Evaluations of Some General Classes of Mordell Integrals by Applying the Mellin-Barnes-Type Contour Integration

Abstract

In this paper, motivated by the contributions of several researchers, including (for example) Riemann, Kronecker, Lerch, Ramanujan, and Glaisher on some special integrals known as Mordell’s integrals, we consider the problem of the evaluation of some generalized Mordell integrals by applying the Mellin-Barnes-type contour integration. In particular, we have evaluated the following integrals: $I_1:={\int }_0^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t\mathrm{and}{I}_2:={\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t$ for real or complex parameters $a,b,c$, and g with $\Re \left(a\right)>0$ in terms of Meijer’s G-function and the generalized hypergeometric function _pF_q.

1. Introduction, Definitions, and Preliminaries

In this paper, we consider the following general case of the Mordell-type integrals (see, for details, [1,2,3]):

$$I_1:={\int }_0^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t,$$

(1)

together with its bilateral form given by

$$I_2:={\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t,$$

(2)

in each of which the parameters $a,b,c$, and g are real or complex numbers and $\Re \left(a\right)>0$. Here, for the sake of simplicity in each case, the path of integration is the real axis (or parallel to the real axis) with indentations, if necessary, whenever the integrand has any singularities on the path of integration.

It is easily verified from the integral (2) that

$$\begin{array}{cc}\hfill I_2& ={\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t\hfill \\ \hfill & ={\int }_{-\infty }^0{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t+{\int }_0^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & =I_3+I_1,\hfill \end{array}$$

(4)

where $I_1$ is defined by (1) and $I_3$ is given by

$$\begin{array}{c}\hfill I_3:={\int }_0^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2-bt}}{{\mathrm{e}}^{-ct}+g}}\mathrm{d}t.\end{array}$$

(5)

In our present investigation, we shall need each of the following integral formulas:

$${\int }_{-\infty }^{\infty }{\mathrm{e}}^{-a{x}^2\pm bx}\mathrm{d}x=\sqrt{{\displaystyle \frac{\pi }{a}}}\exp \left({\displaystyle \frac{b^2}{4a}}\right)\left(\Re \left(a\right)>0;b\in \mathbb{C}\right),$$

(6)

which appeared in the published tables of definite integrals by Bierens de Haan (see ([4] p. 146, Equation (100) (9)); see also ([5] p. 355, Entry 3.323 (2)), Gröbner-Hofreiter ([6] p. 65, Entry 314 (6a)), and ([7] p. 344, Entry 11);

$$\begin{array}{cc}\hfill {\int }_0^{\infty }{\mathrm{e}}^{-{\alpha }^2{x}^2-\beta x}\mathrm{d}x& ={\displaystyle \frac{1}{2\alpha }}\sum _{r=0}^{\infty }{\displaystyle \frac{{\left(-\frac{\beta }{\alpha }\right)}^r}{r!}}\Gamma \left({\displaystyle \frac{r+1}{2}}\right)\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{\sqrt{\pi }}{2\alpha }}\exp \left({\displaystyle \frac{{\beta }^2}{4{\alpha }^2}}\right)-{\displaystyle \frac{\beta }{2{\alpha }^2}}{}_{1}F_1\left[\begin{array}{c}\hfill 1;\\ \\ \hfill {\displaystyle \frac{3}{2}};\end{array}{\displaystyle \frac{{\beta }^2}{4{\alpha }^2}}\right]\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{\sqrt{\pi }}{2\alpha }}\exp \left({\displaystyle \frac{{\beta }^2}{4{\alpha }^2}}\right)\mathrm{erfc}\left({\displaystyle \frac{\beta }{2\alpha }}\right)\hfill \end{array}$$

(9)

$$\left(\Re \left(\alpha \right)>0;\Re \left(\beta \right)\geqq 0\mathrm{when}\Re \left(\alpha \right)=0\mathrm{and}\Im \left(\alpha \right)\ne 0\right),$$

which can be traced back to Nielsen’s classic work on the Gamma function $\Gamma (·)$ (see [8]; see also ([9] p. 302, Equation (7.4.2)), ([6] p. 57, Entry 10 (a)), and ([10] p. 43, Equation (5.41))). The function ${}_{1}F_1$, occurring in (8), is Kummer’s confluent hypergeometric function, for which it is known that

$${}_{1}F_1\left[\begin{array}{c}\hfill a;\\ \\ \hfill c;\end{array}z\right]=\underset{\left|b\right|\to \infty }{lim}\left\{{}_{2}F_1\left[\begin{array}{c}\hfill a,b;\\ \\ \hfill c;\end{array}{\displaystyle \frac{z}{b}}\right]\right\}$$

in terms of the familiar Gauss hypergeometric function ${}_{2}F_1$. Moreover, the function $\mathrm{erfc}(·)$ on the right-hand side of (9) is the complementary error function, which is given by

$$\begin{array}{cc}\hfill \mathrm{erfc}\left(z\right)& :={\displaystyle \frac{2}{\sqrt{\pi }}}{\int }_z^{\infty }\exp (-t^2)\mathrm{d}t=:1-\mathrm{erf}\left(z\right)\hfill \\ \hfill & =1-{\displaystyle \frac{2z}{\sqrt{\pi }}}{}_{1}F_1\left[\begin{array}{c}\hfill {\displaystyle \frac{1}{2}};\\ \\ \hfill {\displaystyle \frac{3}{2}};\end{array}-z^2\right].\hfill \end{array}$$

Each of the above-mentioned functions ${}_{1}F_1$ and ${}_{2}F_1$ is the special case of the generalized hypergeometric function ${}_{p}F_q$, with p numerator parameters ${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_p$ and q denominator parameters ${\beta }_1,{\beta }_2,\cdots ,{\beta }_q$, defined as follows (see ([11] pp. 71–72) and ([12] p. 41 et seq.)):

$${}_{p}F_q\left[\begin{array}{c}\hfill {\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_p;\\ \\ \hfill {\beta }_1,{\beta }_2,\cdots ,{\beta }_q;\end{array}z\right]=\sum _{n=0}^{\infty }{\displaystyle \frac{{\displaystyle \prod _{j=1}^p}{\left({\alpha }_j\right)}_n}{{\displaystyle \prod _{j=1}^q}{\left({\beta }_j\right)}_n}}{\displaystyle \frac{z^n}{n!}}.$$

(10)

$$\begin{array}{c}\hfill (p,q\in {\mathbb{N}}_0;p\leqq q+1;p\leqq q\mathrm{and}\left|z\right|<\infty ;p=q+1\mathrm{and}\left|z\right|<1;\end{array}$$

$$\begin{array}{c}\hfill p=q+1,\left|z\right|=1\mathrm{and}\Re \left(\omega \right)>0;p=q+1,\left|z\right|=1,z\ne 1\mathrm{and}-1<\Re \left(\omega \right)\leqq 0),\end{array}$$

where

$$\omega :=\sum _{j=1}^q{\beta }_j-\sum _{j=1}^p{\alpha }_j$$

$$\left({\alpha }_j\in \mathbb{C}(j=1,2,3,\cdots ,p);{\beta }_j\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}(j=1,2,3,\cdots ,q)\right).$$

One of the ways in which the generalized hypergeometric function ${}_{p}F_q\left(z\right)$$(\mathrm{with}p=q+1)$ can be continued analytically to the domain $|arg(1-z\left)\right|<\pi$, that is, to the whole z-plane cut along the real axis from $z=1$ to $z=\infty$, is by using (10) in conjunction with its Mellin-Barnes-type contour integral representation when $p\leqq q+1$ (see, for details, ([11] p. 94 et seq., Section 54); see also ([12] p. 43))

