Analytical Expressions of Infinite Fourier Sine and Cosine Transform-Based Ramanujan Integrals R_S,C(m, n) in Terms of Hypergeometric Series ₂F₃(⋅)

Abstract

In this chapter, we obtain analytical expressions of infinite Fourier sine and cosine transform-based Ramanujan integrals, RS,Cmn=∫0∞xm−1+exp2πxsincosπnxdx, in an infinite series of hypergeometric functions 2F3⋅, using the hypergeometric technique. Also, we have given some generalizations of the Ramanujan’s integrals RS,Cmn in the form of integrals denoted by IS,C∗υbcλy,JS,Cυbcλy,KS,Cυbcλy and IS,Cυbλy. These generalized definite integrals are expressed in terms of ordinary hypergeometric functions 2F3⋅, with suitable convergence conditions. Moreover, as applications of Ramanujan’s integrals RS,Cmn, some closed form of infinite summation formulas involving hypergeometric functions 1F2, 2F3⋅, and 0F1 are derived.

1. Introduction

Naturally, we call a function”special” when the function, just as the logarithm, the exponential and trigonometric functions (the elementary transcendental functions), belongs to the toolbox of the applied mathematician, the physicist, or the engineer. This branch of mathematics has a good history with great names such as Gauss, Euler, Fourier, Legendre, and Bessel. This chapter includes definitions, namely infinite Fourier sine and cosine transforms, Pochhammer’s symbol and related results, generalized Gauss hypergeometric function and its special cases, Fox-Wright hypergeometric function and its convergence conditions, Hypergeometric form of elementary functions, Gauss-Legendre multiplication formula and infinite series decomposition identity. In the literature [1, 2, 3, 4, 5, 6], the analytical expressions of the Fourier sine and cosine transforms of $x^{\upsilon -1}\backslash \left(exp\left(\mathit{bx}\right)\pm 1\right)$ are available in terms of Riemann’s zeta function, the Psi function (Digamma function), hyperbolic function and Beta function. The analytical solution of the following infinite Fourier sine and cosine transforms based Ramanujan integrals ([7], p. 85, eq. (49) last line):

$${\mathbf{R}}_{S,C}\left(m,n\right)={\int }_0^{\infty }\frac{x^m}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\begin{array}{c}sin\\ cos\end{array}\left(\mathit{\pi nx}\right)\mathit{dx},$$E1

are not given for all positive rational values of $n$ and non-negative integral values of $m$.

2. Definitions and preliminaries

2.1 Fourier sine and cosine transforms

The infinite Fourier sine and cosine transforms of $g\left(x\right)$ over the interval $\left[0,\infty \right)$ are defined by

$$F_{S,C}\left(g\left(x\right);b\right)={\int }_0^{\infty }g\left(x\right)\begin{array}{c}sin\\ cos\end{array}\left(\mathit{bx}\right)\mathit{dx}=G_{S,C}\left(b\right),\left(b>0\right).$$E2

For example, if $y>0$, $0<\mathit{Re}\left(\upsilon \right)<2$ for Fourier sine transform of $x^{-\upsilon }$ and $y>0$, $0<\mathit{Re}\left(\upsilon \right)<1$ for Fourier cosine transform of $x^{-\upsilon }$, then the infinite Fourier sine and cosine transforms of $x^{-\upsilon }$ ([3], p. 68) are given by

$${\int }_0^{\infty }{x}^{-\upsilon }\begin{array}{c}sin\\ cos\end{array}\left(\mathit{xy}\right)\mathit{dx}=y^{\upsilon -1}\mathrm{\Gamma }\left(1-\upsilon \right)\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\upsilon \pi }}{2}\right)$$E3

Further, if $b>0$, $-1<\Re \left(s\right)<1$ for Fourier sine transform and $b>0$, $0<\Re \left(s\right)<1$ for Fourier cosine transform, then the infinite Fourier sine and cosine transform of $x^{s-1}$ are given by [3, 5, 8].

$${\int }_0^{\infty }{x}^{s-1}\begin{array}{c}sin\\ cos\end{array}\left(\mathit{bx}\right)\mathit{dx}=\frac{\mathrm{\Gamma }\left(s\right)\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\pi s}}{2}\right)}{b^s}.$$E4

Moreover, if $\Re \left(\mu \right)>-2$ for Fourier sine transform and $\Re \left(\mu \right)>-1$ for Fourier cosine transform, then we can prove the following integral by using Maclaurin’s expansion of $exp\left(-{\mathit{ax}}^{\xi }\right)$ and term by term integrating with the help of the result (4)

$${\int }_0^{\infty }{x}^{\mu }\mathit{exp}\left(-{\mathit{ax}}^{\xi }\right)\begin{array}{c}sin\\ cos\end{array}\left(\mathit{xy}\right)\mathit{dx}=y^{-\mu -1}\sum _{\mathrm{\ell }=0}^{\infty }{\left(-\frac{a}{y^{\xi }}\right)}^{\mathrm{\ell }}\frac{1}{\mathrm{\ell }!}\mathrm{\Gamma }\left(\mu +1+\xi \mathrm{\ell }\right)\begin{array}{c}sin\\ cos\end{array}\left\{\frac{\pi }{2}\left(\mu +\xi \mathrm{\ell }\right)\right\}.$$E5

where $0<\xi <1$, $a>0$ and $y>0$. The conditions $\Re \left(\mu \right)>-2$ and $\Re \left(\mu \right)>-1$ stated in the integrals (5) follows from the theory of analytic continuation [5, 8]. We have also verified the conditions $\Re \left(\mu \right)>-2$ and $\Re \left(\mu \right)>-1$, using Wolfram Mathematica software.

2.2 Generalized gauss hypergeometric function

A natural generalization of the Gauss hypergeometric function ${}_2{F}_1\left(z\right)$ is the generalized hypergeometric function ${}_p{F}_q\left(z\right)$ with $p$ numerator parameters ${\alpha }_1,\dots ,{\alpha }_p$ and $q$ denominator parameters ${\beta }_1,\dots ,{\beta }_q$ defined by [9].

$${}_p{F}_q\left(\begin{array}{r}{\alpha }_1,\dots ,{\alpha }_p;\\ {\beta }_1,\dots ,{\beta }_q;\end{array}z\right)=\sum _{n=0}^{\infty }\frac{{\left({\alpha }_1\right)}_n\dots {\left({\alpha }_p\right)}_n}{{\left({\beta }_1\right)}_n\dots {\left({\beta }_q\right)}_n}\frac{z^n}{n!},$$E6

where ${\alpha }_j\in \mathbb{C}\left(j=1,\dots ,p\right)$, ${\beta }_j\in \mathbb{C}\backslash {\mathbb{Z}}_0^{-}\left(j=1,\dots ,q\right)$ and $p,q\in {\mathbb{N}}_0$. Then the hypergeometric ${}_p{F}_q\left(z\right)$ function in (6) converges absolutely for $\mid z\mid <\infty$ when $p\le q$ and for $\mid z\mid <1$ when $p=q+1$. Furthermore, if we set,

$$\omega ≔\left(\sum _{j=1}^q{\beta }_j-\sum _{j=1}^p{\alpha }_j\right),$$E7

it is known that when $p=q+1$ the function ${}_p{F}_q\left(z\right)$ is absolutely convergent for $\mid z\mid =1$ if $\Re \left(\omega \right)>0$, conditionally convergent for $\mid z\mid =1$ ($z\ne 1$) if $-1<\Re \left(\omega \right)<0$ and divergent for $\mid z\mid =1$ if $\Re \left(\omega \right)\le -1$.

2.3 Hypergeometric form of elementary functions

The important special cases of ${}_p{F}_q\left(z\right)$ include (for example) the binomial series ${}_1{F}_0\left(z\right)$ given by [9].

$${\left(1-z\right)}^{-a}{=}_1{F}_0\left(\begin{array}{r}a;\\ \overline{};\end{array}z\right)=\sum _{n=0}^{\infty }\frac{{\left(a\right)}_n}{n!}{z}^n,$$E8

where $\mid z\mid <1$, $a\in \mathbb{C}$.

