1 Introduction

Let \(s \in {\mathbb {C}}\). The main object of this paper is the following Dirichlet series:

$$\begin{aligned} {\zeta }_{\exp }(s; \lambda ; z) := \sum \limits _{n=0}^{\infty }\dfrac{\exp \left\{ {-\lambda }/{(n+z)} \right\} }{(n+z)^{s}}, \end{aligned}$$
(1.1)

where \(\lambda \in {\mathbb {C}}\setminus \{0\}\) and \(z \in {\mathbb {C}} \) with \(0 < \mathrm{{Re}}(z) \le 1\). The Dirichlet series (1.1) is known as the exponential-type generating function of the Riemann zeta-function \(\zeta (s)\) by the relation

$$\begin{aligned} {\zeta }_{\exp }(s; \lambda ; 1)= \sum \limits _{m=0}^{\infty }\dfrac{(-\lambda )^{m}}{m!}\,\zeta (s+m) \end{aligned}$$
(1.2)

for \(\lambda \in {\mathbb {R}}\) and \(s \in {\mathbb {C}}\setminus \{1,0,-1,-2, \ldots \}\).

In the present paper, we describe an integral representation of (1.1) along a Hankel contour, which gives a functional relation or transformation formula (Theorems 1.1 and 1.2). As a natural consequence of the functional relation [Theorem 1.2 (1.4)], we obtain a new proof of the Fourier series expansion of the holomorphic Poincaré series attached to \(\mathrm{SL}(2,{\mathbb {Z}})\) (Corollary 1.3). For the exponential-type generating function (1.2), our transformation formula [Theorem 1.2 (1.5)] is equivalent to the Voronoï-type summation formula indicated by Katsurada [14, Theorems 3.1 and 3.2].

Historically, the origin of the Dirichlet series (1.1) goes back to the study of Hardy and Littlewood [7]. They proved the functions

$$\begin{aligned} \sum \limits _{n\le x}\dfrac{1}{n}\cos \dfrac{x}{n}, \quad \sum \limits _{n\le x}\dfrac{1}{n}\sin \dfrac{x}{n} \end{aligned}$$

are unbounded as \(x\rightarrow \infty \) to show their Tauberian conditions being best possible. Segal [20] proved one Voronoï-type summation formula for the integral mean value of \(Q(x)=\sum \nolimits _{n=1}^{\infty } \sin (x/n)n^{-1} \), and discussed the existence of the Bessel series expansion for Q(x) itself. Based on the character analog of the Poisson summation formula due to Berndt [1], Kano [13] derived suitable expressions for \( \sum \nolimits _{n=1}^{\infty }{\chi (n)}\cos ({x/n}){n}^{-1} \) and \( \sum \nolimits _{n=1}^{\infty }{\chi (n)}\sin ({x}/{n}){n}^{-1} \) with primitive and non-principal Dirichlet character \(\chi \). We note that asymptotic approximations for Q(x) were recently developed by Kuznetsov [15].

In the context of generating functions of the Riemann zeta-function, the Hurwitz zeta-function, the binomial-type generating function

$$\begin{aligned} \zeta (s, 1+x)=\sum \limits _{m=0}^{\infty } \dfrac{\varGamma (s+m)}{\varGamma (s)\,m!}\zeta (s+m)(-x)^m \end{aligned}$$

for \(|x|<1\) and \(s \in {{\mathbb {C}}}\setminus \{1\}\), was employed by Ramanujan [19] in the generalization of Glaisher’s formula [6]

$$\begin{aligned} \gamma = 1- 2\left\{ \dfrac{\zeta (3)}{3\cdot 4} +\dfrac{\zeta (5)}{5\cdot 6} +\dfrac{\zeta (7)}{7\cdot 8} + \cdots \right\} . \end{aligned}$$

Here \(\gamma \) is Euler’s constant, \(\zeta (s,x)\) is the Hurwitz zeta-function, and \(\varGamma (s)\) is the Gamma function.

The exponential-type series (1.2) was initially studied by Chowla and Hawkins [3]. They derived an asymptotic formula for \( \sum \nolimits _{m>1}^{\infty }{(-\lambda )^{m}}\zeta (m)/{m!}\) as \(\lambda \rightarrow \infty \), and suggested the existence of the asymptotic expansion in terms of certain Bessel functions (cf. [2]). In [14, Theorems 3.1 and 3.2], Katsurada accomplished the Voronoï-type summation formula for

$$\begin{aligned} G_\nu (x)=\sum \limits _{m>\mathrm{{Re}}(\nu )-1}^{\infty } \dfrac{\zeta (m-\nu )}{m!}(-x)^m \quad (\nu \in {\mathbb {C}}), \end{aligned}$$

via the Mellin–Barnes integral formula. Our transformation formula of \({\zeta }_{\exp }(s; \lambda ; z)\) [Theorem 1.2 (1.5)] is equivalent to Katsurada’s formula, when \(\lambda =x \in {\mathbb {R}}\), \(z=1\) and \(s= -\nu \in {\mathbb {C}}\setminus \{1, 0, -1, -2, \dots \}\).