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\cdots \Gamma \left({\alpha }_p\right)}{\Gamma \left({\beta }_1\right)\Gamma \left({\beta }_2\right)\cdots \Gamma \left({\beta }_q\right)}}{}_{p}F_q\left[\begin{array}{c}\hfill {\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_p;\\ \\ \hfill {\beta }_1,{\beta }_2,\cdots ,{\beta }_q;\end{array}z\right]\hfill \\ \hfill & {\displaystyle =\frac{1}{2\pi \mathrm{i}}{\int }_{\mathcal{L}}\frac{{\displaystyle \Gamma (-\zeta )\prod _{j=1}^p\Gamma ({\alpha }_j+\zeta )}}{{\displaystyle \prod _{j=1}^q\Gamma ({\beta }_j+\zeta )}}{(-z)}^{\zeta }\mathrm{d}\zeta (z\ne 0)}\hfill \end{array}$$

(11)

$$({\alpha }_j\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}(j=1,2,3,\cdots ,p);|arg(-z)|<\pi \mathrm{when}p=q+1;$$

$$|arg(-z)|<{\displaystyle \frac{\pi }{2}}[\mathrm{that}\mathrm{is},\mathrm{in}\mathrm{the}\mathrm{half}-\mathrm{plane}\Re \left(z\right)<0]\mathrm{when}p=q),$$

where the path $\mathcal{L}$ of integration is the imaginary axis (in the complex $\zeta$-plane) from $-\mathrm{i}\infty$ to $\mathrm{i}\infty$ with indentations, if necessary, in order to put the poles of $\Gamma ({\alpha }_j+\zeta )(j=1,2,3,\cdots ,p)$ to the left of the path and to put the poles of $\Gamma (-\zeta )$ to the right of the path. The assumption that there exists such a path of integration in (11) precludes the possibility that any of the parameters ${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_p$ is zero or a negative integer, the parameter constraints ${\beta }_j\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}(j=1,2,3,\cdots ,q)$ being already assumed in the definition (10).

The generalized hypergeometric function ${}_{p}F_q$ encompasses a large number of special functions that are frequently encountered in many branches of the applied mathematical, physical, and engineering sciences. For many (if not all) these special functions, the Mellin-Barnes-type contour integral representations can be derived by appropriately applying the general result (11). In this regard in the present investigation, we provide the following very specialized form of (11), which is associated with the familiar binomial theorem:

$${(1-z)}^{-\lambda }={}_{1}F_0\left[\begin{array}{c}\hfill \lambda ;\\ \\ \hfill -;\end{array}z\right]={\displaystyle \frac{1}{2\pi \mathrm{i}\Gamma \left(\lambda \right)}}{\int }_{\mathcal{L}}\Gamma (-\zeta )\Gamma (\lambda +\zeta ){(-z)}^{\zeta }\mathrm{d}\zeta (z\ne 0)$$

(12)

$$\left(|arg(-z)|<\pi \mathrm{and}\lambda \in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}\right),$$

where the path $\mathcal{L}$ of integration is specified as in the general result (11) for

$$p-1=q=0({\alpha }_1=\lambda ).$$

We now turn our attention towards the G-function, which was introduced by Meijer ([13] p. 83) in the year 1941, arising essentially from an attempt to provide an interpretation of the symbol ${}_{p}F_q$ in the case when $p>q+1$. It is defined, in terms of the Mellin-Barnes-type integral, by (see, for details, ([14] p. 206 et seq., Section 5.3) and ([12] p. 45, Section 1.5))

$$\begin{array}{cc}{G}_{p,q}^{m,n}\left(z\right)\hfill & =G_{p,q}^{m,n}\left(z\left|\begin{array}{c}\hfill {\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_p\\ \\ \hfill {\beta }_1,{\beta }_2,\cdots ,{\beta }_q\end{array}\right.\right)\hfill \\ & {\displaystyle :=\frac{1}{2\pi \mathrm{i}}{\int }_{\mathfrak{L}}\frac{{\displaystyle \prod _{j=1}^m}\Gamma ({\beta }_j-\zeta ){\displaystyle \prod _{j=1}^n}\Gamma (1-{\alpha }_j+\zeta )}{{\displaystyle \prod _{j=m+1}^q}\Gamma (1-{\beta }_j+\zeta ){\displaystyle \prod _{j=n+1}^p}\Gamma ({\alpha }_j-\zeta )}{z}^{\zeta }\mathrm{d}\zeta (z\ne 0).}\hfill \end{array}$$

(13)

There are three remarkably different contours $\mathfrak{L}$ of integration in (13), which are discussed by, among others, Erdéyi et al. ([14] p. 207) and Luke ([15] p. 144 et seq.). For the integrand of the Mellin-Barnes-type contour integral in (13), it is tacitly assumed that an empty product is interpreted as 1, the integers $m,n,p,q$ are such that $1\leqq m\leqq q$ and $0\leqq n\leqq p$, and the parameters $\alpha$’s and $\beta$’s are so constrained that no pole of $\Gamma ({\beta }_j-\zeta )$$(j=1,2,\cdots ,m)$ coincides with any pole of $\Gamma (1-{\alpha }_j+\zeta )$$(j=1,2,\cdots ,n)$. We set

$$\zeta =\sigma +\mathrm{i}\tau \left(\sigma ,\tau \in \mathbb{R};\mathrm{i}=\sqrt{-1}\right)$$

and choose $\sigma$ so that, for $\tau \to \pm \infty$, the following inequality holds true:

$$\begin{array}{c}\hfill (q-p)\sigma >\Re \left(\omega \right)+1-{\displaystyle \frac{1}{2}}(q-p),\end{array}$$

(14)

where, as also with (10),

$$\omega :=\sum _{j=1}^q{\beta }_j-\sum _{j=1}^p{\alpha }_j.$$

Thus, with $\sigma \in \mathbb{R}$ constrained by the inequality (14), the path $\mathfrak{L}$ of integration in (13) is a Mellin-Barnes-type contour that runs from $\sigma -\mathrm{i}\tau$ to $\sigma +\mathrm{i}\tau$ with indentations, if necessary, so that all poles of $\Gamma ({\beta }_j-\zeta )(j=1,2,\cdots ,m)$ lie to the right of the contour and all poles of $\Gamma (1-{\alpha }_j+\zeta )(j=1,2,\cdots ,n)$ lie to the left of the contour. In this case, the integral (13) converges if

$$\Omega :=(m+n)-{\displaystyle \frac{1}{2}}(p+q)>0\mathrm{and}|arg\left(z\right)|<\Omega \pi .$$

In the case when $|arg(z\left)\right|=\Omega \pi (\Omega \geqq 0)$, the integral (13) converges absolutely when $p=q$ if $\Re \left(\omega \right)<-1$, and also when $p\ne q$ if $\sigma \in \mathbb{R}$ is constrained, as above, by the inequality (14).