Elementary trigonometric functions ([10], p. 44, eq. (9) and eq. (10)) are given by

$$cosz{=}_0{F}_1\left(\begin{array}{r}\overline{};\\ \frac{1}{2};\end{array}\frac{-z^2}{4}\right),$$E9

$$sinz=z_0{F}_1\left(\begin{array}{r}\overline{};\\ \frac{3}{2};\end{array}\frac{-z^2}{4}\right).$$E10

Lommel function ([10], p. 44, eq. (13)) is given by

$$s_{\mu ,\upsilon }\left(z\right)=\frac{z^{\mu +1}}{\left(\mu -\upsilon +1\right)\left(\mu +\upsilon +1\right)}{}_1{F}_2\left(\begin{array}{r}1;\\ \frac{\mu -\upsilon +3}{2},\frac{\mu +\upsilon +3}{2};\end{array}\frac{-z^2}{4}\right),$$E11

where $\mu \pm \upsilon \in \mathbb{C}\backslash \left\{-1,-3,-5,-7,\dots \right\}$.

Struve function ([10], p. 44, eq. (16)) is given by

$$H_{\upsilon }\left(z\right)=\frac{2{\left(\frac{z}{2}\right)}^{\upsilon +1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\upsilon +\frac{3}{2}\right)}{}_1{F}_2\left(\begin{array}{r}1;\\ \frac{3}{2},\upsilon +\frac{3}{2};\end{array}\frac{-z^2}{4}\right).$$E12

Modified Struve function ([10], p. 45, eq. (17)) is given by

$$L_{\upsilon }\left(z\right)=\frac{2{\left(\frac{z}{2}\right)}^{\upsilon +1}}{\sqrt{\pi }\mathrm{\Gamma }\left(\upsilon +\frac{3}{2}\right)}{}_1{F}_2\left(\begin{array}{r}1;\\ \frac{3}{2},\upsilon +\frac{3}{2};\end{array}\frac{z^2}{4}\right).$$E13

2.4 Pochhammer’s symbol

Here ${\left(\lambda \right)}_{\upsilon }\left(\lambda ,\upsilon \in \mathbb{C}\right)$ denotes the Pochhammer’s symbol (or the shifted factorial, since ${\left(1\right)}_n=n!)$ is defined, in general, by [10].

$${\left(\lambda \right)}_{\upsilon }≔\frac{\mathrm{\Gamma }\left(\lambda +\upsilon \right)}{\mathrm{\Gamma }\left(\lambda \right)}=\left\{\begin{array}{lll}1,& \left(\upsilon =0;\lambda \in \mathbb{C}\backslash \left\{0\right\}\right)& \\ \lambda \left(\lambda +1\right)\dots \left(\lambda +n-1\right),& \left(\upsilon =n\in \mathbb{N};\lambda \in \mathbb{C}\right).\end{array}\right.$$E14

Algebraic property of Pochhammer symbol:

$${\left(\lambda \right)}_{m+n}={\left(\lambda \right)}_m{\left(\lambda +m\right)}_n={\left(\lambda \right)}_n{\left(\lambda +n\right)}_m.$$E15

2.5 Gauss-Legendre multiplication formula

For every positive integer $m$ ([10], p. 22, eq. (26)), we have

$${\left(\lambda \right)}_{\mathit{mn}}=m^{\mathit{mn}}\prod _{j=1}^m{\left(\frac{\lambda +j-1}{m}\right)}_n;m\in \mathbb{N},n\in {\mathbb{N}}_0.$$E16

From the above result (16) with $\lambda =\mathit{mz}$, it can be proved that

$$\mathrm{\Gamma }\left(\mathit{mz}\right)={\left(2\pi \right)}^{\frac{\left(1-m\right)}{2}}{m}^{\mathit{mz}-\frac{1}{2}}\prod _{j=1}^m\mathrm{\Gamma }\left(z+\frac{j-1}{m}\right),$$E17

where $z\ne 0,-\frac{1}{m},-\frac{2}{m},.\dots ;m\in \mathbb{N}$.

The eq. (17) is known as Gauss-Legendre multiplication formula for Gamma function.

2.6 Legendre’s duplication formula

When we put $m=2$ in the eq. (17), we get

$$\sqrt{\pi }\mathrm{\Gamma }\left(2z\right)=2^{2z-1}\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(z+\frac{1}{2}\right),2z\in \mathbb{C}\backslash {\mathbb{Z}}_0^{-},$$E18

which is known as Legendre’s duplication formula.

2.7 Infinite series decomposition identity

An infinite series decomposition identity ([11], p. 193, eq. (8)) is given by

$$\sum _{\mathrm{\ell }=0}^{\infty }\mathrm{\Omega }\left(\mathrm{\ell }\right)=\sum _{j=0}^{N-1}\left\{\sum _{\mathrm{\ell }=0}^{\infty }\mathrm{\Omega }\left(N\mathrm{\ell }+j\right)\right\},$$E19

where $N$ is an arbitrary positive integer. Put $N=4$ in the above eq. (19), we get

$$\sum _{\mathrm{\ell }=0}^{\infty }\mathrm{\Omega }\left(\mathrm{\ell }\right)=\sum _{j=0}^3\left\{\sum _{\mathrm{\ell }=0}^{\infty }\mathrm{\Omega }\left(4\mathrm{\ell }+j\right)\right\},$$E20

$$=\sum _{\mathrm{\ell }=0}^{\infty }\mathrm{\Omega }\left(4\mathrm{\ell }\right)+\sum _{\mathrm{\ell }=0}^{\infty }\mathrm{\Omega }\left(4\mathrm{\ell }+1\right)+\sum _{\mathrm{\ell }=0}^{\infty }\mathrm{\Omega }\left(4\mathrm{\ell }+2\right)+\sum _{\mathrm{\ell }=0}^{\infty }\mathrm{\Omega }\left(4\mathrm{\ell }+3\right),$$E21

provided that all involved infinite series are absolutely convergent.

2.8 Fox-Wright psi function of one variable

A natural generalization of the hypergeometric function ${}_p{F}_q\left(z\right)$ is the Fox-Wright psi function of one variable with $p$ pairs of numerator parameters $\left({\alpha }_1,A_1\right),\dots ,\left({\alpha }_p,A_p\right)$ and $q$ pairs of denominator parameters $\left({\beta }_1,B_1\right),\dots ,\left({\beta }_q,B_q\right)$, defined by [12, 13].

$${}_p{\mathrm{\Psi }}_q\left[\begin{array}{lll}\left({\alpha }_1,A_1\right),\dots ,\left({\alpha }_p,A_p\right);& & \\ \left({\beta }_1,B_1\right),\dots ,\left({\beta }_q,B_q\right);& & \end{array}z\right]=\sum _{k=0}^{\infty }\frac{\mathrm{\Gamma }\left({\alpha }_1+{\mathit{kA}}_1\right)\dots \mathrm{\Gamma }\left({\alpha }_p+{\mathit{kA}}_p\right)}{\mathrm{\Gamma }\left({\beta }_1+{\mathit{kB}}_1\right)\dots \mathrm{\Gamma }\left({\beta }_q+{\mathit{kB}}_q\right)}\frac{z^k}{k!},$$E22

$$=\frac{1}{2\mathit{\pi \rho }}{\int }_L\frac{\mathrm{\Gamma }\left(\zeta \right){\prod }_{i=1}^p\mathrm{\Gamma }\left({\alpha }_i-A_i\zeta \right)}{{\prod }_{j=1}^q\mathrm{\Gamma }\left({\beta }_j-B_j\zeta \right)}{\left(-z\right)}^{-\zeta }\mathit{d\zeta },$$E23

where $\rho =\sqrt{-1},z\in \mathbb{C};$ parameters ${\alpha }_i,{\beta }_j\in \mathbb{C}$; coefficients $A_i,B_j\in \mathbb{R}=\left(-\infty ,+\infty \right)$ in case of series (22) (or $A_i,B_j\in {\mathbb{R}}_{+}=\left(0,+\infty \right)$ in case of contour integral (23)), $A_i\ne 0\left(i=1,2,\dots ,p\right),B_j\ne 0\left(j=1,2,\dots ,q\right)$. In eq. (22), the parameters ${\alpha }_i,{\beta }_j$ and coefficients $A_i,B_j$ are adjusted in such a way that the product of Gamma functions in numerator and denominator should be well defined.