In recent works [16, 17, 18], we derived some functional properties of Bessel zeta-functions and a confluent hypergeometric zeta-function. The J-Bessel zeta-function appears in the Fourier series expansion of the Poincaré series attached to \(\mathrm{SL}(2,{\mathbb {Z}})\) by the inverse Mellin transform. This fact strongly suggested that our zeta-functions should have a kind of functional equation. The inverse Laplace transform of Weber’s first exponential integral was the key ingredient in the proof of the integral expression, which led to the expected transformation formula [16, Theorem 1.1 (1.4)]. Kaczorowski and Perelli treated more general zeta-functions twisted by hypergeometric or Bessel functions, and derived meromorphic continuations of these zeta-functions via the properties of the non-linear twists [9, 10, 11, 12].

Following the techniques employed in [16, 17], we drive an integral representation of (1.1) via the inverse Laplace transform of \(T^{\alpha }\mathrm{e}^{-1/T}\). The integral expression of \({\zeta }_{\exp }(s; \lambda ; z)\) gives a transformation formula when \(z \in {\mathbb {R}}\) or functional relation when \(z \in {\mathbb {C}}\setminus {\mathbb {R}} \) (Theorem 1.2).

Throughout this paper, H denotes the complex upper half-plane. Let \(J_\nu (z)\) be the Bessel function of the first kind, and \(I_\nu (z)\) and \(K_\nu (z)\) be modified Bessel functions (cf. [5, 7.2.1 (2), 7.2.2 (12), (13)]). Let \(\int _{-\infty }^{(0+)}\) denote integration over a Hankel contour, starting at negative infinity on the real axis, encircling the origin with a small radius in the positive direction, and returning to the starting point. We write \(\int _{\infty \mathrm{e}^{i\theta }}^{(0+)}\) for an integration taken along a rotated Hankel contour, starting at \(\infty \mathrm{e}^{i(-2\pi +\theta )}\), encircling the origin in the positive direction, and returning to the point \(\infty \mathrm{e}^{i\theta }\). We also denote \(\int _{(c)}\) for an integral over the vertical straight path from \(c-i\infty \) to \(c+i\infty \).

Theorem 1.1

Let \(\lambda \in {\mathbb {C}}\setminus \{0\},\) and let \(z \in {\mathbb {C}}\) with \(|\arg z| < \pi /2\) and \(0 < \mathrm{{Re}}(z) \le 1\). Then, the integral representation

$$\begin{aligned} {\zeta }_{\exp }(s; \lambda ; z) = \lambda ^{1-s}\varGamma (s-1) + \dfrac{\lambda ^{-s}}{\pi i} \int _{\infty \mathrm{e}^{i\theta }}^{(0+)} \dfrac{u^{(s-1)/2}\mathrm{e}^{zu/\lambda }}{1-\mathrm{e}^{u/\lambda }} K_{s-1}(2\sqrt{u})\mathrm{d}u \end{aligned}$$
(1.3)

holds, where \(\theta \) is chosen so as \({\pi }/{2} -\min \{0, \arg z\}<\theta - \arg \lambda <{3\pi }/{2} -\max \{0, \arg z\}\). The integral representation above gives a holomorphic continuation to the whole s-plane except on \(s\in \{1, 0, -1, -2, \dots \}\).

Theorem 1.2

For \(z \in H\) and \(\lambda \in {\mathbb {C}}\setminus \{0\},\) the following functional relation holds.

$$\begin{aligned} \begin{array}{ll} &{} {\zeta }_{\exp }(s; \lambda ; z) +(-1)^s{\zeta }_{\exp }(s; -\lambda ; 1-z) \\ &{} \quad = 2\pi i\mathrm{e}^{-\pi is}\displaystyle \sum \limits _{n=1}^{\infty } (2\pi in/\lambda )^{(s-1)/2} \mathrm{e}^{2\pi izn}I_{s-1}(2\sqrt{2\pi in\lambda }). \end{array} \end{aligned}$$
(1.4)

For \(x \in {\mathbb {R}}\) with \(0 < x \le 1\) and \(\lambda \in {\mathbb {R}}_{>0},\) the transformation formula

$$\begin{aligned} \begin{array}{ll} {\zeta }_{\exp }(s; \lambda ; x) &{}= \lambda ^{1-s}\varGamma (s-1) \\ &{} \quad +2\displaystyle \sum \limits _ {n \in {\mathbb {Z}},\, n\not =0} \left( {2\pi in}/{\lambda }\right) ^{(s-1)/2} \mathrm{e}^{2\pi ixn}K_{s-1}(2\sqrt{2\pi in\lambda }) \end{array} \end{aligned}$$
(1.5)

holds.

