\setminus [1,\infty)$$.

The notation $$F(\theta,s)$$ is used for the polylogarithm $$\mathit{Li}_{s}(e^{2\mathrm{i}n\pi \theta})$$ with θ real, called the periodic zeta function (see , Section 25.13) and is given by the Dirichlet series

$$F(\theta,s )=\sum_{n=1}^{\infty}\frac{e^{2\mathrm{i}n\pi \theta}}{n^{s}}, \quad \theta\in\mathbb{R},$$
(2.3)

it converges for $$\operatorname{Re} s>1$$ if $$\theta\in\mathbb{Z}$$, and for $$\operatorname{Re} s>0$$ if $$\theta\in\mathbb{R}/\mathbb{Z}$$. This function may be expressed in terms of the Clausen functions $$\mathit{Ci}_{s}(\theta)$$ and $$\mathit{Si}_{s}(\theta)$$, and vice versa (see , Section 27.8):

$$\mathit{Li}_{s}\bigl(e^{\pm i \theta}\bigr) = \mathit{Ci}_{s}( \theta) \pm \mathrm{i} \mathit{Si}_{s}(\theta) .$$
(2.4)

In , Koornwinder defines the q-analog of the logarithm function

$$-\log(1-z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n},\quad |z|< 1,$$

as follows:

$$\log_{q}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{1-q^{n}},\quad |z|< 1, 0< q< 1.$$
(2.5)

Recall that the q-analog of the ordinary integral (called Jackson’s integral) is defined by

$$\int_{0}^{z}f(t) \, d_{q}t=(1-q)z\sum _{n=0}^{\infty}f\bigl(zq^{n} \bigr)q^{n}.$$
(2.6)

One can recover the ordinary Riemann integral as the limit of the Jackson integral for $$q \uparrow1$$.

### Lemma 2.1

The function $$\log_{q}(z)$$ has the following q-integral representation:

$$(1-q)\log_{q}(z)= \int_{0}^{z}\frac{1}{1-t}\, d_{q}t, \quad |z|< 1.$$
(2.7)

Moreover, it can be extended to any analytic function on $$\mathbb{C}-\{ q^{-n}, n\in\mathbb{N}_{0}\}$$.

### Proof

Assume that $$|z|<1$$, then from (2.5) we have

\begin{aligned} (1-q)\log_{q}(z) &=(1-q)\sum_{n=1}^{\infty}\sum_{m=0}^{\infty}z^{n} q^{nm} \\ &=(1-q)z\sum_{m=0}^{\infty}q^{m}\sum _{n=0}^{\infty}z^{n} q^{nm} \\ &=(1-q)z\sum_{m=0}^{\infty}\frac{q^{m}}{1-zq^{m}}. \end{aligned}

The inversion of the order of summation is permitted, since the double series converges absolutely when $$|z|<1$$.

Let K be a compact subset of $$\mathbb{C}-\{q^{-n}, n\in\mathbb {N}_{0}\}$$. There exists $$N\in\mathbb{N}$$ such that, for all $$z\in K$$, $$|q^{N}z|< q$$. Then for $$n\geq N$$ we have

$$\biggl\vert \frac{q^{m}}{1-zq^{m}}\biggr\vert \leq \frac{q^{m}}{1-q}.$$
(2.8)

Hence, the series $$\sum_{m=N}^{\infty}\frac{q^{m}}{1-zq^{m}}$$ converges uniformly in K. □

The q-dilogarithm (1.2) is related to Koornwinder’s q-logarithm (2.5) by

$$\mathit{Li}_{2}(z,q)= \int_{0}^{z}\frac{\log_{q}(t)}{t}\, dt.$$
(2.9)

It follows that, for $$n \geq2$$, we can also define

$$\mathit{Li}_{n}(z,q)= \int_{0}^{z}\frac{\mathit{Li}_{n-1}(t,q)}{t} \, dt.$$
(2.10)

This integral formula proves by induction that $$\mathit{Li}_{n}(z,q)$$ has an analytic continuation on $$\mathbb {C}-[1, \infty)$$. Moreover, for $$|z|<1$$, we have

$$\mathit{Li}_{n}(z,q)=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{n}(1-q^{k})}.$$