Suppose:

$${\mathrm{\Delta }}^{\ast }=\left(\sum _{j=1}^q{B}_j-\sum _{i=1}^p{A}_i\right),$$E24

$${\delta }^{\ast }=\left(\prod _{i=1}^p{\left|A_i\right|}^{-A_i}\right)\left(\prod _{j=1}^q{\left|B_j\right|}^{B_j}\right),$$E25

$${\mu }^{\ast }=\sum _{j=1}^q{\beta }_j-\sum _{i=1}^p{\alpha }_i+\left(\frac{p-q}{2}\right),$$E26

and

$${\sigma }^{\ast }=\left(1+A_1+\dots +A_p\right)-\left(B_1+\dots +B_q\right)=1-{\mathrm{\Delta }}^{\ast }.$$E27

Then we have the following convergence conditions of (22) or (23).

Case (1): When contour $\left(L\right)$ is a left loop beginning and ending at $-\infty$, then ${}_p{\mathrm{\Psi }}_q\left[\cdot \right]$ given by (22) or (23) holds the following convergence conditions.

1. When ${\mathrm{\Delta }}^{\ast }>-1,$$0<\mid z\mid <\infty$, $z\ne 0$.

2. When ${\mathrm{\Delta }}^{\ast }=-1$, $0<\mid z\mid <{\delta }^{\ast }$.

3. When ${\mathrm{\Delta }}^{\ast }=-1$,$\mid z\mid ={\delta }^{\ast }$, and $\Re \left({\mu }^{\ast }\right)>\frac{1}{2}$.

Case (2): When contour $\left(L\right)$ is a right loop beginning and ending at $+\infty$, then ${}_p{\mathrm{\Psi }}_q\left[\cdot \right]$ given by (22) or (23) holds the following convergence conditions.

1. When ${\mathrm{\Delta }}^{\ast }<-1$, $0<\mid z\mid <\infty ,z\ne 0$.

2. When ${\mathrm{\Delta }}^{\ast }=-1$, $\mid z\mid >{\delta }^{\ast }$.

3. When ${\mathrm{\Delta }}^{\ast }=-1$,$\mid z\mid ={\delta }^{\ast }$, and $\Re \left({\mu }^{\ast }\right)>\frac{1}{2}$.

Case (3): When contour $\left(L\right)$ is starting from $\gamma -i\infty$ and ending at $\gamma +i\infty$ where $\gamma \in \mathbb{R}=\left(-\infty ,+\infty \right)$, then ${}_p{\mathrm{\Psi }}_q\left[\cdot \right]$ is also convergent under the following conditions.

1. When ${\sigma }^{\ast }>0$, $\mid \mathit{arg}\left(-z\right)\mid <\frac{\pi }{2}{\sigma }^{\ast }$, $0<\mid z\mid <\infty ,z\ne 0$.

2. When ${\sigma }^{\ast }=0$, $\mathit{arg}\left(-z\right)=0$, $0<\mid z\mid <\infty ,z\ne 0$ such that $-\gamma {\mathrm{\Delta }}^{\ast }+\Re \left({\mu }^{\ast }\right)>\frac{1}{2}+\gamma$.

3. When $\gamma =0$, ${\sigma }^{\ast }=0,\mathit{arg}\left(-z\right)=0$, $0<\mid z\mid <\infty ,z\ne 0$, such that $\Re \left({\mu }^{\ast }\right)>\frac{1}{2}$.

In the available literature [7, 14, 15, 16, 17, 18] on Ramanujan’s Mathematics, the analytical expression of Ramanujan’s integrals ${\mathbf{R}}_{S,C}\left(m,n\right)$ are not given. Therefore, the main object of this chapter is to evaluate the representation of ${\mathbf{R}}_{S,C}\left(m,n\right)$ in an ordinary hypergeometric function ${}_2{F}_3\left(\cdot \right)$. Also, our contribution to Ramanujan’s Mathematics is determined by the result in [19, 20]. Here in this chapter, we generalize Ramanujan’s integrals ${\mathbf{R}}_{S,C}\left(m,n\right)$ in the following forms:

$${\mathbf{I}}_{S,C}^{\ast }\left(\upsilon ,b,c,\lambda ,y\right)=\sum _{k=0}^{\infty }\frac{\mathrm{\Theta }\left(k\right)}{k!}{\int }_0^{\infty }{x}^{\upsilon -1}{e}^{-\left(\mathit{\lambda b}+\mathit{ck}\right)\sqrt{x}}\begin{array}{c}sin\\ cos\end{array}\left(\mathit{xy}\right)\mathit{dx},$$

$${\mathbf{J}}_{S,C}\left(\upsilon ,b,c,\lambda ,y\right)={\int }_0^{\infty }{x}^{\upsilon -1}{e^{-\mathit{b\lambda }\sqrt{x}}}_r{\mathrm{\Psi }}_s\left[\begin{array}{lll}\left({\alpha }_1,A_1\right),\dots ,\left({\alpha }_r,A_r\right);& & \\ \left({\beta }_1,B_1\right),\dots ,\left({\beta }_s,B_s\right);& & \end{array}{e}^{-c\sqrt{x}}\right]\begin{array}{c}sin\\ cos\end{array}\left(\mathit{xy}\right)\mathit{dx},$$

$${\mathbf{K}}_{S,C}\left(\upsilon ,b,c,\lambda ,y\right)={\int }_0^{\infty }{x}^{\upsilon -1}{e^{-\mathit{b\lambda }\sqrt{x}}}_r{F}_s\left(\begin{array}{lll}{\alpha }_1,\dots ,{\alpha }_r;& & \\ {\beta }_1,\dots ,{\beta }_s;& & \end{array}{e}^{-c\sqrt{x}}\right)\begin{array}{c}sin\\ cos\end{array}\left(\mathit{xy}\right)\mathit{dx},$$

$${\mathbf{I}}_{S,C}\left(\upsilon ,b,\lambda ,y\right)={\int }_0^{\infty }{x}^{\upsilon -1}{\left\{-1+exp\left(b\sqrt{x}\right)\right\}}^{-\lambda }\begin{array}{c}sin\\ cos\end{array}\left(\mathit{xy}\right)\mathit{dx},$$

where ${\left\{\mathrm{\Theta }\left(k\right)\right\}}_{k=0}^{\infty }$ is a fixed sequence of the arbitrary real or complex numbers. Moreover, we also show how the main general theorem given below applies to obtaining new interesting results by suitable adjustments in parameters and variables.

3. Ramanujan’s integrals

The analytical solution of the following integral of Ramanujan ([7], p. 85, eq. (49) last line):

$${\mathbf{R}}_C\left(m,n\right)={\int }_0^{\infty }{x}^m\frac{cos\left(\mathit{\pi nx}\right)}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\mathit{dx},$$E28

is not given for all positive rational values of $n$ and non-negative integral values of $m$.

For particular values of $m$ and $n$ in Ramanujan’s integral ${\mathbf{R}}_C\left(m,n\right)$, the following three integrals are given by ([7], p. 86, eq. (50)):

$${\mathbf{R}}_C\left(1,1/2\right)={\int }_0^{\infty }\frac{xcos\left(\frac{\mathit{\pi x}}{2}\right)}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\mathit{dx}=\frac{13-4\pi }{8{\pi }^2},$$E29

$${\mathbf{R}}_C\left(1,2\right)={\int }_0^{\infty }\frac{xcos\left(2\mathit{\pi x}\right)}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\mathit{dx}=\frac{1}{64}\left(\frac{1}{2}-\frac{3}{\pi }+\frac{5}{{\pi }^2}\right),$$E30

$${\mathbf{R}}_C\left(2,2\right)={\int }_0^{\infty }\frac{x^{2}cos\left(2\mathit{\pi x}\right)}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\mathit{dx}=\frac{1}{256}\left(1-\frac{5}{\pi }+\frac{5}{{\pi }^2}\right).$$E31

The following theorem is proved by Ramanujan ([7], p. 76, 77, eq. (10 and 10′)).