The G-function is an analytic function of z with a branch point at the origin $(z=0)$. It is symmetric in the following parameters:

$${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n;{\alpha }_{n+1},{\alpha }_{n+2},\cdots ,{\alpha }_p;{\beta }_1,{\beta }_2,\cdots ,{\beta }_m;{\beta }_{m+1},{\beta }_{m+2},\cdots ,{\beta }_q.$$

In several readily accessible books and monographs, one can find systematic lists of various properties and relationships involving the G-function as well as its many interesting particular cases (see, among other studies, ([14] pp. 208–222)). For example, by a close comparison between the definitions (11) and (13), one is led easily to the relationships (see ([14] p. 215, Equation (5).6 (1)), ([15] p. 147, Equations (5).2 (14) and (5).2 (17)), and ([12] p. 47, Equation (1).5 (9))):

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\Gamma \left({\alpha }_1\right)\Gamma \left({\alpha }_2\right)\cdots \Gamma \left({\alpha }_p\right)}{\Gamma \left({\beta }_1\right)\Gamma \left({\beta }_2\right)\cdots \Gamma \left({\beta }_q\right)}}{}_{p}F_q\left[\begin{array}{c}\hfill {\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_p;\\ \\ \hfill {\beta }_1,{\beta }_2,\cdots ,{\beta }_q;\end{array}z\right]\hfill \\ \hfill & =G_{p,q+1}^{1,p}\left(-z\left|\begin{array}{c}\hfill 1-{\alpha }_1,1-{\alpha }_2,\cdots ,1-{\alpha }_p\\ \\ \hfill 0,1-{\beta }_1,1-{\beta }_2,\cdots ,1-{\beta }_q\end{array}\right.\right)\hfill \\ \hfill & =G_{q+1,p}^{p,1}\left(-{\displaystyle \frac{1}{z}}\left|\begin{array}{c}\hfill 1,{\beta }_1,{\beta }_2,\cdots ,{\beta }_q\\ \\ \hfill {\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_p\end{array}\right.\right)\hfill \end{array}$$

(15)

$$\left(0<\left|z\right|<\infty \mathrm{when}p\leqq q;0<\left|z\right|<1\mathrm{when}p=q+1\right).$$

Our plan to present the section-wise content of this paper is as follows. In Section 2, we have briefly discussed two special cases of the general forms (1) and (2) of the Mordell integral, which were considered by [16]. We have also shown the equivalence of the convergence conditions of these general forms of the Mordell integral and those of the specialized form of the Mordell–Ramanujan integral. In Section 3, in terms of Meijer’s G-function (13) and the generalized hypergeometric function ${}_{p}F_q(·)(\mathrm{with}p=q+1)$, we have evaluated some new potentially useful integrals, (20) to (22) and (26) to (28), by applying Nielsen’s integral (7) and the binomial theorem (12). In Section 4, in terms of Meijer’s G-function (13) and the generalized hypergeometric function ${}_{p}F_q(·)(\mathrm{with}p-1=q=2r)$, we have evaluated the general form $I_2$ of the Mordell integral, given by (2), by means of the integral (6) and the binomial theorem (12). Additionally, in Section 4 itself, we have again evaluated the general form $I_2$ of the Mordell integral by applying the relationship (4) with the general form $I_1$ given by (1) in conjunction with some suitable combinations of the integrals (20) to (22) and (26) to (28) considered in Section 3. In Section 5, we consider some transformations involving Meijer’s G-function, which are derivable by means of a comparative study of the integrals discussed in the preceding sections. Finally, in the concluding section (Section 6), we have presented some remarks and observations that pertain to the results in this paper.

2. A Discussion About Specialized Forms of the Mordell Integral

For particular values of the parameters $a,b,c$, and g, the Mordell integral $I_2$ in (2) occurs in various contexts in the works of Ramanujan (see, for details, [17,18,19]). Integrals of the type $I_2$ were first studied by Ramanujan and then in some detail by Siegel [20]. However, in his above-cited papers [1,2,3], Mordell did classify different standard forms of the integral $I_2$ according to the value of the parameters $a,b,c$, and g. Mordell evaluated the integral $I_2$ in terms of Jacobi’s theta and other related functions.

If, in the Mordell integral (2), we put

$$-a=\pi \mathrm{i}\eta ,b=-2\pi x,c=2\pi \mathrm{and}g=-1,$$

the resulting integral

$${\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{\pi \mathrm{i}\eta t^2-2\pi xt}}{{\mathrm{e}}^{2\pi t}-1}}\mathrm{d}t$$

is evaluated firstly in terms of the following functions:

$$f(x,\eta ),{\vartheta }_1(x,\eta )\mathrm{and}f\left({\displaystyle \frac{x}{\eta }},-{\displaystyle \frac{1}{\eta }}\right),$$

where the Mordell integral function $f(x,\eta )$ is defined by (see ([16] p. 237, Equation (1)))

$$\mathrm{i}f(x,\eta )=\sum _{n=-\infty }^{\infty }{\displaystyle \frac{{(-1)}^{n-1}{q}^{\frac{1}{4}{(2n-1)}^2}{\mathrm{e}}^{(2n-1)\pi \mathrm{i}x}}{1+q^{2n-1}}}\left(q={\mathrm{e}}^{\pi \mathrm{i}\eta };\Im \left(\eta \right)>0\right)$$

(16)

and ${\vartheta }_1(x,\eta )$ is the Jacobi theta function of the first kind given by ([16] p. 236, Equation (2)):

$${\vartheta }_1(x,\eta )=\mathrm{i}\sum _{n=-\infty }^{\infty }{(-1)}^n{q}^{{\left(n-{\displaystyle \frac{1}{2}}\right)}^2}{\mathrm{e}}^{\mathrm{i}\pi (2n-1)x}\left(q={\mathrm{e}}^{\pi \mathrm{i}\eta };\Im \left(\eta \right)>0\right).$$

(17)

If, in the Mordell integral (2), we set

$$-a=\pi \mathrm{i}\eta ,b=-2\pi x,c=2\pi \eta \mathrm{and}g=-1,$$

the second evaluation of the resulting integral

$${\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{\pi \mathrm{i}\eta t^2-2\pi xt}}{{\mathrm{e}}^{2\pi t\eta }-1}}\mathrm{d}t$$

is given by combining two of the following functions at a time:

$$\theta (x,\eta ),\theta \left({\displaystyle \frac{x}{\eta }},-{\displaystyle \frac{1}{\eta }}\right),\varphi (x,\eta )\mathrm{and}\varphi \left({\displaystyle \frac{x}{\eta }},-{\displaystyle \frac{1}{\eta }}\right),$$

where the two functions $\theta (x,\eta )$ and $\varphi (x,\eta )$ are given by (see ([16] p. 242, Equations (2) and (3)))

$$\theta (x,\eta )=\sum _{n=1}^{\infty }{q}^{n^2}{\mathrm{e}}^{2\mathrm{i}\pi nx}\left(q={\mathrm{e}}^{\pi \mathrm{i}\eta };\Im \left(\eta \right)>0\right)$$

(18)

and

$$\varphi (x,\eta )=\sum _{n=0}^{\infty }{q}^{n^2}{\mathrm{e}}^{-2\mathrm{i}\pi nx}\left(q={\mathrm{e}}^{\pi \mathrm{i}\eta };\Im \left(\eta \right)>0\right).$$

(19)

The following integrals of Ramanujan (see, for example, ([16] pp. 249–250))

$${\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{\mathrm{i}\pi \eta t^2-2\pi xt}}{{\mathrm{e}}^{2\pi t}+1}}\mathrm{d}t\mathrm{and}{\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{\pm \mathrm{i}\pi \left(\frac{a}{b}\right)t^2-2\pi xt}}{{\mathrm{e}}^{2\pi t}\pm 1}}\mathrm{d}t,$$

and also the following integral of Andrews [21] (see, for example, ([16] p. 255, Equation (9)))

$${\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-2\gamma t^2+\gamma t}}{{\mathrm{e}}^{2\gamma t}+1}}\mathrm{d}t$$

were evaluated in terms of Laurent’s series and power series.

In order to show the equivalence of the convergence conditions in the general form of the Mordell integral and the specialized form of Ramanujan’s integral, we observe that, in the first and second evaluations of the specialized form of the Mordell integral, the convergence condition is stated as $\Im \left(\eta \right)>0$, whereas, in our general form, the convergence condition is $\Re \left(a\right)>0$, where $a=-\mathrm{i}\pi \eta$. If we suppose that $\eta =\alpha +\mathrm{i}\beta$$(\beta >0)$, then

$$\Im \left(\eta \right)=\beta \mathrm{and}a=-\mathrm{i}\pi (\alpha +\mathrm{i}\beta )=-\mathrm{i}\pi \alpha +\pi \beta ,$$

which means that $\Im \left(\eta \right)=\beta >0$ since $\Re \left(a\right)=\pi \beta >0$ and $\beta >0$. The equivalence of both convergence conditions is thus verified.