This converges absolutely for $$|z| < 1$$ and defines a germ of a holomorphic function in the neighborhood of the origin. Note that

\begin{aligned}& \lim_{q\uparrow1}(1-q)\mathit{Li}_{2}\bigl((1-q)z,q \bigr)=\mathit{Li}_{2}(z), \\& \lim_{q\downarrow0}(1-q)\mathit{Li}_{2}(z,q)=-\operatorname{Log}(1-z),\quad |z|< 1. \end{aligned}

Let $$\omega=e^{-zx+2\mathrm{i}\pi\theta}$$, $$\theta\in\mathbb{R}$$, and $$\operatorname{Re} z>0$$, we define

\begin{aligned}& \mathit{Ci}_{2}(\omega,q)=\sum_{n=1}^{\infty}\frac{e^{-zx}\cos(2\pi n\theta)}{n(1-q^{n})}, \end{aligned}
(2.11)
\begin{aligned}& \mathit{Si}_{2}(\omega,q)= \sum_{n=1}^{\infty}\frac{e^{-zx}\sin(2\pi n \theta)}{n(1-q^{n})}. \end{aligned}
(2.12)

Note that these functions can be considered as q-analogs of the Clausen functions (2.4) and are related to the q-dilogarithm by

$$\mathit{Li}_{2}(\omega,q)=\mathit{Ci}_{2}(\omega,q)+\mathrm{i} \mathit{Si}_{2}(\omega,q).$$
(2.13)

Now, we will use the Mellin transform method to obtain the integral representation

$$\mathit{Li}_{2}\bigl(\omega,e^{-x}\bigr)=\frac{1}{2\mathrm{i}\pi} \int_{c-\mathrm{i}\infty }^{c+\mathrm{i}\infty} \zeta(s,z)F(\theta,s) \Gamma(s)x^{-s} \, ds, \quad c>1,$$
(2.14)

where

$$\omega=e^{-zx+2\mathrm{i}\pi\theta},\qquad x>0, \qquad \operatorname{Re} z>1,\qquad 0< \theta< 1.$$

Recall that the Mellin transform for a locally integrable function $$f(x)$$ on $$(0,\infty)$$ is defined by

$$M(f,s)= \int_{0}^{\infty}f(x)x^{s-1}\, dx,$$
(2.15)

which converges absolutely and defines an analytic function in the strip

$$a< \operatorname{Re} s< b,$$

where a and b are real constants (with $$a< b$$) such that, for $$\varepsilon>0$$,

$$f(x)= \textstyle\begin{cases} \mathcal{O}(x^{-a-\varepsilon}) & \text{as } x\rightarrow 0^{+}, \\ \mathcal{O}(x^{-b-\varepsilon}) & \text{as } x\rightarrow +\infty. \end{cases}$$
(2.16)

$$f(x)=\frac{1}{2\mathrm{i}\pi} \int_{c-\mathrm{i}\infty}^{c+\mathrm{i}\infty} M(f,s)x^{-s} ds,$$
(2.17)

where c satisfies $$a< c< b$$. Equation (2.17) is valid at all points $$x\geq0$$ where $$f(x)$$ is continuous.

We first compute the Mellin transform $$M(\psi_{n}(x),s)$$, where

$$\psi_{n}(x)=\frac{e^{-nzx}}{n(1-e^{-nx})}, \quad x>0, \operatorname{Re} z>1, n\in\mathbb{N}.$$
(2.18)

Since

\begin{aligned}& \psi_{n}(x)\sim\frac{1}{nx}, \quad x\rightarrow0^{+}, \end{aligned}
(2.19)
\begin{aligned}& \psi_{n}(x)\sim\frac{1}{n}e^{-nx(z-1)}, \quad x\rightarrow+ \infty. \end{aligned}
(2.20)

We concluded that $$M(\psi_{n}(x),s)$$ is defined in the half-plane $$\operatorname{Re} s>0$$. That is, the constants a and b satisfy $$a=1$$ and $$b=+\infty$$, which values can be used for all $$n\geq1$$ and $$\operatorname{Re} z>1$$. The Mellin transform of $$\psi_{n}(x)$$ can be obtained from the following integral representation of the Hurwitz zeta function $$\zeta (s, z)$$:

$$\zeta(s,z)=\frac{1}{\Gamma(s)} \int_{0}^{\infty}\frac {e^{-zx}}{1-e^{-x}}x^{s-1} \, dx \quad \bigl(\operatorname{Re} s>0, \bigl\vert \arg(1-z)\bigr\vert < \pi; \operatorname{Re} s>1, z=1\bigr).$$
(2.21)