Theorem 1.3.1. Let n be real and positive. Then if

$$R_C\left(0,n\right)=\mathrm{\Phi }\left(n\right)={\int }_0^{\infty }\frac{cos\left(\mathit{\pi nx}\right)}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\mathit{dx},$$E32

and

$$\mathrm{\Upsilon }\left(n\right)-\frac{1}{2\mathit{\pi n}}={\int }_0^{\infty }\frac{sin\left(\mathit{\pi nx}\right)}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\mathit{dx}=R_S\left(0,n\right),$$E33

then

$${\mathit{R}}_C\left(0,n\right)=\mathrm{\Phi }\left(n\right)=\frac{1}{n}\sqrt{\left(\frac{2}{n}\right)}\mathrm{\Upsilon }\left(\frac{1}{n}\right)-\mathrm{\Upsilon }\left(n\right),$$E34

and

$$\mathrm{\Upsilon }\left(n\right)=\frac{1}{n}\sqrt{\left(\frac{2}{n}\right)}\mathrm{\Phi }\left(\frac{1}{n}\right)+\mathrm{\Phi }\left(n\right),$$E35

For particular values of $n$, some values of Ramanujan’s integral ([7], p. 85 (eq. 48)) are given below

$${\mathbf{R}}_C\left(0,1\right)=\mathrm{\Phi }\left(1\right)={\int }_0^{\infty }\frac{cos\left(\mathit{\pi x}\right)}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\mathit{dx}=\frac{2-\sqrt{2}}{8},$$E36

$${\mathbf{R}}_C\left(0,2\right)=\mathrm{\Phi }\left(2\right)={\int }_0^{\infty }\frac{cos\left(2\mathit{\pi x}\right)}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\mathit{dx}=\frac{1}{16},$$E37

$${\mathbf{R}}_C\left(0,4\right)=\mathrm{\Phi }\left(4\right)={\int }_0^{\infty }\frac{cos\left(4\mathit{\pi x}\right)}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\mathit{dx}=\frac{3-\sqrt{2}}{32},$$E38

$${\mathbf{R}}_C\left(0,6\right)=\mathrm{\Phi }\left(6\right)={\int }_0^{\infty }\frac{cos\left(6\mathit{\pi x}\right)}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\mathit{dx}=\frac{13-4\sqrt{3}}{144},$$E39

$${\mathbf{R}}_C\left(0,1/2\right)=\mathrm{\Phi }\left(\frac{1}{2}\right)={\int }_0^{\infty }\frac{cos\left(\frac{\mathit{\pi x}}{2}\right)}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\mathit{dx}=\frac{1}{4\pi },$$E40

$${\mathbf{R}}_C\left(0,2/5\right)=\mathrm{\Phi }\left(\frac{2}{5}\right)={\int }_0^{\infty }\frac{cos\left(\frac{2\mathit{\pi x}}{5}\right)}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\mathit{dx}=\frac{8-3\sqrt{5}}{16}.$$E41

By calculation of (7) and (9), we get the following infinite fourier sine transform of $-1+exp\left(2\pi \sqrt{x}\right)$

$${\int }_0^{\infty }\frac{sin\left(\mathit{\pi nx}\right)}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\mathit{dx}=\frac{1}{n}\sqrt{\left(\frac{2}{n}\right)}\mathrm{\Phi }\left(\frac{1}{n}\right)+\mathrm{\Phi }\left(n\right)-\frac{1}{2\mathit{\pi n}}.$$E42

For special values of $n=1,2,\frac{1}{2}$ in the above eq. (16) and using $\mathrm{\Phi }\left(1\right),\mathrm{\Phi }\left(2\right)$ and $\mathrm{\Phi }\left(\frac{1}{2}\right)$, we get after simplification the following three results:

$${\mathbf{R}}_S\left(0,1\right)={\int }_0^{\infty }\frac{sin\left(\mathit{\pi x}\right)}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\mathit{dx}=\frac{\pi \sqrt{2}-4}{8\pi },$$E43

$${\mathbf{R}}_S\left(0,2\right)={\int }_0^{\infty }\frac{sin\left(2\mathit{\pi x}\right)}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\mathit{dx}=\frac{\pi -2}{16\pi },$$E44

$${\mathbf{R}}_S\left(0,1/2\right)={\int }_0^{\infty }\frac{sin\left(\frac{\mathit{\pi x}}{2}\right)}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\mathit{dx}=\frac{\pi -3}{4\pi }.$$E45

4. Main general theorems on infinite Fourier sine and cosine transform

In this section, we give some generalizations of the infinite Fourier sine and cosine transform-based Ramanujan integrals ${\mathbf{R}}_{S,C}\left(\cdot \right)$ in the form of infinite series of hypergeometric functions ${}_2{F}_3\left(\cdot \right)$. Moreover, we denote these generalizations by ${\mathbf{I}}_{S,C}^{\ast }\left(\cdot \right)$, ${\mathbf{J}}_{S,C}\left(\cdot \right)$, ${\mathbf{K}}_{S,C}\left(\cdot \right)$ and ${\mathbf{I}}_{S,C}\left(\cdot \right)$ [21, 22].

Theorem 1.4.1. Suppose${\left\{\mathrm{\Theta }\left(k\right)\right\}}_{k=0}^{\infty }$is a fixed sequence of arbitrary real or complex numbers and satisfy the conditions$\Re \left(\upsilon \right)>-1,c>0,y>0;\lambda >0,b>0\left(\mathit{\text{or}}\lambda <0,b<0\right)$.

then we have

$${\mathbf{I}}_{S,C}^{\ast }\left(\upsilon ,b,c,\lambda ,y\right)=\sum _{k=0}^{\infty }\frac{\mathrm{\Theta }\left(k\right)}{k!}{\int }_0^{\infty }{x}^{\upsilon -1}{e}^{-\left(\mathit{\lambda b}+\mathit{ck}\right)\sqrt{x}}\begin{array}{c}sin\\ cos\end{array}\left(\mathit{xy}\right)\mathit{dx},$$E46

$$=y^{-\upsilon }\sum _{k=0}^{\infty }\frac{\mathrm{\Theta }\left(k\right)}{k!}\sum _{\mathrm{\ell }=0}^{\infty }\frac{{\left(-1\right)}^{\mathrm{\ell }}{\left(\mathit{\lambda b}+\mathit{ck}\right)}^{\mathrm{\ell }}\mathrm{\Gamma }\left(\upsilon +\frac{\mathrm{\ell }}{2}\right)}{y^{\frac{\mathrm{\ell }}{2}}\mathrm{\ell }!}\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\upsilon \pi }}{2}+\frac{\mathrm{\ell }\pi }{4}\right),$$E47

Now replacing $\mathrm{\ell }$ by $4\mathrm{\ell }+j$, after simplification we get

$$\begin{array}{l}{\mathbf{I}}_{S,C}^{\ast }\left(\upsilon ,b,c,\lambda ,y\right)=y^{-\upsilon }\sum _{k=0}^{\infty }\frac{\mathrm{\Theta }\left(k\right)}{k!}\sum _{j=0}^3\frac{{\left(-1\right)}^j{\left(\mathit{\lambda b}+\mathit{ck}\right)}^j\mathrm{\Gamma }\left(\upsilon +\frac{j}{2}\right)}{y^{\frac{j}{2}}j!}\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\upsilon \pi }}{2}+\frac{\mathit{j\pi }}{4}\right)\\ {\times }_2{F}_3\left(\begin{array}{lll}\mathrm{\Delta }\left(2;\frac{2\upsilon +j}{2}\right);& & \\ {\mathrm{\Delta }}^{\ast }\left(4;1+j\right);& & \end{array}\frac{-1}{64y^2}{\left\{\frac{\left(\mathit{\lambda b}\right){\left(\frac{\mathit{\lambda b}+c}{c}\right)}_k}{{\left(\frac{\mathit{\lambda b}}{c}\right)}_k}\right\}}^4\right),\end{array}$$E48

$$\begin{array}{l}=y^{-\upsilon }\sum _{k=0}^{\infty }\frac{\mathrm{\Theta }\left(k\right)}{k!}\sum _{j=0}^3\frac{{\left(-1\right)}^j\mathrm{\Gamma }\left(\upsilon +\frac{j}{2}\right)}{j!}\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\upsilon \pi }}{2}+\frac{\mathit{j\pi }}{4}\right){\left(\frac{\mathit{\lambda b}}{\sqrt{y}}\right)}^j\\ \times {\left\{\frac{{\left(\frac{\mathit{\lambda b}+c}{c}\right)}_k}{{\left(\frac{\mathit{\lambda b}}{c}\right)}_k}\right\}}^j{}_2{F}_3\left(\begin{array}{lll}\mathrm{\Delta }\left(2;\frac{2\upsilon +j}{2}\right);& & \\ {\mathrm{\Delta }}^{\ast }\left(4;1+j\right);& & \end{array}\frac{-1}{64y^2}{\left\{\frac{\left(\mathit{\lambda b}\right){\left(\frac{\mathit{\lambda b}+c}{c}\right)}_k}{{\left(\frac{\mathit{\lambda b}}{c}\right)}_k}\right\}}^4\right),\end{array}$$E49