3. Evaluations of the Generalized Mordell Integral $I_1$

In this section, we propose to evaluate the potentially useful generalized Mordell integral $I_1$ in terms of Meijer’s G-function and the generalized hypergeometric function _p$F_q(·)(\mathrm{with}p=q+1)$.

Theorem 1.

Let $a,b,c,g\in \mathbb{C}$ such that $c\ne 0,$ $g\ne 0,-1$ and $\Re \left(a\right)>0$. Suppose also that the generalized Mordell integral $I_1$ is defined by (1). Then, each of the following integral formulas holds true:

$$\begin{array}{cc}\hfill I_1& :={\int }_0^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t={\displaystyle \frac{\sqrt{\pi }}{2(g+1)\sqrt{a}}}\hfill \\ \hfill & +{\displaystyle \frac{1}{2\sqrt{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{\Gamma \left(\frac{r+1}{2}\right)c^r}{r!a^{\frac{r}{2}}}}{\displaystyle \frac{1}{g}}{G}_{r+1,r+1}^{1,r+1}\left({\displaystyle \frac{1}{g}}\left|\begin{array}{c}0,\stackrel{r\mathrm{times}}{\overbrace{-{\displaystyle \frac{b}{c}},\cdots ,-{\displaystyle \frac{b}{c}}}}\hfill \\ \\ 0,\underset{r\mathrm{times}}{\underbrace{1-{\displaystyle \frac{b}{c}},\cdots ,1-{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right)\hfill \end{array}$$

(20)

and

$$\begin{array}{cc}\hfill I_1& ={\int }_0^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t={\displaystyle \frac{\sqrt{\pi }}{2(1+g)\sqrt{a}}}\hfill \\ \hfill & +{\displaystyle \frac{1}{2\sqrt{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{\Gamma \left(\frac{r+1}{2}\right)c^r}{r!a^{\frac{r}{2}}}}{G}_{r+1,r+1}^{r+1,1}\left(g\left|\begin{array}{c}0,\stackrel{r\mathrm{times}}{\overbrace{{\displaystyle \frac{b}{c}}-1,\cdots ,{\displaystyle \frac{b}{c}}-1}}\hfill \\ \\ 0,\underset{r\mathrm{times}}{\underbrace{{\displaystyle \frac{b}{c}},\cdots ,{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right).\hfill \end{array}$$

(21)

Furthermore, in terms of the generalized hypergeometric function ${}_{r+1}F_r(·),$ the integral Formula (20) implies when $\left|g\right|>1$ that

$$\begin{array}{cc}\hfill I_1& ={\int }_0^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t={\displaystyle \frac{\sqrt{\pi }}{2(g+1)\sqrt{a}}}\hfill \\ \hfill & +{\displaystyle \frac{1}{2g\sqrt{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{\Gamma \left(\frac{r+1}{2}\right)b^r}{r!a^{\frac{r}{2}}}}{}_{r+1}F_r\left[\begin{array}{c}\hfill 1,\stackrel{r\mathrm{times}}{\overbrace{{\displaystyle \frac{b}{c}}+1,\cdots ,{\displaystyle \frac{b}{c}}+1}};\\ \\ \hfill \underset{r\mathrm{times}}{\underbrace{{\displaystyle \frac{b}{c}},\cdots ,{\displaystyle \frac{b}{c}}}};\end{array}-{\displaystyle \frac{1}{g}}\right].\hfill \end{array}$$

(22)

Proof. 

First of all, by using the Mellin-Barnes-type contour integral (12) for the familiar binomial theorem, we see that

$$\begin{array}{cc}\hfill I_1& ={\int }_0^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t\hfill \\ \hfill & ={\displaystyle \frac{1}{g}}{\int }_0^{\infty }{\left(1+{\displaystyle \frac{{\mathrm{e}}^{ct}}{g}}\right)}^{-1}{\mathrm{e}}^{-a{t}^2+bt}\mathrm{d}t\hfill \\ \hfill & ={\displaystyle \frac{1}{g}}{\int }_0^{\infty }\left[{\displaystyle \frac{1}{2\pi \mathrm{i}}}{\int }_{\sigma -\mathrm{i}\infty }^{\sigma +\mathrm{i}\infty }\Gamma (-\zeta )\Gamma (1+\zeta ){\left({\displaystyle \frac{{\mathrm{e}}^{ct}}{g}}\right)}^{\zeta }\mathrm{d}\zeta \right]{\mathrm{e}}^{-a{t}^2+bt}\mathrm{d}t,\hfill \end{array}$$

which, upon the inversion of the order of integration (justified by de la Vallée Poussin’s theorem ([22] p. 504)), yields

$$I_1={\displaystyle \frac{1}{2g\pi \mathrm{i}}}{\int }_{\sigma -\mathrm{i}\infty }^{\sigma +\mathrm{i}\infty }{\displaystyle \frac{\Gamma (-\zeta )\Gamma (1+\zeta )}{g^{\zeta }}}\left({\int }_0^{\infty }{\mathrm{e}}^{-a{t}^2+bt+c\zeta t}\mathrm{d}t\right)\mathrm{d}\zeta .$$

A short check of the conditions under which the binomial theorem holds true is in order here.

Now, if we apply Nielsen’s integral (7), we find that

$$\begin{array}{cc}\hfill I_1& ={\displaystyle \frac{1}{2g\pi \mathrm{i}}}{\int }_{\sigma -\mathrm{i}\infty }^{\sigma +\mathrm{i}\infty }{\displaystyle \frac{\Gamma (-\zeta )\Gamma (1+\zeta )}{g^{\zeta }}}\left({\displaystyle \frac{1}{2\sqrt{a}}}\sum _{r=0}^{\infty }{\displaystyle \frac{{(b+c\zeta )}^r\Gamma \left(\frac{r+1}{2}\right)}{r!a^{\frac{r}{2}}}}\right)\mathrm{d}\zeta \hfill \\ \hfill & ={\displaystyle \frac{1}{2g\pi \mathrm{i}}}{\int }_{\sigma -\mathrm{i}\infty }^{\sigma +\mathrm{i}\infty }{\displaystyle \frac{\Gamma (-\zeta )\Gamma (1+\zeta )}{g^{\zeta }}}{\displaystyle \frac{1}{2\sqrt{a}}}\left(\sqrt{\pi }+\sum _{r=1}^{\infty }{\displaystyle \frac{{(b+c\zeta )}^r\Gamma \left(\frac{r+1}{2}\right)}{r!a^{\frac{r}{2}}}}\right)\mathrm{d}\zeta \hfill \\ \hfill & ={\displaystyle \frac{1}{2g}}{\displaystyle \frac{\sqrt{\pi }}{\sqrt{a}}}\left({\displaystyle \frac{1}{2\pi \mathrm{i}}}{\int }_{\sigma -\mathrm{i}\infty }^{\sigma +\mathrm{i}\infty }{\displaystyle \frac{\Gamma (-\zeta )\Gamma (1+\zeta )}{g^{\zeta }}}\mathrm{d}\zeta \right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2g\sqrt{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{\Gamma \left(\frac{r+1}{2}\right)}{r!a^{\frac{r}{2}}}}\left({\displaystyle \frac{1}{2\pi \mathrm{i}}}{\int }_{\sigma -\mathrm{i}\infty }^{\sigma +\mathrm{i}\infty }{\displaystyle \frac{\Gamma (-\zeta )\Gamma (1+\zeta ){(b+c\zeta )}^r}{g^{\zeta }}}\mathrm{d}\zeta \right).\hfill \end{array}$$

(23)

The first integral Formula (20) would follow from (23) when we make use of the contour integral (12) for the binomial theorem (23) once again and interpret the resulting contour single integral by means of the definition (13) of the G-function.