Let \(m\in {\mathbb {Z}}_{>0}\) and \(k \ge 4\) be an integer. We denote \(\gamma (z)=(az+b)/(cz+d)\) for \(\gamma = \bigl ( {\begin{matrix} a &{} b \\ c &{} d \end{matrix}} \bigr ) \in \mathrm{SL}(2,{\mathbb {Z}})\) and \(z \in H\), and use the notation \({{e}}(z)=\exp (2\pi i z)\). In a usual manner, we define the m-th Poincaré series attached to \(\mathrm{SL}(2,{\mathbb {Z}})\) of weight k by

$$\begin{aligned} P_m^{\,k}(z):=(-1)^k \sum \limits _{\{c,d\}} \dfrac{{e} (m\gamma (z))}{(cz+d)^k}. \end{aligned}$$
(1.6)

Here the summation is taken over \(\gamma = \bigl ( {\begin{matrix} * &{} * \\ c &{} d \end{matrix}} \bigr ) \), a complete system of representation of \(\left\{ \bigl ( {\begin{matrix} * &{} * \\ 0 &{} * \end{matrix}} \bigr ) \in \mathrm{SL}(2,{\mathbb {Z}}) \right\} \backslash \mathrm{SL}(2,{\mathbb {Z}})\) \( \cong \{ (c,d) \in {\mathbb {Z}}^2 \mid \gcd (c,d)=1,\; c>0 \;\mathrm{{or}}\; c=0,\, d=1\}\). Theorem 1.2 (1.5) leads to

Corollary 1.3

Fourier series expansion of the holomorphic Poincaré series

$$\begin{aligned} P_m^{\,k}(z)&= (-1)^k{{{e}} (mz)}+ (-1)^{\frac{k}{2}} \,2\pi \sum \limits _{n=1}^{\infty }\! \left( \dfrac{n}{m }\right) ^{\frac{k-1}{2}}\\&\quad \sum \limits _{c=1}^{\infty } \frac{1}{c} K_c(m,n) J_{k-1}\left( \frac{4\pi }{c} \sqrt{mn}\right) {{e}}(nz). \end{aligned}$$

Here, the Kloosterman sum is defined as follows:

$$\begin{aligned} K_c(m,n) =\sum \limits _ {{\begin{array}{c} d \!\!\!\!\! \mod c \\ \gcd (c,d)=1\end{array}}} \!\!\!\!\!\! {e} \left( \frac{m{\bar{d}}+nd}{c}\right) , \quad ({\bar{d}}{d}\equiv 1 \mod c). \end{aligned}$$

Corollary 1.4

Under the same assumptions of Theorem 1.1, the integral representation

$$\begin{aligned} {\zeta }_{\exp }(s; \lambda ; z)= \dfrac{-\varGamma (s-1)\varGamma (-s)}{\pi i \lambda ^s} \int _{\infty e^{i\theta }}^{(0+)} \dfrac{u^{(s-1)/2}\mathrm{e}^{zu/\lambda }}{1-\mathrm{e}^{u/ \lambda }} I_{s-1}(2\sqrt{u})\mathrm{d}u \end{aligned}$$
(1.7)

holds for \(s \in {\mathbb {C}} \setminus {\mathbb {Z}} ,\) and

$$\begin{aligned} {\zeta }_{\exp }(n; \lambda ; z)= \dfrac{(-1)^n}{\pi i \lambda ^n} \int _{\infty \mathrm{e}^{i\theta }}^{(0+)} \dfrac{u^{(n-1)/2}\mathrm{e}^{zu/\lambda }}{1-\mathrm{e}^{u/ \lambda }} I_{n-1}(2\sqrt{u})\log \sqrt{u}\mathrm{d}u \end{aligned}$$
(1.8)

holds for \(n =2, 3, 4, \dots \) The above integral representations also provide a holomorphic continuation to the whole s-plane except on \(s\in \{1, 0, -1, -2, \dots \},\) and give the functional relation (1.4).

2 Preliminary results

In this section, we establish some Fourier–Mellin integral of K-Bessel function (Lemma 2.2), which are equivalent to the inverse Laplace transforms of \(T^{-\alpha } \exp (-1/T)\). We examine this integral transform directly starting from the Mellin–Barnes integrals of the Bessel function.

First, we quote Barnes’ integral representation of the modified Bessel function of the second kind.

Lemma 2.1

The K-Bessel function has the expression

$$\begin{aligned} K_{\nu -1}(Z)= \dfrac{1}{4\pi i} \int _{(c_1)} \varGamma (w) \varGamma (w-\nu +1) \left( {Z}/{2} \right) ^{-2w+\nu -1} \mathrm{d}w \end{aligned}$$
(2.1)

for \(|\arg {Z}|<\pi /2\) and \(c_1>\max \{0, \mathrm{{Re}}(\nu )-1 \}\).