Note that $$\zeta(s, z)$$ is expressed also by the series

$$\zeta(s,z)=\sum_{k=1}^{\infty}\frac{1}{(z+k)^{s}}, \quad \operatorname{Re} s>1, z\neq-1, -2, \ldots .$$
(2.22)

For the other values of z, $$\zeta(s,z)$$ is defined by analytic continuation. It has a meromorphic continuation in the s-plane, its only singularity in $$\mathbb{C}$$ being a simple pole at $$s=1$$,

$$\zeta(s,z)=\frac{1}{s-1}-\psi(z)+ \mathcal{O}(s-1).$$
(2.23)

Applying the Mellin inversion theorem to the integral (2.21), we then find

$$\psi_{n}(x)=\frac{1}{2\mathrm{i}\pi} \int_{c-\mathrm{i}\infty}^{c+\mathrm{i}\infty} \zeta(s,z)\Gamma(s) (nx)^{-s}\, ds.$$
(2.24)

We use the Stirling formula, which shows that, for finite σ,

$$\Gamma(\sigma+\mathrm{i} t)=\mathcal{O}\bigl(|t|^{\sigma-1}e^{-\frac {1}{2}\pi|t|} \bigr)\quad \bigl(\vert t\vert \rightarrow+\infty\bigr)$$
(2.25)

and the well-known behavior of $$\zeta(s,z)$$ (see )

$$\zeta(s,z)=\mathcal{O}\bigl(|t|^{\tau(\sigma)}\log|t|\bigr),$$
(2.26)

where

$$\tau(\sigma)= \textstyle\begin{cases} \frac{1}{2}-\sigma, &\sigma\leq0, \\ \frac{1}{2},& 0\leq\sigma\leq\frac{1}{2}, \\ 1-\sigma,& \frac{1}{2}\leq\sigma\leq1, \\ 0,& \sigma\geq1. \end{cases}$$

Then we obtain the following majorization of the modulus of the integrand in (2.24):

$$\mathcal{O}\bigl(|t|^{\tau(\sigma)+\sigma-1}\log|t|\bigr).$$
(2.27)

Consequently, the integral (2.24) converges absolutely in the whole vertical strip of the half-plane $$\operatorname{Re} s>0$$. Then we replace x by nx, where n is a positive integer, and sum over n, and we then obtain

$$\mathit{Li}_{2}\bigl(\omega,e^{-x}\bigr) =\frac{1}{2\mathrm{i}\pi} \int_{c-\mathrm{i}\infty}^{c+\mathrm{i}\infty} \zeta(s,z)F(\theta,s+1) \Gamma(s)x^{-s}\, ds, \quad c>1,$$
(2.28)

where

$$\omega=e^{-zx+2\mathrm{i}\pi\theta},\qquad x>0,\qquad \operatorname{Re} z>1, \qquad 0< \theta< 1.$$

## 3 Asymptotic at $$q=1$$

The integral (2.28) will be used to derive asymptotic expansions of the q-dilogarithm. The contour of integration is moved at first to the left to obtain an asymptotic expansion at $$q=1$$ and then to the right to get an asymptotic expansion at $$q=0$$.

Let us consider the function

$$g(s)=\zeta(s,z)F(\theta,s+1)\Gamma(s).$$
(3.1)

The periodic function zeta function $$F(\theta,s)$$ has an extension to an entire function in the s-plane (see ). Hence, the function $$g(s)$$ has a meromorphic continuation in the s-plane, its only singularity in $$\mathbb{C}$$ coincides with the pole of $$\Gamma (s)$$ and $$\zeta(s,z)$$ being a simple pole at $$s=1, 0,-1,-2,\ldots$$ .

Now we compute the residues of the poles. The special values at $$s=-1, -2 \ldots$$ of the periodic zeta function are reduced to the Apostol-Bernoulli polynomials (see ),

$$F(\theta,-n )=-\frac{B_{n+1}(1,e^{2\mathrm{i}\pi\theta})}{n+1}.$$
(3.2)

We need also the following asymptotic expansions of $$\Gamma(s)$$ and $$\zeta(s)$$ at $$s=0$$:

\begin{aligned}& \Gamma(s)=\frac{1}{s}-\gamma+ \mathcal{O}\bigl(s^{2} \bigr), \end{aligned}
(3.3)
\begin{aligned}& \zeta(s)=\frac{1}{2}-z+s\log\frac{\Gamma(z)}{2\pi} + \mathcal{O} \bigl(s^{2}\bigr). \end{aligned}
(3.4)