Our derivation of the second integral Formula (21) runs parallel to that of the first integral Formula (20). In this case, we multiply both numerator and denominator of the integrand in the definition (2) of $I_1$ by ${\mathrm{e}}^{ct}$ and then proceed as above. The routine details are being skipped here.

The third integral Formula (22) can be deduced fairly easily from the first integral Formula (20) by applying the known transformation formula ([15] p. 146, Equation (12)):

$$\begin{array}{cc}\hfill & G_{p,q}^{1,n}\left(z\left|\begin{array}{c}{a}_1,a_2,\cdots ,a_p\hfill \\ \\ b_1,b_2,\cdots ,b_q\hfill \end{array}\right.\right)={\displaystyle \frac{{\displaystyle \prod _{j=1}^n}\Gamma (1+b_1-a_j)z^{b_1}}{{\displaystyle \prod _{j=2}^q}\Gamma (1+b_1-b_j){\displaystyle \prod _{j=n+1}^p}\Gamma (a_j-b_1)}}\hfill \\ \hfill & ·{}_{p}F_{q-1}\left[\begin{array}{c}\hfill 1+b_1-a_1,\cdots ,1+b_1-a_p;\\ \\ \hfill 1+b_1-b_2,\cdots ,1+b_1-b_q;\end{array}{(-1)}^{p+n+1}z\right]\hfill \end{array}$$

(24)

$$\left(p<q\mathrm{and}0<\left|z\right|<\infty ;p=q\mathrm{and}0<\left|z\right|<1\right)$$

or, alternatively, from the second integral Formula (21) by applying the known transformation formula ([15] p. 147, Equation (15)):

$$\begin{array}{cc}\hfill & G_{p,q}^{m,1}\left(z\left|\begin{array}{c}{a}_1,a_2,\cdots ,a_p\hfill \\ \\ b_1,b_2,\cdots ,b_q\hfill \end{array}\right.\right)={\displaystyle \frac{{\displaystyle \prod _{j=1}^m}\Gamma (1+b_j-a_1)z^{a_1-1}}{{\displaystyle \prod _{j=2}^p}\Gamma (1+a_j-a_1){\displaystyle \prod _{j=m+1}^q}\Gamma (a_1-b_j)}}\hfill \\ \hfill & ·{}_{q}F_{p-1}\left[\begin{array}{c}\hfill 1+b_1-a_1,\cdots ,1+b_q-a_1;\\ \\ \hfill 1+a_2-a_1,\cdots ,1+a_p-a_1;\end{array}{\displaystyle \frac{{(-1)}^{q+m+1}}{z}}\right]\hfill \end{array}$$

(25)

$$\left(q<p\mathrm{and}0<\left|z\right|<\infty ;q=p\mathrm{and}\left|z\right|>1\right).$$

□

Remark 1.

Under the following parametric constraints,

$$a,b,c,g\in \mathbb{C}\mathrm{with}c\ne 0,g\ne 0,-1\mathrm{and}\Re \left(a\right)>0,$$

we find, for the Mordell-type integral $I_3$ defined by (5), that

$$\begin{array}{cc}\hfill I_3& :={\int }_0^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2-bt}}{{\mathrm{e}}^{-ct}+g}}\mathrm{d}t={\displaystyle \frac{\sqrt{\pi }}{2(g+1)\sqrt{a}}}+{\displaystyle \frac{1}{2g\sqrt{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{\Gamma \left(\frac{r+1}{2}\right){(-c)}^r}{r!a^{\frac{r}{2}}}}\hfill \\ \hfill & ·G_{r+1,r+1}^{1,r+1}\left({\displaystyle \frac{1}{g}}\left|\begin{array}{c}0,\stackrel{r\mathrm{times}}{\overbrace{-{\displaystyle \frac{b}{c}},\cdots ,-{\displaystyle \frac{b}{c}}}}\hfill \\ \\ 0,\underset{r\mathrm{times}}{\underbrace{1-{\displaystyle \frac{b}{c}},\cdots ,1-{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right),\hfill \end{array}$$

(26)

which results from the integral Formula (20) when we replace the parameters b with $-b$ and c with $-c$.

The following evaluations (27) and (28) of the integral $I_3$ would result from the integral Formulas (21) and (22), respectively, when we replace the parameters b with $-b$ and c with $-c$:

$$\begin{array}{cc}\hfill I_3& ={\displaystyle \frac{\sqrt{\pi }}{2(1+g)\sqrt{a}}}+{\displaystyle \frac{1}{2\sqrt{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{\Gamma \left(\frac{r+1}{2}\right){(-c)}^r}{r!a^{\frac{r}{2}}}}\hfill \\ \hfill & ·G_{r+1,r+1}^{r+1,1}\left(g\left|\begin{array}{c}0,\stackrel{r\mathrm{times}}{\overbrace{{\displaystyle \frac{b}{c}}-1,\cdots ,{\displaystyle \frac{b}{c}}-1}}\hfill \\ \\ 0,\underset{r\mathrm{times}}{\underbrace{{\displaystyle \frac{b}{c}},\cdots ,{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right)\hfill \end{array}$$

(27)

and

$$\begin{array}{cc}\hfill I_3& ={\displaystyle \frac{\sqrt{\pi }}{2(g+1)\sqrt{a}}}+{\displaystyle \frac{1}{2g\sqrt{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{\Gamma \left(\frac{r+1}{2}\right){(-b)}^r}{r!a^{\frac{r}{2}}}}\hfill \\ \hfill & ·{}_{r+1}F_r\left[\begin{array}{c}\hfill 1,\stackrel{r\mathrm{times}}{\overbrace{{\displaystyle \frac{b}{c}}+1,\cdots ,{\displaystyle \frac{b}{c}}+1}};\\ \\ \hfill \underset{r\mathrm{times}}{\underbrace{{\displaystyle \frac{b}{c}}}};\end{array}-{\displaystyle \frac{1}{g}}\right]\left(\right|g|>1).\hfill \end{array}$$

(28)

4. Evaluations of the Generalized Mordell Integral $I_2$

We propose in this section to consider two approaches to the problem involving evaluation of the generalized Mordell integral $I_2$ defined by (2). Our first approach to this problem leads us to Theorem 2 below.

Theorem 2.

If the parameters $a,b,c,g\in \mathbb{C}$ are such that $c\ne 0,$$g\ne 0,-1$ and $\Re \left(a\right)>0,$ then each of the following integral formulas holds true for the generalized Mordell integral $I_2$ defined by (2)