Proof

The equality (2.1) is equivalent to [21, 6.5 (6)] by substituting [21, 3.7 (8)], taking \(s=-w\) and replacing \(\nu \) by \(\nu -1\). \(\square \)

The following Lemma is crucial in the proof of our main results.

Lemma 2.2

Let \(|\arg {T}|<\pi \). For every \( \alpha \in {\mathbb {C}},\) the integral representation

$$\begin{aligned} T^{-\alpha }\mathrm{e}^{-1/{T}}= \dfrac{1}{\pi i} \int _{\infty \mathrm{e}^{i\theta }}^{(0+)} u^{(\alpha -1)/2 }\mathrm{e}^{Tu}K_{\alpha -1}(2\sqrt{u})\mathrm{d}u, \end{aligned}$$
(2.2)

holds for \(\theta \) with \(\pi /2<\theta + \arg {T}<3\pi /2 \).

Proof

Temporarily, we assume \(|\arg {T}|<\pi /2 \), \(\pi /2<\theta <\pi \), and \(\pi /2<\theta \pm \arg {T}<3\pi /2 \). For \(r>0\), we let \(\varLambda _{r,\theta }\) denote the contour starting on the ray from \(\infty \mathrm{e}^{-i\theta }\) to \(r\mathrm{e}^{-i\theta }\), encircling the origin with radius r in the positive direction from \(r\mathrm{e}^{-i\theta }\) to \(r\mathrm{e}^{i\theta }\), and returning on the ray from \(r\mathrm{e}^{i\theta }\) to \(\infty \mathrm{e}^{i\theta }\). First, we prove the following expression

$$\begin{aligned} T^{-\alpha }\mathrm{e}^{-1/{T}}= \dfrac{1}{\pi i} \int _{\varLambda _{r,\theta } } u^{(\alpha -1)/2 }\mathrm{e}^{Tu}K_{\alpha -1}(2\sqrt{u})\mathrm{d}u. \end{aligned}$$
(2.3)

Taking \(\nu =\alpha \) and \(Z=2\sqrt{u}\) in (2.1), we see the right side of (2.3) is equal to

$$\begin{aligned} \begin{array}{ll} &{} \dfrac{1}{(2\pi i)^2 } \displaystyle \int _{\varLambda _{r,\theta } } {\mathrm{e}^{Tu}u^{(\alpha -1)/2 }} \int _{(c_1)} \varGamma (w) \varGamma (w -\alpha +1) \left( \sqrt{u} \right) ^{-2w +\alpha -1} \mathrm{d}w\mathrm{d}u \\ &{}\quad = \dfrac{1}{(2\pi i)^2 } \displaystyle \int _{(c_1)} {\varGamma (w)\varGamma ( w -\alpha +1)} \int _{\varLambda _{r,\theta } } {\mathrm{e}^{Tu}u^{- w +\alpha -1}} \mathrm{d}u\mathrm{d}w. \end{array} \end{aligned}$$
(2.4)

The interchange of the order of integration is justified by the absolute convergence of double integrals in (2.4) due to Stirling’s formula under the conditions \(\pi /2<\theta <\pi \) and \(\pi /2<\theta \pm \arg {T}<3\pi /2 \) for given \(|\arg {T}|<\pi /2\). After changing variable Tu to u and shifting back to the integration path to \( \varLambda _{r,\theta }\), the u-integral in the right side of (2.4) is evaluated by means of (cf. [8, Theorem 8.4b])

$$\begin{aligned} \dfrac{1}{2\pi i} \int _{\varLambda _{r,\theta } } \mathrm{e}^{u}u^{-s}\mathrm{d}u =\dfrac{1}{\varGamma (s)}. \end{aligned}$$

Hence the right side of (2.4) is equal to

$$\begin{aligned} \dfrac{T^{-\alpha }}{2\pi i } \int _{(c_1)} {\varGamma (w)} T^{w} \mathrm{d}w, \end{aligned}$$
(2.5)

and the Mellin inversion integral transformation

$$\begin{aligned} \mathrm{e}^{-Z}= \dfrac{1}{2\pi i} \int _{(c)} \varGamma (w) Z^{-w}\mathrm{d}w \end{aligned}$$

for any \(c>0\) and \(Z\in {\mathbb {C}}\) such that \(|\arg Z|<\pi /2\) gives the equality (2.3).