$$\begin{array}{l}=\frac{\mathrm{\Gamma }\left(\upsilon \right)\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\upsilon \pi }}{2}\right)}{y^{\upsilon }}\sum _{k=0}^{\infty }\frac{\mathrm{\Theta }\left(k\right)}{k!}{}_2{F}_3\left(\begin{array}{lll}\frac{\upsilon }{2},\frac{\upsilon +1}{2};& & \\ \frac{1}{4},\frac{1}{2},\frac{3}{4};& & \end{array}\frac{-1}{64y^2}{\left\{\frac{\left(\mathit{\lambda b}\right){\left(\frac{\mathit{\lambda b}+c}{c}\right)}_k}{{\left(\frac{\mathit{\lambda b}}{c}\right)}_k}\right\}}^4\right)\\ -\frac{\left(\mathit{\lambda b}\right)\mathrm{\Gamma }\left(\upsilon +\frac{1}{2}\right)\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\upsilon \pi }}{2}+\frac{\pi }{4}\right)}{y^{\upsilon +\frac{1}{2}}}\sum _{k=0}^{\infty }\frac{\mathrm{\Theta }\left(k\right)}{k!}\left\{\frac{{\left(\frac{\mathit{\lambda b}+c}{c}\right)}_k}{{\left(\frac{\mathit{\lambda b}}{c}\right)}_k}\right\}\\ {\times }_2{F}_3\left(\begin{array}{lll}\frac{2\upsilon +1}{4},\frac{2\upsilon +3}{4};& & \\ \frac{1}{2},\frac{3}{4},\frac{5}{4};& & \end{array}\frac{-1}{64y^2}{\left\{\frac{\left(\mathit{\lambda b}\right){\left(\frac{\mathit{\lambda b}+c}{c}\right)}_k}{{\left(\frac{\mathit{\lambda b}}{c}\right)}_k}\right\}}^4\right)\\ +\frac{{\left(\mathit{\lambda b}\right)}^2\mathrm{\Gamma }\left(\upsilon +1\right)\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\upsilon \pi }}{2}\right)}{2y^{\upsilon +1}}\sum _{k=0}^{\infty }\frac{\mathrm{\Theta }\left(k\right)}{k!}{\left\{\frac{{\left(\frac{\mathit{\lambda b}+c}{c}\right)}_k}{{\left(\frac{\mathit{\lambda b}}{c}\right)}_k}\right\}}^2\\ {\times }_2{F}_3\left(\begin{array}{lll}\frac{\upsilon +1}{2},\frac{\upsilon +2}{2};& & \\ \frac{3}{4},\frac{5}{4},\frac{3}{2};& & \end{array}\frac{-1}{64y^2}{\left\{\frac{\left(\mathit{\lambda b}\right){\left(\frac{\mathit{\lambda b}+c}{c}\right)}_k}{{\left(\frac{\mathit{\lambda b}}{c}\right)}_k}\right\}}^4\right)\\ -\frac{{\left(\mathit{\lambda b}\right)}^3\mathrm{\Gamma }\left(\upsilon +\frac{3}{2}\right)\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\upsilon \pi }}{2}+\frac{\pi }{4}\right)}{6y^{\upsilon +\frac{3}{2}}}\sum _{k=0}^{\infty }\frac{\mathrm{\Theta }\left(k\right)}{k!}{\left\{\frac{{\left(\frac{\mathit{\lambda b}+c}{c}\right)}_k}{{\left(\frac{\mathit{\lambda b}}{c}\right)}_k}\right\}}^3\\ {\times }_2{F}_3\left(\begin{array}{lll}\frac{2\upsilon +3}{4},\frac{2\upsilon +5}{4};& & \\ \frac{5}{4},\frac{3}{2},\frac{7}{4};& & \end{array}\frac{-1}{64y^2}{\left\{\frac{\left(\mathit{\lambda b}\right){\left(\frac{\mathit{\lambda b}+c}{c}\right)}_k}{{\left(\frac{\mathit{\lambda b}}{c}\right)}_k}\right\}}^4\right).\end{array}$$E50

Our result (30)or (31) or (32) is convergent in view of the convergence condition of ${}_p{F}_q\left(\cdot \right)$ series, when $p\le q$, and $\forall \mid z\mid <\infty$.

Proof: The result (29) is obtained by the application of the integral (49) [with substitutions $\mu =\upsilon -1,a=\mathit{\lambda b}+\mathit{ck},\xi =\frac{1}{2}$] in the R.H.S. of eq. (28). Also, we calculate the results (30) to (32) by using the infinite series decomposition identity (20) and (21) and algebraic properties of Pochhammer’s symbols.

4.1 Analytical expressions of infinite Fourier sine and cosine transforms

Theorem 1.4.2. Analytical expressions of the infinite Fourier sine and cosine transforms of$x^{\upsilon -1}{e^{-\mathit{b\lambda }\sqrt{x}}}_r{\mathrm{\Psi }}_s\left[\cdot \right]$holds true for$\Re \left(\upsilon \right)>-1;c>0,y>0;\lambda >0,b>0\left(\mathit{\text{or}}\lambda <0,b<0\right)$then we have

$${\mathit{J}}_{S,C}\left(\upsilon ,b,c,\lambda ,y\right)={\int }_0^{\infty }{x}^{\upsilon -1}{e^{-\mathit{b\lambda }\sqrt{x}}}_r{\mathrm{\Psi }}_s\left[\begin{array}{lll}\left({\alpha }_1,A_1\right),\dots ,\left({\alpha }_r,A_r\right);& & \\ \left({\beta }_1,B_1\right),\dots ,\left({\beta }_s,B_s\right);& & \end{array}{e}^{-c\sqrt{x}}\right]\begin{array}{c}sin\\ cos\end{array}\left(\mathit{xy}\right)\mathit{dx},$$E51

$$\begin{array}{l}=y^{-\upsilon }\sum _{k=0}^{\infty }\frac{\mathrm{\Gamma }\left({\alpha }_1+{\mathit{kA}}_1\right)\dots \mathrm{\Gamma }\left({\alpha }_r+{\mathit{kA}}_r\right)}{\mathrm{\Gamma }\left({\beta }_1+{\mathit{kB}}_1\right)\dots \mathrm{\Gamma }\left({\beta }_s+{\mathit{kB}}_s\right)k!}\sum _{j=0}^3\frac{{\left(-1\right)}^j{\left(\mathit{\lambda b}+\mathit{ck}\right)}^j\mathrm{\Gamma }\left(\upsilon +\frac{j}{2}\right)}{y^{\frac{j}{2}}j!}\times \\ \times \begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\upsilon \pi }}{2}+\frac{\mathit{j\pi }}{4}\right){}_2{F}_3\left(\begin{array}{lll}\mathrm{\Delta }\left(2;\frac{2\upsilon +j}{2}\right);& & \\ {\mathrm{\Delta }}^{\ast }\left(4;1+j\right);& & \end{array}\frac{-1}{64y^2}{\left\{\frac{\left(\mathit{\lambda b}\right){\left(\frac{\mathit{\lambda b}+c}{c}\right)}_k}{{\left(\frac{\mathit{\lambda b}}{c}\right)}_k}\right\}}^4\right),\end{array}$$E52

Here the parameters ${\alpha }_i,{\beta }_j\in \mathbb{C}$ and coefficients $A_i,B_j\in \mathbb{R}=\left(-\infty ,+\infty \right);A_i\ne 0\left(i=1,2,\dots ,r\right),B_j\ne 0\left(j=1,2,\dots ,s\right)$ and ${}_r{\mathrm{\Psi }}_s\left[\cdot \right]$ is the Fox-Wright psi function of one variable subject to suitable convergence conditions derived from conditions discussed in case (1) or case (2) or case (3) of the function ${}_p{\mathrm{\Psi }}_q\left[\cdot \right]$ given by (22) and (23). When $N$ is positive integer then $\mathrm{\Delta }\left(N;\lambda \right)$ denotes the array of $N$ parameters given by $\frac{\lambda }{N},\frac{\lambda +1}{N},\dots ,\frac{\lambda +N-1}{N}$. When $N$ and $j$ are independent variables then the notation $\mathrm{\Delta }\left(N;j+1\right)$ denotes the set of $N$ parameters given by $\frac{j+1}{N},\frac{j+2}{N},\dots ,\frac{j+N}{N}$. When $j$ is dependent variable that is $j=\mathrm{0,1,2,3},\dots ,N-1$, then the asterisk in ${\mathrm{\Delta }}^{\ast }\left(N;j+1\right)$ represents the fact that the (denominator) parameters $\frac{N}{N}$ is always omitted (due to the need of factorial in denominator in the power series form of hypergeometric function) so that the set ${\mathrm{\Delta }}^{\ast }\left(N;j+1\right)$ obviously contains only $\left(N-1\right)$ parameters ([10], Chap. 3, p. 214).