$$\begin{array}{cc}\hfill I_2& ={\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t={\displaystyle \frac{1}{1+g}}\sqrt{{\displaystyle \frac{\pi }{a}}}+{\displaystyle \frac{1}{g}}\sqrt{{\displaystyle \frac{\pi }{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{c^{2r}}{r!{\left(4a\right)}^r}}\hfill \\ \hfill & ·G_{2r+1,2r+1}^{1,2r+1}\left({\displaystyle \frac{1}{g}}\left|\begin{array}{c}0,\stackrel{2r\mathrm{times}}{\overbrace{-{\displaystyle \frac{b}{c}},\cdots ,-{\displaystyle \frac{b}{c}}}}\hfill \\ \\ 0,\underset{2r\mathrm{times}}{\underbrace{1-{\displaystyle \frac{b}{c}},\cdots ,1-{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill I_2& ={\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t={\displaystyle \frac{1}{1+g}}\sqrt{{\displaystyle \frac{\pi }{a}}}+\sqrt{{\displaystyle \frac{\pi }{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{c^{2r}}{r!{\left(4a\right)}^r}}\hfill \\ \hfill & ·G_{2r+1,2r+1}^{1,2r+1}\left(g\left|\begin{array}{c}0,\stackrel{2r\mathrm{times}}{\overbrace{{\displaystyle \frac{b}{c}}-1,\cdots ,{\displaystyle \frac{b}{c}}-1}}\hfill \\ \\ 0,\underset{2r\mathrm{times}}{\underbrace{{\displaystyle \frac{b}{c}},\cdots ,{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right),\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill I_2& ={\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t={\displaystyle \frac{1}{1+g}}\sqrt{{\displaystyle \frac{\pi }{a}}}+\sqrt{{\displaystyle \frac{\pi }{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{c^{2r}}{r!{\left(4a\right)}^r}}\hfill \\ \hfill & ·G_{2r+1,2r+1}^{2r+1,1}\left(g\left|\begin{array}{c}0,\stackrel{2r\mathrm{times}}{\overbrace{{\displaystyle \frac{b}{c}}-1,\cdots ,{\displaystyle \frac{b}{c}}-1}}\hfill \\ \\ 0,\underset{2r\mathrm{times}}{\underbrace{{\displaystyle \frac{b}{c}},\cdots ,{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right),\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill I_2& ={\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t={\displaystyle \frac{1}{1+g}}\sqrt{{\displaystyle \frac{\pi }{a}}}+{\displaystyle \frac{1}{g}}\sqrt{{\displaystyle \frac{\pi }{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{b^{2r}}{r!{\left(4a\right)}^r}}\hfill \\ \hfill & ·{}_{2r+1}F_{2r}\left[\begin{array}{c}\hfill 1,\stackrel{2r\mathrm{times}}{\overbrace{{\displaystyle \frac{b}{c}}+1,\cdots ,{\displaystyle \frac{b}{c}}+1}};\\ \\ \hfill \underset{2r\mathrm{times}}{\underbrace{{\displaystyle \frac{b}{c}},\cdots ,{\displaystyle \frac{b}{c}}}};\end{array}-{\displaystyle \frac{1}{g}}\right]\left(\right|g|>1)\hfill \end{array}$$

(32)

and

$$\begin{array}{cc}\hfill I_2& ={\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t={\displaystyle \frac{1}{1+g}}\sqrt{{\displaystyle \frac{\pi }{a}}}+{\displaystyle \frac{1}{2\sqrt{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{{(c-b)}^{2r}}{r!{\left(4a\right)}^r}}\hfill \\ \hfill & ·{}_{2r+1}F_{2r}\left[\begin{array}{c}\hfill 1,\stackrel{2r}{\overbrace{2-{\displaystyle \frac{b}{c}},\cdots ,2-{\displaystyle \frac{b}{c}}}};\\ \\ \hfill \underset{2r\mathrm{times}}{\underbrace{1-{\displaystyle \frac{b}{c}},\cdots ,1-{\displaystyle \frac{b}{c}}}};\end{array}-g\right](0<|g|<1).\hfill \end{array}$$

(33)

Proof. 

By means of the contour integral for the binomial theorem in (12), it is easily seen for the generalized Mordell integral $I_2$ that

$$\begin{array}{c}\hfill I_2={\displaystyle \frac{1}{g}}{\int }_{-\infty }^{\infty }\left({\displaystyle \frac{1}{2\pi \mathrm{i}}}{\int }_{\sigma -\mathrm{i}\infty }^{\sigma +\mathrm{i}\infty }{\displaystyle \frac{\Gamma (-\zeta )\Gamma (1+\zeta ){\mathrm{e}}^{ct\zeta }}{g^{\zeta }}}\mathrm{d}\zeta \right){\mathrm{e}}^{-a{t}^2+bt}\mathrm{d}t.\end{array}$$

(34)

Upon the justifiable inversion of the order of the absolutely convergent double integral in (34) by appealing the Fubini-type theorems, if we apply the known result (6), we are led to the following evaluation:

$$\begin{array}{cc}\hfill I_2& ={\displaystyle \frac{1}{2g\pi \mathrm{i}}}{\int }_{\sigma -\mathrm{i}\infty }^{\sigma +\mathrm{i}\infty }{\displaystyle \frac{\Gamma (-\zeta )\Gamma (1+\zeta )}{g^{\zeta }}}\left[\sqrt{{\displaystyle \frac{\pi }{a}}}exp\left({\displaystyle \frac{{(b+c\zeta )}^2}{4a}}\right)\right]\mathrm{d}\zeta \hfill \\ \hfill & ={\displaystyle \frac{1}{2g\pi \mathrm{i}}}\sqrt{{\displaystyle \frac{\pi }{a}}}{\int }_{\sigma -\mathrm{i}\infty }^{\sigma +\mathrm{i}\infty }{\displaystyle \frac{\Gamma (-\zeta )\Gamma (1+\zeta )}{g^{\zeta }}}\left(\sum _{r=0}^{\infty }{\displaystyle \frac{{(b+c\zeta )}^{2r}}{r!{\left(4a\right)}^r}}\right)\mathrm{d}\zeta \hfill \\ \hfill & ={\displaystyle \frac{1}{2g\pi \mathrm{i}}}\sqrt{{\displaystyle \frac{\pi }{a}}}{\int }_{\sigma -\mathrm{i}\infty }^{\sigma +\mathrm{i}\infty }{\displaystyle \frac{\Gamma (-\zeta )\Gamma (1+\zeta )}{g^{\zeta }}}\left(1+\sum _{r=1}^{\infty }{\displaystyle \frac{{(b+c\zeta )}^{2r}}{r!{\left(4a\right)}^r}}\right)\mathrm{d}\zeta ,\hfill \end{array}$$

so that

$$\begin{array}{cc}\hfill I_2& ={\displaystyle \frac{1}{g}}\left({\displaystyle \frac{1}{2\pi \mathrm{i}}}\sqrt{{\displaystyle \frac{\pi }{a}}}{\int }_{\sigma -\mathrm{i}\infty }^{\sigma +\mathrm{i}\infty }{\displaystyle \frac{\Gamma (-\zeta )\Gamma (1+\zeta )}{g^{\zeta }}}\mathrm{d}\zeta \right)\hfill \\ \hfill & +{\displaystyle \frac{1}{g}}\sqrt{{\displaystyle \frac{\pi }{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{1}{r!{\left(4a\right)}^r}}\left({\displaystyle \frac{1}{2\pi \mathrm{i}}}{\int }_{\sigma -\mathrm{i}\infty }^{\sigma +\mathrm{i}\infty }{\displaystyle \frac{\Gamma (0-\zeta )\Gamma (1+\zeta ){(b+c\zeta )}^{2r}}{g^{\zeta }}}\mathrm{d}\zeta \right)\hfill \\ \hfill & ={\displaystyle \frac{1}{1+g}}\sqrt{{\displaystyle \frac{\pi }{a}}}+{\displaystyle \frac{1}{g}}\sqrt{{\displaystyle \frac{\pi }{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{c^{2r}}{r!{\left(4a\right)}^r}}\hfill \\ \hfill & ·\left({\displaystyle \frac{1}{2\pi \mathrm{i}}}{\int }_{\sigma -\mathrm{i}\infty }^{\sigma +\mathrm{i}\infty }{\displaystyle \frac{\Gamma (-\zeta )\Gamma (1+\zeta ){\left[\Gamma \left(1+\frac{b}{c}+\zeta \right)\right]}^{2r}}{g^{\zeta }{\left[\Gamma \left(\frac{b}{c}+\zeta \right)\right]}^{2r}}}\mathrm{d}\zeta \right).\hfill \end{array}$$

(35)

The integral Formula (29) would result when we interpret the last contour integral in (35) by applying the definition (13) as a G-function.

The proofs of the integral Formulas (30) and (31) are much akin to the detailed proof of the integral Formula (29) and we choose to omit them here.