The asymptotic expansion of the K-Bessel function (cf. [21, 3.71 (12)], [5, 7.4.1. (1)]) yields

$$\begin{aligned} K_{\nu -1}(Z) = \left( \frac{\pi }{2Z}\right) ^{\frac{1}{2}} \mathrm{e}^{-Z} \left\{ 1+ O (Z^{-1})\right\} \end{aligned}$$
(2.6)

for \(|\arg (Z)|<3\pi /2\), and O-constant depends only on \(\nu \). Due to the exponential decay (2.6), the integrand \(u^{(\alpha -1)/2 }\mathrm{e}^{Tu}K_{\alpha -1}(2\sqrt{u})\) in (2.3) decreases rapidly when \(|u|\rightarrow \infty \) for every \(\theta \) such that \(\pi /2<\theta + \arg {T}<3\pi /2 \). Therefore, the path of integration (2.3) may be shifted to the rotated Hankel contour as in the integral representation (2.2), which gives the analytic continuation to the domain \(|\arg {T}|<\pi \) with \(\pi /2<\theta + \arg {T}<3\pi /2 \). This completes the proof of Lemma 2.2. \(\square \)

In (2.2), it is possible to replace the integrand by using the \(I_{\nu }\).

Lemma 2.3

Under the same assumptions of Lemma 2.2,

$$\begin{aligned} T^{-\alpha }\mathrm{e}^{-1/{T}}= \dfrac{-\varGamma (\alpha -1)\varGamma (2-\alpha )}{2\pi i} \int _{\infty \mathrm{e}^{i\theta }}^{(0+)} u^{(\alpha -1)/2 }\mathrm{e}^{Tu}I_{\alpha -1}(2\sqrt{u})\mathrm{d}u, \end{aligned}$$
(2.7)

holds for \( \alpha \in {\mathbb {C}} \setminus {\mathbb {Z}} ,\) and

$$\begin{aligned} T^{-n }\mathrm{e}^{-1/{T}}= \dfrac{(-1)^n}{\pi i} \int _{\infty \mathrm{e}^{i\theta }}^{(0+)} u^{(n-1)/2 }\mathrm{e}^{Tu}I_{n-1}(2\sqrt{u})\log (\sqrt{u})\mathrm{d}u, \end{aligned}$$
(2.8)

holds for \(n =2, 3, 4, \dots \)

Proof

Using the relation

$$\begin{aligned} K_{\nu }(Z)=\dfrac{\pi }{2}\cdot \dfrac{I_{-\nu }(Z)-I_{\nu }(Z)}{\sin (\pi \nu )} \end{aligned}$$
(2.9)

(cf. [5, 7.2.2 (13)]) to the integrand in (2.2), and noting that

$$\begin{aligned} u^{(\alpha -1)/2 }I_{1-\alpha }(2\sqrt{u})= \sum \limits _{m=0}^{\infty } \dfrac{u^m}{\varGamma (m -\alpha +2)\,m!} \end{aligned}$$
(2.10)

(cf. [5, 7.2.2 (12)]) is an entire function of u, we have

$$\begin{aligned} \begin{array}{ll} &{} T^{-\alpha }\mathrm{e}^{-1/{T}} \\ &{}\quad = \dfrac{1}{\pi i} \displaystyle \int _{\infty \mathrm{e}^{i\theta }}^{(0+)} u^{(\alpha -1)/2 }\mathrm{e}^{Tu} \left\{ K_{\alpha -1}(2\sqrt{u})\mathrm{d}u - \dfrac{\pi }{2\sin (\alpha -1)\pi } I_{1-\alpha }(2\sqrt{u}) \right\} \mathrm{d}u \end{array} \end{aligned}$$
(2.11)

for \( \alpha \in {\mathbb {C}} \setminus {\mathbb {Z}} \). Hence the equality (2.7) holds with \({\pi }/{\sin (\pi \nu )}=\varGamma (\nu )\varGamma (1-\nu )\).

For positive integers \(n=1, 2, 3, \dots \),

$$\begin{aligned} \begin{array}{ll} &{} K_{n}(2\sqrt{u})\\ &{}\quad = (-1)^{n+1}I_{n}(2\sqrt{u})\log (\sqrt{u}) +\dfrac{1}{2} \displaystyle \sum \limits _{m=0}^{n-1}(-1)^{n}(\sqrt{u})^{2m-n} \dfrac{(n-m-1)!}{m!} \\ &{} \qquad + \dfrac{1}{2} (-1)^{n} \displaystyle \sum \limits _{m=0}^{\infty } (\sqrt{u})^{2m+n} \dfrac{\psi (n+m+1) \psi (m+1) }{(m+n)!\,m!} \end{array} \end{aligned}$$
(2.12)

holds (cf. [5, 7.2.5 (37)]). Here \(\psi (s)={\varGamma '(s)}/{\varGamma (s)}\) is the logarithmic derivative of the gamma function. This gives the integral expression (2.8). \(\square \)

3 Proofs of theorems

Proof of Theorem 1.1

Let \(\lambda \in {\mathbb {C}}\setminus \{0\}\) and \(z \in {\mathbb {C}}\) with \(0 < \mathrm{{Re}}(z) \le 1\). For \(s \in {\mathbb {C}}\setminus {\mathbb {Z}}\), we define the function

$$\begin{aligned} {P}_{\exp }(s; \lambda ; z; Z):= \sum \limits _{n=0}^{\infty }\dfrac{\exp \left\{ {-\lambda }/{(n+z)} \right\} }{(n+z)^{s}}Z^n. \end{aligned}$$
(3.1)