Proof: Let us consider $\mathrm{\Theta }\left(k\right)=\frac{\mathrm{\Gamma }\left({\alpha }_1+{\mathit{kA}}_1\right)\dots \mathrm{\Gamma }\left({\alpha }_r+{\mathit{kA}}_r\right)}{\mathrm{\Gamma }\left({\beta }_1+{\mathit{kB}}_1\right)\dots \mathrm{\Gamma }\left({\beta }_s+{\mathit{kB}}_s\right)}\left(k=\mathrm{0,1,2,3},\dots \right)$ in the eqs. (28)and (30), then after evaluation we get integral expressions (33) involving the Fox-Wright Psi function in the form of infinite series of an ordinary ${}_2{F}_3$ hypergeometric function (34).

Theorem 1.4.3. Analytical expressions of the infinite Fourier sine and cosine transforms of$x^{\upsilon -1}{e^{-\mathit{b\lambda }\sqrt{x}}}_r{F}_s\left(\cdot \right)$holds true for$\Re \left(\upsilon \right)>-1;c>0,y>0;\lambda >0,b>0$($\mathit{\text{or}}\lambda <0,b<0$) then we have

$${\mathit{K}}_{S,C}\left(\upsilon ,b,c,\lambda ,y\right)={\int }_0^{\infty }{x}^{\upsilon -1}{e^{-\mathit{b\lambda }\sqrt{x}}}_r{F}_s\left(\begin{array}{lll}{\alpha }_1,\dots ,{\alpha }_r;& & \\ {\beta }_1,\dots ,{\beta }_s;& & \end{array}{e}^{-c\sqrt{x}}\right)\begin{array}{c}sin\\ cos\end{array}\left(\mathit{xy}\right)\mathit{dx},$$E53

$$\begin{array}{l}=y^{-\upsilon }\sum _{k=0}^{\infty }\frac{{\left({\alpha }_1\right)}_k\dots {\left({\alpha }_r\right)}_k}{{\left({\beta }_1\right)}_k\dots {\left({\beta }_s\right)}_{k}k!}\sum _{j=0}^3\frac{{\left(-1\right)}^j{\left(\mathit{\lambda b}+\mathit{ck}\right)}^j\mathrm{\Gamma }\left(\upsilon +\frac{j}{2}\right)}{y^{\frac{j}{2}}j!}\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\upsilon \pi }}{2}+\frac{\mathit{j\pi }}{4}\right)\\ {\times }_2{F}_3\left(\begin{array}{lll}\mathrm{\Delta }\left(2;\frac{2\upsilon +j}{2}\right);& & \\ {\mathrm{\Delta }}^{\ast }\left(4;1+j\right);& & \end{array}\frac{-1}{64y^2}{\left\{\frac{\left(\mathit{\lambda b}\right){\left(\frac{\mathit{\lambda b}+c}{c}\right)}_k}{{\left(\frac{\mathit{\lambda b}}{c}\right)}_k}\right\}}^4\right),\end{array}$$E54

where the parameters ${\alpha }_i,{\beta }_j\in \mathbb{C}\left(i=1,2,\dots ,r\right),\left(j=1,2,\dots ,s\right)$ and $r\le s+1$.

Proof: If $A_1=\dots =A_r=B_1=\dots =B_s=1$ in (33) and (34), then we get the above integral expressions involving generalized hypergeometric function in the form of infinite series of an ordinary ${}_2{F}_3$ hypergeometric function (36).

Corollary 1.4.4. An infinite Fourier sine and cosine transforms of$x^{\upsilon -1}{\left\{-1+exp\left(b\sqrt{x}\right)\right\}}^{-\lambda }$holds true for$\Re \left(\upsilon \right)>-1;\lambda >0,b>0$and$y>0$then we have

$${\mathit{I}}_{S,C}\left(\upsilon ,b,\lambda ,y\right)={\int }_0^{\infty }\frac{x^{\upsilon -1}}{{\left\{-1+exp\left(b\sqrt{x}\right)\right\}}^{\lambda }}\begin{array}{c}sin\\ cos\end{array}\left(\mathit{xy}\right)\mathit{dx},$$E55

$$=y^{-\upsilon }\sum _{k=0}^{\infty }\frac{{\left(\lambda \right)}_k}{k!}\sum _{\mathrm{\ell }=0}^{\infty }\frac{{\left(-1\right)}^{\mathrm{\ell }}{\left(\mathit{\lambda b}+\mathit{bk}\right)}^{\mathrm{\ell }}\mathrm{\Gamma }\left(\upsilon +\frac{\mathrm{\ell }}{2}\right)}{y^{\frac{\mathrm{\ell }}{2}}\mathrm{\ell }!}\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\upsilon \pi }}{2}+\frac{\mathrm{\ell }\pi }{4}\right),$$E56

$$\begin{array}{l}=y^{-\upsilon }\sum _{k=0}^{\infty }\frac{{\left(\lambda \right)}_k}{k!}\sum _{j=0}^3\frac{{\left(-1\right)}^j{\left(\mathit{\lambda b}+\mathit{bk}\right)}^j\mathrm{\Gamma }\left(\upsilon +\frac{j}{2}\right)}{y^{\frac{j}{2}}j!}\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\upsilon \pi }}{2}+\frac{\mathit{j\pi }}{4}\right)\times \\ {\times }_2{F}_3\left(\begin{array}{lll}\mathrm{\Delta }\left(2;\frac{2\upsilon +j}{2}\right);& & \\ {\mathrm{\Delta }}^{\ast }\left(4;1+j\right);& & \end{array}\frac{-1}{64y^2}{\left\{\frac{\left(\mathit{\lambda b}\right){\left(\lambda +1\right)}_k}{{\left(\lambda \right)}_k}\right\}}^4\right),\end{array}$$E57

$$=\begin{array}{l}\frac{\mathrm{\Gamma }\left(\upsilon \right)\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\upsilon \pi }}{2}\right)}{y^{\upsilon }}\sum _{k=0}^{\infty }\frac{{\left(\lambda \right)}_k}{k!}{}_2{F}_3\left(\begin{array}{l}\frac{\upsilon }{2},\frac{\upsilon +1}{2};\\ \frac{1}{4},\frac{1}{2},\frac{3}{4};\end{array}\frac{-1}{64y^2}{\left\{\frac{\left(\mathit{\lambda b}\right){\left(\lambda +1\right)}_k}{{\left(\lambda \right)}_k}\right\}}^4\right)\\ -\frac{\left(\mathit{\lambda b}\right)\mathrm{\Gamma }\left(\upsilon +\frac{1}{2}\right)\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\upsilon \pi }}{2}+\frac{\pi }{4}\right)}{y^{\upsilon +\frac{1}{2}}}\sum _{k=0}^{\infty }\frac{{\left(\lambda +1\right)}_k}{k!}{}_2{F}_3\left(\begin{array}{l}\frac{2\upsilon +1}{4},\frac{2\upsilon +3}{4};\\ \frac{1}{2},\frac{3}{4},\frac{5}{4};\end{array}\frac{-1}{64y^2}{\left\{\frac{\left(\mathit{\lambda b}\right){\left(\lambda +1\right)}_k}{{\left(\lambda \right)}_k}\right\}}^4\right)\\ +\frac{{\left(\mathit{\lambda b}\right)}^2\mathrm{\Gamma }\left(\upsilon +1\right)\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\upsilon \pi }}{2}\right)}{2y^{\upsilon +1}}\sum _{k=0}^{\infty }\frac{{\left\{{\left(\lambda +1\right)}_k\right\}}^2}{{\left(\lambda \right)}_{k}k!}{}_2{F}_3\left(\begin{array}{l}\frac{\upsilon +1}{2},\frac{\upsilon +2}{2};\\ \frac{3}{4},\frac{5}{4},\frac{3}{2};\end{array}\frac{-1}{64y^2}{\left\{\frac{\left(\mathit{\lambda b}\right){\left(\lambda +1\right)}_k}{{\left(\lambda \right)}_k}\right\}}^4\right)\\ -\frac{{\left(\mathit{\lambda b}\right)}^3\mathrm{\Gamma }\left(\upsilon +\frac{3}{2}\right)\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{\upsilon \pi }}{2}+\frac{\pi }{4}\right)}{6y^{\upsilon +\frac{3}{2}}}\sum _{k=0}^{\infty }\frac{{\left\{{\left(\lambda +1\right)}_k\right\}}^3}{k!{\left\{{\left(\lambda \right)}_k\right\}}^2}{}_2{F}_3\left(\begin{array}{l}\frac{2\upsilon +3}{4},\frac{2\upsilon +5}{4};\\ \frac{5}{4},\frac{3}{2},\frac{7}{4};\end{array}\frac{-1}{64y^2}{\left\{\frac{\left(\mathit{\lambda b}\right){\left(\lambda +1\right)}_k}{{\left(\lambda \right)}_k}\right\}}^4\right),\end{array}$$E58