When we use the transformations (24) in (29) and (25) in (31), we obtain the integral Formula (32). Moreover, if we apply the transformation (24) in (30), we are led to the integral Formula (33). □

The second approach to the proposed problem involves the evaluation of the generalized Mordell integral $I_2$ by breaking the integral $I_2$ into the sum of two integrals $I_1$ and $I_3$, as in the relationship (4), where $I_1$ and $I_3$ are defined by (1) and (5), respectively. The resulting integral formulas are stated as the following theorem.

Theorem 3.

Under the hypotheses of Theorem $2,$ each of the following results holds true:

$$\begin{array}{cc}\hfill I_2& ={\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t={\displaystyle \frac{\sqrt{\pi }}{(1+g)\sqrt{a}}}+{\displaystyle \frac{1}{g\sqrt{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{\Gamma \left(\frac{2r+1}{2}\right)c^{2r}}{\left(2r\right)!a^r}}\hfill \\ \hfill & ·G_{2r+1,2r+1}^{1,2r+1}\left({\displaystyle \frac{1}{g}}\left|\begin{array}{c}0,\stackrel{2r\mathrm{times}}{\overbrace{-{\displaystyle \frac{b}{c}},\cdots ,-{\displaystyle \frac{b}{c}}}}\hfill \\ \\ 0,\underset{2r\mathrm{times}}{\underbrace{1-{\displaystyle \frac{b}{c}},\cdots ,1-{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right),\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill I_2& ={\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t={\displaystyle \frac{\sqrt{\pi }}{(1+g)\sqrt{a}}}+{\displaystyle \frac{1}{2\sqrt{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{\Gamma \left(\frac{r+1}{2}\right)c^r}{r!a^{\frac{r}{2}}}}\hfill \\ \hfill & ·\left[{\displaystyle \frac{1}{g}}{G}_{r+1,r+1}^{1,r+1}\left({\displaystyle \frac{1}{g}}\left|\begin{array}{c}0,\stackrel{r\mathrm{times}}{\overbrace{-{\displaystyle \frac{b}{c}},\cdots ,-{\displaystyle \frac{b}{c}}}}\hfill \\ \\ 0,\underset{r\mathrm{times}}{\underbrace{1-{\displaystyle \frac{b}{c}},\cdots ,1-{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right)\right.\hfill \\ \hfill & \left.+{(-1)}^r{G}_{r+1,r+1}^{r+1,1}\left(g\left|\begin{array}{c}0,\stackrel{r\mathrm{times}}{\overbrace{{\displaystyle \frac{b}{c}}-1,\cdots ,{\displaystyle \frac{b}{c}}-1}}\hfill \\ 0,\underset{r\mathrm{times}}{\underbrace{{\displaystyle \frac{b}{c}},\cdots ,{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right)\right],\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill I_2& ={\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t={\displaystyle \frac{\sqrt{\pi }}{(1+g)\sqrt{a}}}+{\displaystyle \frac{1}{2\sqrt{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{\Gamma \left(\frac{r+1}{2}\right)c^r}{r!a^{\frac{r}{2}}}}\hfill \\ \hfill & ·\left[G_{r+1,r+1}^{r+1,1}\left(g\left|\begin{array}{c}0,\stackrel{r\mathrm{times}}{\overbrace{{\displaystyle \frac{b}{c}}-1,\cdots ,{\displaystyle \frac{b}{c}}-1}}\hfill \\ \\ 0,\underset{r\mathrm{times}}{\underbrace{{\displaystyle \frac{b}{c}},\cdots ,{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right)\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{{(-1)}^r}{g}}{G}_{r+1,r+1}^{1,r+1}\left({\displaystyle \frac{1}{g}}\left|\begin{array}{c}0,\stackrel{r\mathrm{times}}{\overbrace{-{\displaystyle \frac{b}{c}},\cdots ,-{\displaystyle \frac{b}{c}}}}\hfill \\ \\ 0,\underset{r\mathrm{times}}{\underbrace{1-{\displaystyle \frac{b}{c}},\cdots ,1-{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right)\right],\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill I_2& ={\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t={\displaystyle \frac{\sqrt{\pi }}{(1+g)\sqrt{a}}}+{\displaystyle \frac{1}{\sqrt{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{\Gamma \left(\frac{2r+1}{2}\right)c^{2r}}{\left(2r\right)!a^r}}\hfill \\ \hfill & ·G_{2r+1,2r+1}^{2r+1,1}\left(g\left|\begin{array}{c}0,\stackrel{2r\mathrm{times}}{\overbrace{{\displaystyle \frac{b}{c}}-1,\cdots ,{\displaystyle \frac{b}{c}}-1}}\hfill \\ 0,\underset{2r\mathrm{times}}{\underbrace{{\displaystyle \frac{b}{c}},\cdots ,{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right)\hfill \end{array}$$

(39)

and

$$\begin{array}{cc}\hfill I_2& ={\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t={\displaystyle \frac{1}{g+1}}{\displaystyle \frac{\sqrt{\pi }}{\sqrt{a}}}+{\displaystyle \frac{1}{g\sqrt{a}}}\sum _{r=1}^{\infty }{\displaystyle \frac{\Gamma \left(\frac{2r+1}{2}\right)b^{2r}}{\left(2r\right)!a^r}}\hfill \\ \hfill & ·{}_{2r+1}F_{2r}\left[\begin{array}{c}\hfill 1,\stackrel{2r\mathrm{times}}{\overbrace{{\displaystyle \frac{b}{c}}+1,\cdots ,{\displaystyle \frac{b}{c}}+1}};\\ \hfill \underset{2r\mathrm{times}}{\underbrace{{\displaystyle \frac{b}{c}},\cdots ,{\displaystyle \frac{b}{c}}}};\end{array}-{\displaystyle \frac{1}{g}}\right]\left(\right|g|>1).\hfill \end{array}$$

(40)

Proof of the Integrals 

(36)–(40). We apply relationship (4) to break the generalized Mordell integral $I_2$ into the sum of two integrals as follows:

$$\begin{array}{cc}\hfill I_2& =I_1+I_3\hfill \\ \hfill & ={\int }_0^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2+bt}}{{\mathrm{e}}^{ct}+g}}\mathrm{d}t+{\int }_0^{\infty }{\displaystyle \frac{{\mathrm{e}}^{-a{t}^2-bt}}{{\mathrm{e}}^{-ct}+g}}\mathrm{d}t.\hfill \end{array}$$

(41)

We also make use of the following rather elementary series identity:

$$\sum _{r=1}^{\infty }\left[{(-1)}^r+1\right]\Phi \left(r\right)=2\sum _{r=1}^{\infty }\Phi \left(2r\right),$$

(42)

provided that the series on each side of the identity (42) is absolutely convergent.

The derivation of series identity (42) is based essentially upon the idea of separating a power series into its even and odd terms (see, for example, ([12] Chapter 3)). Indeed, it is easily observed that

$$\begin{array}{cc}\hfill \sum _{r=1}^{\infty }\left[{(-1)}^r+1\right]\Phi \left(r\right)& =\sum _{r=0}^{\infty }\left[{(-1)}^{r+1}+1\right]\Phi (r+1)\hfill \\ \hfill & =\sum _{r=0}^{\infty }\left[{(-1)}^{2r+1}+1\right]\Phi (2r+1)\hfill \\ \hfill & +\sum _{r=0}^{\infty }\left[{(-1)}^{2r+2}+1\right]\Phi (2r+2)\hfill \\ \hfill & =0+2\sum _{r=0}^{\infty }\Phi (2r+2)\hfill \\ \hfill & =2\sum _{r=1}^{\infty }\Phi \left(2r\right),\hfill \end{array}$$

(43)

which is precisely the series on the right-hand side of the identity (42).