The right side of (3.1) converges absolutely when \(|Z|<1\). By (2.11), the right side of (3.1) is equal to

$$\begin{aligned} \dfrac{1}{\pi i\lambda ^s} \displaystyle \sum \limits _{n=0}^{\infty } \displaystyle \int _{\infty \mathrm{e}^{i\theta }}^{(0+)} Z^n u^{(s-1)/2 } \mathrm{e}^{(n+z)u/\lambda } \left\{ K_{s-1}(2\sqrt{u}) - \dfrac{\pi }{2\sin \pi (s-1)} I_{1-s}(2\sqrt{u})\right\} \mathrm{d}u, \end{aligned}$$
(3.2)

where \(\theta \) is chosen so as \({\pi }/{2}<\arg \{(n+z)/\lambda \} +\theta <{3\pi }/{2}\) for \(n \in {\mathbb {Z}}_{\ge 0}\).

Let r be the small radius around the origin for the path of the above integral. Then the interchange of summation and integration is justified when \(|Z\mathrm{e}^{u/ \lambda }|<1\). Accordingly (3.2) is equal to

$$\begin{aligned} \begin{array}{l} \dfrac{1}{\pi i\lambda ^s} \displaystyle \int _{\infty \mathrm{e}^{i\theta }}^{(0+)} \dfrac{u^{(s-1)/2}\mathrm{e}^{zu/\lambda }}{1-Z\mathrm{e}^{u/\lambda }} K_{s-1}(2\sqrt{u})\mathrm{d}u \\ \quad - \dfrac{\varGamma (s-1)\varGamma (2-s)}{2\pi i\lambda ^s} \displaystyle \int _{\infty \mathrm{e}^{i\theta }}^{(0+)} \dfrac{u^{(s-1)/2}\mathrm{e}^{zu/\lambda }}{1-Z\mathrm{e}^{u/\lambda }} I_{1-s}(2\sqrt{u})\mathrm{d}u. \end{array} \end{aligned}$$
(3.3)

Here we have used \({\pi }/{\sin (\pi \nu )}=\varGamma (\nu )\varGamma (1-\nu )\) (cf. [4,  1.2 (6)]). The integrals above converge absolutely if \(Z \not = \mathrm{e}^{-u/\lambda } \) for \( u \in U_r =\left\{ u\in {\mathbb {C}}\; | \; u \in \int _{\infty \mathrm{e}^{i\theta }}^{(0+)} \right\} \), hence (3.3) is a holomorphic function of Z and a meromorphic function of s when \(Z \not =\mathrm{e}^{-u/\lambda } \) for \(u \in U_r\). Therefore, (3.3) gives an analytic continuation of \({P}_{\exp }(s; \lambda ; z; Z) \) in both Z and s. In particular, \({P}_{\exp }(s; \lambda ; z; Z) \) is holomorphic at \(Z=1\). By (2.10), the second integrand in (3.3) is holomorphic except at \(u=0\), and its residue is

$$\begin{aligned} \mathrm{{Res}}_{u=0} \left\{ \dfrac{u^{(s-1)/2}\mathrm{e}^{zu/\lambda }}{1-\mathrm{e}^{u/\lambda }} I_{1-s}(2\sqrt{u}) \right\} =\dfrac{-\lambda }{\varGamma (2-s)}. \end{aligned}$$

This provides (1.3) for \(s \in {\mathbb {C}}\setminus \{1, 0, -1, -2, \dots \}\). \(\square \)

Proof of Theorem 1.2

In the same manner as the proof of Theorem 1.1 (1.3), we obtain

$$\begin{aligned} \begin{array}{ll} &{} (-1)^s{\zeta }_{\exp }(s; -\lambda ; 1-z) \\ &{}\quad = (-1)^s \displaystyle \sum \limits _{n=0}^{\infty }\dfrac{\exp \left\{ \lambda /{(n+1-z)} \right\} }{(n+1-z)^{s}} \\ &{}\quad = \dfrac{-\varGamma (s-1) }{ \lambda ^{s-1}} + \dfrac{1}{\pi i\lambda ^{s}} \displaystyle \int _{\infty \mathrm{e}^{i\rho }}^{(0+)} \dfrac{u^{(s-1)/2}\mathrm{e}^{(1-z)u/(- \lambda )}}{1-\mathrm{e}^{u/(- \lambda )}} K_{s-1}(2\sqrt{u})\mathrm{d}u. \end{array} \end{aligned}$$
(3.4)

Here \(\rho \), the argument of the Hankel contour, was chosen so as \(\pi /2< \rho + \arg \{(n+1-z)/(-\lambda )\}<3\pi /2 \) for \(n \in {\mathbb {Z}}_{\ge 0}\). For \(z \in H\) with \(0<\mathrm{{Re}}(z) \le 1\), we may choose \(\rho \) as \(0<\rho -\arg \lambda <\pi /2 \) in (3.4), and may choose \(\theta \) as \(\pi /2<\theta -\arg \lambda <\pi \) in (1.3).