Hence,

\begin{aligned}& \lim_{s\rightarrow1}(s-1) g(s)=\mathit{Li}_{2}\bigl(e^{2\mathrm{i}\pi\theta} \bigr), \\& \lim_{s\rightarrow-n}(s+n) g(s)=\frac{(-1)^{n}}{(n+1)(n+1)!}B_{n+1} (z)B_{n+1}\bigl(1,e^{2\mathrm{i}\pi\theta}\bigr). \end{aligned}

Here $$B_{n}(z)$$ is the Bernoulli polynomial (see ).

Let N be an integer and d real number such that $$-N-1< d<-N$$. We consider the integral taken round the rectangular contour with vertices at $$d \pm \mathrm{i}A$$ and $$c \pm \mathrm{i}A$$, so that the side in $$\operatorname{Re}(s) < 0$$ parallel to the imaginary axis passes midway between the poles $$s=1, 0 -1, -2, \ldots, -N$$. The contribution from the upper and lower sides $$s = \sigma\pm\mathrm{i}A$$ vanishes as $$|A|\rightarrow+\infty$$, since the modulus of the integrand is controlled by

$$\mathcal{O}\bigl(|A|^{\tau(\sigma)+\sigma-1/2}\log |A| e^{-\frac{1}{2}\pi|A|}\bigr).$$
(3.5)

This follows from Stirling’s formula (2.25), the behavior $$\zeta (s,z)$$ being given by (2.26), and the following estimation:

$$\bigl\vert F(\theta,s+1)\bigr\vert \leq\zeta(\sigma+1)=\mathcal{O}(1),\quad \vert A\vert \rightarrow+\infty.$$

Displacement of the contour (2.28) to the left then yields

\begin{aligned} \mathit{Li}_{2}\bigl(\omega,e^{-x}\bigr) =& \mathit{Ci}_{2}(\theta) \frac{1}{x}+\biggl(\frac{1}{2}-z\biggr)\mathit{Ci}_{1}(\theta) \\ &{}+\sum _{n=1} ^{N}\frac{(-1)^{n+1}}{(n+1)(n+1)!}B_{n+1} (z)B_{n+1}\bigl(1,e^{2\mathrm{i}\pi\theta}\bigr)x^{n} + R_{N}(x), \end{aligned}
(3.6)

where the remainder integral $$R_{N}(z)$$ is given by

$$R_{N}(x)=\frac{1}{2\mathrm{i}\pi} \int_{d-\mathrm{i}\infty}^{d+\mathrm{i}\infty} \zeta(s,z)F(\theta,s+1) \Gamma(s)x^{-s} \, ds, \quad x>0, \operatorname{Re} z>1.$$
(3.7)

From (3.5), we find

$$\bigl\vert R_{N}(x)\bigr\vert =\mathcal{O}\biggl( \frac{1}{x^{N+1}}\biggr).$$

## 4 Asymptotic at $$q=0$$

Recall that the periodic zeta function satisfies the functional equation (see )

\begin{aligned}& F(\theta,s)=\frac{\Gamma(1-s)}{(2\pi )^{1-s}} \bigl\{ e^{\frac{\pi\mathrm{i}(1-s)}{2}}\zeta(1-s, \theta)+ e^{\frac{\pi\mathrm{i}(s-1)}{2}}\zeta(1-s,1-\theta) \bigr\} \\& \quad (\operatorname{Re} s>1, 0< \theta< 1), \end{aligned}
(4.1)

first given by Lerch, whose proof follows the lines of the first Riemann proof of the functional equation for $$\zeta(x)$$.