In conjunction with the identity (42), we now establish the values of $I_1$ and $I_3$ from the combinations of (20) and (26), of (20) and (27), of (21) and (26), and of (21) and (27). In this way, we obtain the four different evaluations (36), (37), (38), and (39), respectively, of the generalized Mordell integral $I_2$.

In the case when $\left|g\right|>1$, if we use the transformation (24) in (36) and the transformation (25) in (39), we obtain the assertion (40) of Theorem 3. □

5. Transformation Formulas Involving the G-Function

In this section, we apply some of our results to derive transformation formulas for the G-function. Indeed, by comparing the right-hand sides of the Equations (20) and (21), we get

$${\displaystyle \frac{1}{g}}{G}_{r+1,r+1}^{1,r+1}\left({\displaystyle \frac{1}{g}}\left|\begin{array}{c}0,\stackrel{r\mathrm{times}}{\overbrace{-{\displaystyle \frac{b}{c}},\cdots ,-{\displaystyle \frac{b}{c}}}}\hfill \\ \\ 0,\underset{r\mathrm{times}}{\underbrace{1-{\displaystyle \frac{b}{c}},\cdots ,1-{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right)=G_{r+1,r+1}^{r+1,1}\left(g\left|\begin{array}{c}0,\stackrel{r\mathrm{times}}{\overbrace{{\displaystyle \frac{b}{c}}-1,\cdots ,{\displaystyle \frac{b}{c}}-1}}\hfill \\ \\ 0,\underset{r\mathrm{times}}{\underbrace{{\displaystyle \frac{b}{c}},\cdots ,{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right)$$

(44)

$$\left(b,c,g\in \mathbb{C}\mathrm{with}c\ne 0\mathrm{and}g\ne 0\right).$$

Remark 2.

If we multiply each side of (44) by g and use the known relation ([14] p. 209, Equation (5).3.1 (8)) (with $\sigma =1$) for the resulting right-hand side, we obtain a G-function, which, in view of another known relation ([14] p. 209, Equation (5).3.1 (9)), is precisely the same as the G-function occurring in the left-hand side of (44).

If we compare the right-hand sides of Equations (29)–(31), we find that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{g}}{G}_{2r+1,2r+1}^{1,2r+1}\left({\displaystyle \frac{1}{g}}\left|\begin{array}{c}0,\stackrel{2r\mathrm{times}}{\overbrace{-{\displaystyle \frac{b}{c}},\cdots ,-{\displaystyle \frac{b}{c}}}}\hfill \\ \\ 0,\underset{2r\mathrm{times}}{\underbrace{1-{\displaystyle \frac{b}{c}},\cdots ,1-{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right)\hfill \\ \hfill & =G_{2r+1,2r+1}^{1,2r+1}\left(g\left|\begin{array}{c}0,\stackrel{2r\mathrm{times}}{\overbrace{{\displaystyle \frac{b}{c}}-1,\cdots ,{\displaystyle \frac{b}{c}}-1}}\hfill \\ \\ 0,\underset{2r\mathrm{times}}{\underbrace{{\displaystyle \frac{b}{c}},\cdots ,{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right)\hfill \end{array}$$

(45)

$$\left(b,c,g\in \mathbb{C}\mathrm{with}c\ne 0\mathrm{and}g\ne 0\right)$$

and

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{g}}{G}_{2r+1,2r+1}^{1,2r+1}\left({\displaystyle \frac{1}{g}}\left|\begin{array}{c}0,\stackrel{2r\mathrm{times}}{\overbrace{-{\displaystyle \frac{b}{c}},\cdots ,-{\displaystyle \frac{b}{c}}}}\hfill \\ \\ 0,\underset{2r\mathrm{times}}{\underbrace{1-{\displaystyle \frac{b}{c}},\cdots ,1-{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right)\hfill \\ \hfill & =G_{2r+1,2r+1}^{2r+1,1}\left(g\left|\begin{array}{c}0,\stackrel{2r\mathrm{times}}{\overbrace{{\displaystyle \frac{b}{c}}-1,\cdots ,{\displaystyle \frac{b}{c}}-1}}\hfill \\ \\ 0,\underset{2r\mathrm{times}}{\underbrace{{\displaystyle \frac{b}{c}},\cdots ,{\displaystyle \frac{b}{c}}}}\hfill \end{array}\right.\right)\hfill \end{array}$$

(46)

$$\left(b,c,g\in \mathbb{C}\mathrm{with}c\ne 0\mathrm{and}g\ne 0\right),$$

respectively.

Further transformation formulas for the G-function would result when we compare the remaining integrals of Section 4.

Remark 3.

A direct demonstration of each of the transformation Formulas (45) and (46), which is also based upon the known relations ([14] p. 209, Equations (5).3.1 (8) and 5.3.1 (9)), would follow the lines as detailed in Remark 2 above.

6. Concluding Remarks and Observations

Mordell’s pioneering work on the integral (2) (under the condition $\Re \left(a\right)>0)$ laid the foundation for understanding its modular properties in his two influential papers (see [2,3]). Prior studies by Siegel [20], Kronecker (see [23,24]), and Lerch (see [25,26,27]) had explored specific cases, often in the context of important mathematical results like the Riemann zeta function and Gauss sums. However, it was Mordell who systematically classified these integrals into the following two standard forms, which are now known as the Mordell integrals as follows ([2] p. 332, Equation (5); p. 341, Equation (12)):

$$\phi (z,\tau ):=\tau {\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{\pi \mathrm{i}\tau t^2-2\pi zt}}{{\mathrm{e}}^{2\pi t}-1}}\mathrm{d}t\left(\Re \left(\tau \right)<0\right)$$

(47)

and

$$\sigma (z,\tau ):={\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{\pi \mathrm{i}\tau t^2-2\pi zt}}{{\mathrm{e}}^{2\pi \tau t}-1}}\mathrm{d}t\left(\Re \left(\tau \right)<0\right).$$

(48)

The integrals (47) and (48) have significant implications in various areas of mathematics. Mordell’s analysis of their behavior under modular transformations represents a key advancement in the study of these integrals, thereby establishing their relevance in both theoretical and applied contexts. The term Mordell integral, introduced by Bellman [28], highlights the importance and lasting impact of these integrals in the mathematical literature.

The following definition of a Mordell integral $h(z;\tau )$$\left(\Im \left(\tau \right)>0\right)$, which was employed by Zwegers ([29] p. 6), is now standard in the contemporary literature:

$$h(z,\tau ):={\int }_{-\infty }^{\infty }{\displaystyle \frac{{\mathrm{e}}^{\pi \mathrm{i}\tau x^2-2\pi zx}}{cosh\pi x}}\mathrm{d}x\left(\Im \left(\tau \right)>0\right).$$

(49)

For a number of similar integrals, interested readers may refer to the papers by Mordell (see, for example, [1,3]). Moreover, several interesting developments associated with Mordell-type integrals are reported in publications including [21,30,31,32,33,34,35].

One possible direction for further research pertaining to our present investigation that may potentially motivate readers in terms of closed-form analytical evaluations of Mordell-type integrals and various other related integrals is to construct and evaluate similar types of new and useful definite integrals by means of the known integrals given in the tables of Bierens de Hann [4], Erdélyi et al. [14], Gradshteyn and Ryzhik [5], Gröbner-Hofreiter [6], Oberhettinger and Badii [10], Prudnikov et al. (see [7,36]), and so on. Remarkably, for $g=-1$ and particular values of the other parameters, a, b, and c, evaluations of various specialized forms of Mordell-type integrals have already been discussed in Section 2 and Section 6, and also in some of our cited references. Many of the results presented in this paper are fairly significant and potentially beneficial for researchers and practitioners in several diverse fields of pure and applied mathematical, physical, statistical, and engineering sciences.