Proof: If we consider $\mathrm{\Theta }\left(k\right)={\left(\lambda \right)}_k$ and $c=b$ in (28), which yields

$${\mathbf{I}}_{S,C}\left(\upsilon ,b,\lambda ,y\right)={\int }_0^{\infty }{x}^{\upsilon -1}{e}^{-\left(\mathit{\lambda b}\right)\sqrt{x}}\left\{\sum _{k=0}^{\infty }\frac{{\left(\lambda \right)}_k}{k!}{e}^{-\left(\mathit{bk}\right)\sqrt{x}}\right\}\begin{array}{c}sin\\ cos\end{array}\left(\mathit{xy}\right)\mathit{dx}.$$E59

Upon the use of binomial expansion (52) in the above eq. (41), then we get after evaluation (37). The results (38) to (40) are derived from (29) to (32) by putting $\mathrm{\Theta }\left(k\right)={\left(\lambda \right)}_k$ and $c=b$.

Corollary 1.4.5. Analytical expressions of the Ramanujan’s integrals${\mathit{R}}_{S,C}\left(m,n\right)$holds true for non-negative integer$m$and positive rational number$n$[21,22].

$$\begin{array}{l}{\mathit{R}}_{S,C}\left(m,n\right)={\int }_0^{\infty }\frac{x^m}{\left\{-1+exp\left(2\pi \sqrt{x}\right)\right\}}\begin{array}{c}sin\\ cos\end{array}\left(\mathit{\pi nx}\right)\mathit{dx},\\ ={\left(\mathit{n\pi }\right)}^{-m-1}\sum _{k=0}^{\infty }\sum _{\mathrm{\ell }=0}^{\infty }\frac{1}{\mathrm{\ell }!}{\left\{\frac{-\left(2\pi +2\mathit{\pi k}\right)}{\sqrt{\mathit{n\pi }}}\right\}}^{\mathrm{\ell }}\mathrm{\Gamma }\left(m+1+\frac{\mathrm{\ell }}{2}\right)\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{m\pi }}{2}+\frac{\mathrm{\ell }\pi }{4}\right),\end{array}$$E60

$$\begin{array}{l}={\left(\mathit{n\pi }\right)}^{-m-1}\sum _{k=0}^{\infty }\sum _{j=0}^3\frac{1}{j!}{\left\{\frac{-\left(2\pi +2\mathit{\pi k}\right)}{\sqrt{\mathit{n\pi }}}\right\}}^j\mathrm{\Gamma }\left(m+1+\frac{j}{2}\right)\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{m\pi }}{2}+\frac{\mathit{j\pi }}{4}\right)\times \\ {\times }_2{F}_3\left(\begin{array}{lll}\mathrm{\Delta }\left(2;\frac{2m+j+2}{2}\right);& & \\ {\mathrm{\Delta }}^{\ast }\left(4;1+j\right);& & \end{array}\frac{-{\pi }^2}{4n^2}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right),\end{array}$$E61

$$\begin{array}{l}=\frac{m!\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{m\pi }}{2}\right)}{{\left(\mathit{n\pi }\right)}^{m+1}}\sum _{k=0}^{\infty }{}_2{F}_3\left(\begin{array}{lll}\frac{m+1}{2},\frac{m+2}{2};& & \\ \frac{1}{4},\frac{1}{2},\frac{3}{4};& & \end{array}-\frac{{\pi }^2}{4n^2}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\\ -\frac{{\left(\frac{3}{2}\right)}_m\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{m\pi }}{2}+\frac{\pi }{4}\right)}{{\left(\pi \right)}^m{\left(n\right)}^{m+\frac{3}{2}}}\sum _{k=0}^{\infty }\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}{}_2{F}_3\left(\begin{array}{lll}\frac{2m+3}{4},\frac{2m+5}{4};& & \\ \frac{1}{2},\frac{3}{4},\frac{5}{4};& & \end{array}\frac{-{\pi }^2}{4n^2}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\\ -\frac{\left(2\right)\left(m+1\right)!\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{m\pi }}{2}\right)}{{\left(\pi \right)}^m{\left(n\right)}^{m+2}}\sum _{k=0}^{\infty }{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^2{}_2{F}_3\left(\begin{array}{lll}\frac{m+2}{2},\frac{m+3}{2};& & \\ \frac{3}{4},\frac{5}{4},\frac{3}{2};& & \end{array}\frac{-{\pi }^2}{4n^2}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\\ +\frac{{\left(\frac{5}{2}\right)}_m\begin{array}{c}sin\\ cos\end{array}\left(\frac{\mathit{m\pi }}{2}+\frac{\pi }{4}\right)}{{\left(\pi \right)}^{m-1}{\left(n\right)}^{m+\frac{5}{2}}}\sum _{k=0}^{\infty }{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^3{}_2{F}_3\left(\begin{array}{lll}\frac{2m+5}{4},\frac{2m+7}{4};& & \\ \frac{5}{4},\frac{3}{2},\frac{7}{4};& & \end{array}\frac{-{\pi }^2}{4n^2}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right),\end{array}$$E62

Proof: The results (42) to (44) are obtained from (37), (38), (39) and (40) by putting $\upsilon =m+1,b=2\pi$, $\lambda =1$ and $y=\mathit{n\pi }$.

5. Closed form infinite summation formulas

We have derived some closed forms of infinite summation formulas involving hypergeometric functions ${}_0{F}_1$, ${}_1{F}_2$, and ${}_2{F}_3$ [21, 22].

$$\begin{array}{l}\sum _{k=0}^{\infty }\left[{}_1{F}_2\left(\begin{array}{lll}1;& & \\ \frac{1}{4},\frac{3}{4};& & \end{array}\frac{-{\pi }^2}{4}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\frac{\pi }{\sqrt{2}}\sum _{k=0}^{\infty }\left[\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{1}{2};& & \end{array}\frac{-{\pi }^2}{4}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ \left.+\frac{{\pi }^2}{\sqrt{2}}\sum _{k=0}^{\infty }[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^3{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{3}{2};& & \end{array}\frac{-{\pi }^2}{4}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]=\frac{\pi \sqrt{2}-4}{8},\end{array}$$E63

$$\begin{array}{l}\sum _{k=0}^{\infty }\left[{}_1{F}_2\left(\begin{array}{lll}1;& & \\ \frac{1}{4},\frac{3}{4};& & \end{array}\frac{-{\pi }^2}{16}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]-\frac{\pi }{2}\sum _{k=0}^{\infty }\left[\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{1}{2};& & \end{array}\frac{-{\pi }^2}{16}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ +\frac{{\pi }^2}{4}\sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^3{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{3}{2};& & \end{array}\frac{-{\pi }^2}{16}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]=\frac{\pi -2}{8},\end{array}$$E64

$$\begin{array}{l}\sum _{k=0}^{\infty }\left[{}_1{F}_2\left(\begin{array}{lll}1;& & \\ \frac{1}{4},\frac{3}{4};& & \end{array}-{\pi }^2{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]-\pi \sum _{k=0}^{\infty }\left[\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{1}{2};& & \end{array}-{\pi }^2{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ +2{\pi }^2\sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^3{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{3}{2};& & \end{array}-{\pi }^2{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]=\frac{\pi -3}{8}.\end{array}$$E65