Let \( \mathbf{{K}}(s;\lambda ;z;u) \) be the integrand in (3.4). Rotating the Hankel contour from \( \int _{\infty \mathrm{e}^{i\rho }}^{(0+)}\) to \( \int _{\infty \mathrm{e}^{i\theta }}^{(0+)}\) in the positive direction, we obtain

$$\begin{aligned} \begin{array}{ll} &{} \dfrac{1}{\pi i\lambda ^{s}} \displaystyle \int _{\infty \mathrm{e}^{i\rho }}^{(0+)} \dfrac{u^{(s-1)/2}\mathrm{e}^{(1-z)u/(-\lambda )}}{1-\mathrm{e}^{u/(-\lambda )}} K_{s-1}(2\sqrt{u})\mathrm{d}u \\ &{}\quad = \dfrac{2}{\lambda ^{s}} \left\{ \displaystyle \sum \limits _{n=0}^{\infty }\mathrm{{Res}}_{u=2\pi in\lambda } \mathbf{{K}}(s;\lambda ;z;u) - \displaystyle \sum \limits _{n=0}^{\infty }\mathrm{{Res}}_{u=2\pi \mathrm{e}^{-\frac{3}{2}\pi i}n\lambda } \mathbf{{K}}(s;\lambda ;z;u) \right\} \\ &{} \qquad + \dfrac{1}{\pi i\lambda ^{s}} \displaystyle \int _{\infty \mathrm{e}^{i\theta }}^{(0+)} \dfrac{u^{(s-1)/2}\mathrm{e}^{zu/\lambda }}{\mathrm{e}^{u/\lambda }-1} K_{s-1}(2\sqrt{u})\mathrm{d}u. \end{array} \end{aligned}$$
(3.5)

By the integral expressions (1.3) and (3.4), and substituting

$$\begin{aligned}&\mathrm{{Res}}_{u=2\pi in\lambda } \mathbf{{K}}(s;\lambda ;z;u) ={\lambda }(2\pi in\lambda )^{(s-1)/2}\mathrm{e}^{2\pi inz} K_{s-1}(2\sqrt{2\pi in\lambda }), \\&\mathrm{{Res}}_{u=2\pi \mathrm{e}^{-\frac{3}{2}\pi i} n\lambda } \mathbf{{K}}(s;\lambda ;z;u) ={\lambda }\left( 2\pi \mathrm{e}^{-\frac{3}{2}\pi i} n\lambda \right) ^{(s-1)/2}\mathrm{e}^{2\pi inz} K_{s-1}\left( 2\sqrt{2\pi \mathrm{e}^{-\frac{3}{2}\pi i} n\lambda }\right) \end{aligned}$$

in (3.5), we have

$$\begin{aligned}&{\zeta }_{\exp }(s; \lambda ; z) +(-1)^s{\zeta }_{\exp }(s; -\lambda ; 1-z) \\&\quad = 2\displaystyle \sum \limits _{n=1}^{\infty } (2\pi in/\lambda )^{\frac{s-1}{2}} \mathrm{e}^{2\pi izn} \left\{ K_{s-1}(2\sqrt{2\pi i n\lambda }) -\mathrm{e}^{-\pi i (s-1)} K_{s-1}(2\mathrm{e}^{-\pi i}\sqrt{2\pi i n\lambda })\right\} . \end{aligned}$$

Applying (2.9) and \(I_{\nu }(z\mathrm{e}^{in\pi })=\mathrm{e}^{in\pi \nu }I_{\nu }(z)\) for \(n\in {\mathbb {Z}}\) (cf. [5, 7.11 (44)]), we arrive at the relation formula (1.4).

Next, we deform the Hankel contour (1.3) so as to prove the transformation formula (1.5). Let \(x \in {\mathbb {R}}\) with \(0 < x \le 1\) and \(\lambda \in {\mathbb {R}}_{>0}\). Temporarily, we assume \( s<-1\). Let R be a sufficiently large integer and \(C_R\) be the path of integration starting at negative infinity on the real axis, encircling the origin with the radius \(2\pi (R+1/2)\lambda \) in the positive direction, and returning to the starting point. In the transforming the path of integral, the contour \(C_R\) passes simple poles at \(u= 2\pi i n \lambda \) \((n= \pm 1, \pm 2,\dots , \pm R)\). Then, by the residue theorem, we have