It is well known that the asymptotic expansion near infinity via the Mellin transform is obtained by displacement of the contour of integration in the Mellin inversion formulas (2.16) to the right-hand side (see ). However, the integrand (2.28) has no poles in the half-plane $$\operatorname{Re} s>1$$. The periodic zeta function $$F(\theta,s)$$ has an analytic continuation to the whole s-space for $$0<\theta<1$$. Moreover, the poles of $$\Gamma(1-s)$$ in equation (4.1) at $$s=-1, -2 \ldots$$ are canceled by the zeros of the function

$$e^{\frac{\pi\mathrm{i}(1-s)}{2}}\zeta(1-s,\theta)+ e^{\frac{\pi\mathrm{i}(s-1)}{2}}\zeta(1-s,1-\theta).$$

On the other hand from (4.1) we easily obtain

$$\Gamma (s)\bigl\{ F(\theta,s+1)+F(1-\theta,s+1)\bigr\} =-\frac{(2\pi)^{s+1}}{2s \sin\frac{\pi s}{2}}\bigl\{ \zeta(-s,\theta)+ \zeta(-s,1-\theta)\bigr\} ,$$
(4.2)

where we are able to simplify (4.2) by the well-known reflection formulas

$$\frac{\pi}{\sin\pi s}=\Gamma(s)\Gamma(1-s),\qquad \frac{\sin\pi s}{\pi}= \frac{2}{\pi}\sin\frac{\pi s}{2} \sin \frac{\pi(1- s)}{2}.$$

Proceeding similar to above we also obtain

$$\Gamma(s)\bigl\{ F(\theta,s)-F(1-\theta,s)\bigr\} = \frac{(2\pi)^{s+1}}{2s \cos\frac{\pi(s)}{2}}\bigl\{ \zeta(-s,\theta)- \zeta(-s,1-\theta)\bigr\} .$$
(4.3)

Moreover, the integral representation (2.28) is valid for all $$0<\theta<1$$. So we can replace θ by $$1-\theta$$ in its integrand. Using the above decomposition (4.2) and (4.3), we then obtain

$$\mathit{Ci}_{2}\bigl(\omega,e^{-x}\bigr)=-\frac{1}{2\mathrm{i}\pi} \int _{c-\mathrm{i} \infty}^{c+\mathrm{i}\infty}\frac{(2\pi)^{s+1}\zeta(s,z)}{2s\sin \frac{\pi s}{2}}\bigl\{ \zeta(-s, \theta)+ \zeta(-s,1-\theta)\bigr\} \frac{ds}{x^{s}}$$
(4.4)

and

$$\mathit{Si}_{2}\bigl(\omega,e^{-x}\bigr)= \frac{1}{2\mathrm{i}\pi} \int_{c-\mathrm{i} \infty}^{c+\mathrm{i}\infty}\frac{(2\pi)^{s+1}\zeta(s,z)}{2s\cos \frac{\pi s}{2}}\bigl\{ \zeta(-s, \theta)- \zeta(-s,1-\theta)\bigr\} \frac{ds}{x^{s}},$$
(4.5)

where $$\omega=e^{-zx+2\mathrm{i}\pi\theta}$$, $$0< x$$, $$0<\theta<1$$, $$0<\operatorname{Re} z$$, and $$1< c<2$$.

Note that the special values $$\zeta(n,z)$$ ($$n\in\mathbb{N}_{0}$$) are expressed in terms of the polygamma function $$\psi(z)$$,

$$\zeta(n+1,z)=\frac{(-1)^{n+1}}{n!}\psi^{(n)}(z) , \quad z \neq 0, -1, -2, \ldots ,$$
(4.6)

and $$\zeta(-n,z)$$ ($$n\in\mathbb{N}$$) is reduced to the Bernoulli polynomial

$$\zeta(-n,z)=-\frac{B_{n+1}(z)}{n+1}.$$
(4.7)

Applying the identities for the Bernoulli polynomial

$$B_{n}(1-\theta)=(-1)^{n}B_{n}(\theta),$$

we obtain

\begin{aligned}& \zeta(-n,\theta)+\zeta(-n,1-\theta)=\bigl((-1)^{n+1}-1\bigr) \frac {B_{n+1}(\theta)}{n+1}, \end{aligned}
(4.8)
\begin{aligned}& \zeta(-n,\theta)-\zeta(-n,1-\theta)=\bigl((-1)^{n}-1\bigr) \frac {B_{n+1}(\theta)}{n+1}. \end{aligned}
(4.9)