$$\begin{array}{l}\sum _{k=0}^{\infty }\left[{}_2{F}_3\left(\begin{array}{lll}1,\frac{3}{2};& & \\ \frac{1}{4},\frac{1}{2},\frac{3}{4};& & \end{array}-{\pi }^2{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ -\frac{3\pi }{2}\sum _{k=0}^{\infty }\left[\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}{}_1{F}_2\left(\begin{array}{lll}\frac{7}{4};& & \\ \frac{1}{2},\frac{3}{4};& & \end{array}-{\pi }^2{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ +5{\pi }^2\sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^3{}_1{F}_2\left(\begin{array}{lll}\frac{9}{4};& & \\ \frac{5}{4},\frac{3}{2};& & \end{array}-{\pi }^2{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]=\frac{1}{32}\left(4\pi -13\right),\end{array}$$E66

$$\begin{array}{l}\sum _{k=0}^{\infty }\left[{}_2{F}_3\left(\begin{array}{lll}1,\frac{3}{2};& & \\ \frac{1}{4},\frac{1}{2},\frac{3}{4};& & \end{array}\frac{-{\pi }^2}{16}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ -\frac{3\pi }{4}\sum _{k=0}^{\infty }\left[\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}{}_1{F}_2\left(\begin{array}{lll}\frac{7}{4};& & \\ \frac{1}{2},\frac{3}{4};& & \end{array}\frac{-{\pi }^2}{16}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ +\frac{5{\pi }^2}{8}\sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^3{}_1{F}_2\left(\begin{array}{lll}\frac{9}{4};& & \\ \frac{5}{4},\frac{3}{2};& & \end{array}\frac{-{\pi }^2}{16}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]=\frac{{\pi }^2}{16}\left(\frac{3}{\pi }-\frac{1}{2}-\frac{5}{{\pi }^2}\right),\end{array}$$E67

$$\begin{array}{l}\sum _{k=0}^{\infty }\left[\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}{}_2{F}_3\left(\begin{array}{lll}\frac{7}{4},\frac{9}{4};& & \\ \frac{1}{2},\frac{3}{4},\frac{5}{4};& & \end{array}\frac{-{\pi }^2}{16}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ -\frac{16}{5}\sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^2{}_2{F}_3\left(\begin{array}{lll}2,\frac{5}{2};& & \\ \frac{3}{4},\frac{5}{4},\frac{3}{2};& & \end{array}\frac{-{\pi }^2}{16}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ +\frac{7\pi }{6}\sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^3{}_2{F}_3\left(\begin{array}{lll}\frac{9}{4},\frac{11}{4};& & \\ \frac{5}{4},\frac{3}{2},\frac{7}{4};& & \end{array}\frac{-{\pi }^2}{16}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]=\frac{{\pi }^2}{60}\left(\frac{5}{\pi }-\frac{5}{{\pi }^2}-1\right),\end{array}$$E68

$$\begin{array}{l}\sum _{k=0}^{\infty }\left[\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{1}{2};& & \end{array}-\frac{{\pi }^2}{4}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ -2\sqrt{2}\sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^2{}_1{F}_2\left(\begin{array}{lll}1;& & \\ \frac{3}{4},\frac{5}{4};& & \end{array}\frac{-{\pi }^2}{4}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ +\pi \sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^3{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{3}{2};& & \end{array}\frac{-{\pi }^2}{4}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]=\frac{\sqrt{2}-1}{4},\end{array}$$E69

$$\begin{array}{l}\sum _{k=0}^{\infty }\left[\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{1}{2};& & \end{array}\frac{-{\pi }^2}{16}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ -2\sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^2{}_1{F}_2\left(\begin{array}{lll}1;& & \\ \frac{3}{4},\frac{5}{4};& & \end{array}\frac{-{\pi }^2}{16}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ +\frac{\pi }{2}\sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^3{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{3}{2};& & \end{array}\frac{-{\pi }^2}{16}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]=\frac{1}{4},\end{array}$$E70

$$\begin{array}{l}\sum _{k=0}^{\infty }\left[\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{1}{2};& & \end{array}\frac{-{\pi }^2}{64}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ -\sqrt{2}\sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^2{}_1{F}_2\left(\begin{array}{lll}1;& & \\ \frac{3}{4},\frac{5}{4};& & \end{array}\frac{-{\pi }^2}{64}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ +\frac{\pi }{4}\sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^3{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{3}{2};& & \end{array}\frac{-{\pi }^2}{64}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]=\frac{3\sqrt{2}-2}{4},\end{array}$$E71

$$\begin{array}{l}\sum _{k=0}^{\infty }\left[\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{1}{2};& & \end{array}\frac{-{\pi }^2}{144}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ -\frac{2\sqrt{3}}{3}\sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^2{}_1{F}_2\left(\begin{array}{lll}1;& & \\ \frac{3}{4},\frac{5}{4};& & \end{array}\frac{-{\pi }^2}{144}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ +\frac{\pi }{6}\sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^3{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{3}{2};& & \end{array}\frac{-{\pi }^2}{144}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]=\frac{13\sqrt{3}-12}{12},\end{array}$$E72

$$\begin{array}{l}\sum _{k=0}^{\infty }\left[\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{1}{2};& & \end{array}-{\pi }^2{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ -4\sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^2{}_1{F}_2\left(\begin{array}{lll}1;& & \\ \frac{3}{4},\frac{5}{4};& & \end{array}-{\pi }^2{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ +2\pi \sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^3{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{3}{2};& & \end{array}-{\pi }^2{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]=\frac{1}{8\pi },\end{array}$$E73

$$\begin{array}{l}\sum _{k=0}^{\infty }\left[\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{1}{2};& & \end{array}\frac{-25{\pi }^2}{16}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ -2\sqrt{5}\sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^2{}_1{F}_2\left(\begin{array}{lll}1;& & \\ \frac{3}{4},\frac{5}{4};& & \end{array}\frac{-25{\pi }^2}{16}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]\\ +\frac{5\pi }{2}\sum _{k=0}^{\infty }\left[{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^3{}_0{F}_1\left(\begin{array}{lll}\overline{};& & \\ \frac{3}{2};& & \end{array}\frac{-25{\pi }^2}{16}{\left\{\frac{{\left(2\right)}_k}{{\left(1\right)}_k}\right\}}^4\right)\right]=\frac{8\sqrt{5}-15}{100}.\end{array}$$E74

Proof: When $m=0$ with $n=1,2,\frac{1}{2}$ in the eqs. (42) to (44) and comparing with the eqs. (17), (18), and (19), we get the results (46), (47) and (48) respectively. In view of the hypergeometric functions (70), (71) and (72), we can express the above results (46) to (48) in terms of cosine, sine and Lommel functions. Our results (46) to (48) are convergent in view of the convergence condition of ${}_p{F}_q\left(\cdot \right)$ series, when $p\le q$, and for all $\mid z\mid <\infty$.

Similarly, we derive (49) to (51) by putting $m=1,n=\frac{1}{2}$; $m=1,n=2$ and $m=2,n=2$ in the eqs. (42) and (44) and finally comparing with (3), (4) and (5). When $m=0$ with $n=1,2,4,6,\frac{1}{2},\frac{2}{5}$ in the eqs. (42) and (44) and comparing with (10) to (15), we get the rest of results (52) to (57) respectively. In view of the hypergeometric functions (53), (54) and (55), we can express the above results (52) to (57) in terms of cosine, sine and Lommel functions. Our results (49) to (57) are convergent in view of the convergence condition of ${}_p{F}_q\left(\cdot \right)$ series, when $p\le q$, and for all $\mid z\mid <\infty$.

6. Concluding remarks

We have derived analytical expressions of the infinite Fourier sine and cosine transforms related to Ramanujan’s integrals as an infinite sum of ordinary hypergeometric functions ${}_2{F}_3$, with suitable convergence conditions. Moreover, as applications of Ramanujan’s integrals ${\mathbf{R}}_S\left(m,n\right)$, some closed form infinite summation formulas associated with hypergeometric functions ${}_1{F}_2$, ${}_2{F}_3$ and ${}_0{F}_1$ are evaluated. It is hoped that other such integrals can also be evaluated in a similar way. We conclude by remarking that various new results and applications can be obtained from our general theorem by appropriate choice of the parameters $\upsilon ,\lambda ,b,c,y$ and fixed sequence ${\left\{\mathrm{\Theta }\left(k\right)\right\}}_{k=0}^{\infty }$ in $I_C^{\ast }\left(\upsilon ,b,c,\lambda ,y\right)$.

Acknowledgments

One of the authors (S.A. Dar) acknowledges financial support from the University Grants Commission of India for the award of a Dr. D.S. Kothari Post Doctoral Fellowship(DSKPDF) (Grant number F.4-2/2006 (BSR)/MA/20-21/0061).