$$\begin{aligned} \begin{array}{ll} &{} \dfrac{1}{\pi i\lambda ^s} \displaystyle \int _{\infty \mathrm{e}^{i\pi }}^{(0+)} \dfrac{u^{(s-1)/2}\mathrm{e}^{xu/\lambda }}{1-\mathrm{e}^{u/\lambda }} K_{s-1}(2\sqrt{u})\mathrm{d}u \\ &{}\quad = 2 \!\!\!\!\! \displaystyle \sum \limits _{n=-R, \, n\not =0}^{R} (2\pi in/\lambda )^{(s-1)/2} \mathrm{e}^{2\pi ixn}K_{s-1}(2\sqrt{2\pi in\lambda }) \\ &{} \qquad + \dfrac{1}{\pi i\lambda ^s} \displaystyle \int _{C_R} \! \dfrac{u^{(s-1)/2}\mathrm{e}^{xu/\lambda }}{1-\mathrm{e}^{u/\lambda }} K_{s-1}(2\sqrt{u})\mathrm{d}u. \end{array} \end{aligned}$$
(3.6)

Here \((1-\mathrm{e}^{u/\lambda })^{-1}\) is bounded except for the neighborhoods at the points of \(u/\lambda =2\pi im \) \((m \in {{\mathbb {Z}}})\), and by (2.6), there exists a positive constant A such that

$$\begin{aligned} \left| \dfrac{u^{(s-1)/2}\mathrm{e}^{xu/\lambda }}{1-\mathrm{e}^{u/\lambda }} K_{s-1}(2\sqrt{u}) \right| \le A \cdot (R\lambda )^{\mathrm{{Re}}(s/2)-3/4}\mathrm{e}^{\pi |\mathrm{{Im}}(s)|} \end{aligned}$$

on the contour \(C_R\). Taking \(R\rightarrow \infty \) in (3.4), we obtain the transformation formula (1.5) under the assumption \(\mathrm{{Re}}(s)<-1/2\). By (2.6), \( K_{s-1}(2\sqrt{2\pi in\lambda })\) decays exponentially as \(n\rightarrow \pm \infty \), hence (1.5) holds for any \(s\in {\mathbb {C}}\). This completes the proof of Theorem 1.2. \(\square \)

4 Proofs of corollaries

Proposition 4.1

Let \(\mu >0\) and k be a positive integer. Then the following equality holds.

$$\begin{aligned} (-1)^k \!\!\! \sum \limits _{n=-\infty }^{\infty }\!\!\! \dfrac{{{e}}\left( -{\mu }/{(z+n)}\right) }{(z+n)^{k}} =2\pi (-1)^{\frac{k}{2}} \sum \limits _{n=1}^{\infty } \left( \dfrac{n}{\mu }\right) ^{\frac{k-1}{2}}\!\!\!\! J_{k-1}(4\pi \sqrt{\mu n}) {{e}}(nz). \end{aligned}$$

Proof

Noting the relation

$$\begin{aligned} \begin{array}{ll} \sum \limits _{n=-\infty }^{\infty }\!\!\! \dfrac{{{e}}\left( -{\mu }/{(z+n)}\right) }{(z+n)^{k}} &{} = {\zeta }_{\exp }(k; 2\pi i\mu ; z) +(-1)^k{\zeta }_{\exp }(k; -2\pi i\mu ; 1-z) \\ &{} = 2\pi i\mathrm{e}^{-\pi ik}\displaystyle \sum \limits _{n=1}^{\infty } (n/\mu )^{(k-1)/2} \mathrm{e}^{2\pi izn}I_{k-1}(4\pi i\sqrt{n\mu }), \end{array} \end{aligned}$$

and \(I_{s-1}(4\pi i\sqrt{n\mu }) = \mathrm{e}^{-\pi i(s-1)/2}J_{s-1}(4\pi \mathrm{e}^{\pi i}\sqrt{n\mu }) = \mathrm{e}^{\pi i(s-1)/2}J_{s-1}(4\pi \sqrt{n\mu })\) (cf. [5, 7.2.2. (12)]), we obtain the assertion of Proposition 4.1. \(\square \)

Proof of Corollary 1.3

By the definition of \(P_m^{\,k}(z)\) (1.6), we see

$$\begin{aligned} P_m^{\,k}(z) = (-1)^k{{{e}} (mz)}+(-1)^k\!\!\!\!\!\!\!\!\!\!\!\! \sum \limits _ {{ \begin{array}{l}(c,d)\in {\mathbb {Z}}^2,\, c>0 \\ \gcd (c,d)=1\end{array}}} \dfrac{{{e}} (m\gamma (z))}{(cz+d)^k}. \end{aligned}$$
(4.1)

According to standard steps, we rearrange the d-sum into n-sum with finite d-sum modulo c, and apply Proposition 4.1 taking \(\mu =m/c^2\) and replacing z by \(z+d/c\). Thus, we achieve the Fourier series expansion of the Poincaré series. \(\square \)

Proof of Corollary 1.4

In the same way as the proof of Theorem 1.1 (1.3), we obtain the assertions of Corollary 1.4 via the integral transforms (2.7) and (2.8) in Lemma 2.3. \(\square \)