The integrand in (4.4) has a meromorphic continuation in the s-plane, its only singularity in the half-plane $$\operatorname{Re} s>0$$ coincides with the pole of $$1/\sin{\frac{\pi s}{2}}$$ being a simple pole at $$s=2, 4,\ldots$$ . Then by the Cauchy integral, we can shift the contour in (4.4) to the right, picking up the residues at $$s=2, \ldots, 2N$$ , with the result

$$\mathit{Ci}_{2}\bigl(\omega,e^{-x}\bigr)=4\sum _{n=1}^{N}(-1)^{n}\frac{\psi^{(2n-1)}(z)B_{2n+1} (\theta)}{(2n+1)!} \biggl(\frac{2\pi}{x}\biggr)^{2n}+Q_{N}(x),$$
(4.10)

where

$$Q_{N}(x)=-\frac{1}{2\mathrm{i}\pi} \int_{c+2N-\mathrm{i} \infty}^{c+2N+\mathrm{i}\infty}\frac{(2\pi)^{s+1}\zeta(s,z)}{2s\sin \frac{\pi s}{2}}\bigl\{ \zeta(-s, \theta)+ \zeta(-s,1-\theta)\bigr\} \frac{ds}{x^{s}}.$$
(4.11)

Using the following estimations in a vertical strip $$s=\sigma+\mathrm{i}t$$, $$\sigma\neq0, \pm 1, \pm2,\ldots$$ ,

$$\frac{1}{\sin\frac{\pi s}{2}}= \mathcal{O}\bigl(|t|^{-1}e^{-\frac{\pi}{2}|t|} \bigr),$$
(4.12)

we obtain

$$\bigl\vert Q_{N}(x)\bigr\vert =\mathcal{O}\biggl( \frac{1}{x^{2N+1}}\biggr).$$
(4.13)

Similarly,

$$\mathit{Si}_{2}\bigl(\omega,e^{-x}\bigr)=\frac{4\gamma}{\pi}B_{2}( \theta)\frac {1}{x}+4\sum_{n=1}^{N}(-1)^{n} \frac{\psi^{(2n)}(z)B_{2n+2} (\theta)}{(2n+2)!}\biggl(\frac{2\pi}{x}\biggr)^{2n+1}+\mathcal{O} \biggl(\frac{1}{x^{2N+2}}\biggr).$$
(4.14)

### Proposition 4.1

Let $$\omega=e^{-zx+2\mathrm{i}\pi\theta}$$, $$x>0$$, $$\operatorname{Re} z>1$$ and $$0<\theta<1$$. Then

$$\mathit{Li}_{2}\bigl(\omega,e^{-x}\bigr) \sim\frac{4\gamma}{\pi}B_{2}( \theta)\frac{\mathrm{i}}{x}+4\sum_{n=1}^{\infty} \mathrm{i}^{n}\frac{\psi^{(n-1)}(z)B_{n+1} (\theta)}{(n+1)!}\biggl(\frac{2\pi}{x} \biggr)^{n}, \quad x\rightarrow\infty.$$
(4.15)

## References

1. Abramowitz, M, Stegun, IA: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1972). ISBN:0-486-61272-4

2. Kirillov, AN: Dilogarithm identities. Prog. Theor. Phys. Suppl. 118, 61-142 (1995)

3. Berndt, BC: Ramanujan’s Notebooks. Part IV. Springer, Berlin (1994)

4. Hardy, GH, Littlewood, JE: Note on the theory of series (XXIV): a curious power series. Proc. Camb. Philos. Soc. 42, 85-90 (1946)

5. McIntosh, RJ: Some asymptotic formulae for q-shifted factorials. Ramanujan J. 3, 205-214 (1999)

6. Lewin, L: Polylogarithms and Associated Functions. North-Holland, New York (1981). ISBN:0-444-00550-1

7. Olver, FWJ, Lozier, DW, Boisvert, RF, Clark, CW (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010). http://dlmf.nist.gov

8. Koornwinder, TH: Special functions and q-commuting variables. In: Ismail, MEH, Masson, DR, Rahman, M (eds.) Special Functions, q-Series and Related Topics. Fields Institute Communications, vol. 14, pp. 131-166. Am. Math. Soc., Providence (1997)

9. Whittaker, ET, Watson, GN: A Course in Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1990)

10. Apostol, TM: On the Lerch zeta function. Pac. J. Math. 1, 161-167 (1951)

11. Paris, RB, Kaminsky, D: Asymptotics and the Mellin-Barnes Integrals. Cambridge University Press, New York (2001). ISBN:0-521-79001-8

## Acknowledgements

The first author would like to extend his sincere appreciation to the Deanship of Scientific Research at King Saudi University for funding this Research group No. (RG-1437-020).

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Correspondence to Fethi Bouzeffour.

### Competing interests

The authors declare that they have no competing interests.

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All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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