General series identities, some additive theorems on hypergeometric functions and their applications

TL;DR

This paper develops general series identities involving hypergeometric functions and bounded sequences, and applies these results to various special functions such as trigonometric, elliptic, and gamma functions.

Contribution

It introduces new rigorous identities for hypergeometric series with broad applicability to many special functions.

Findings

Established general series identities for hypergeometric functions.

Applied identities to derive results for various special functions.

Enhanced understanding of relationships among special functions.

Abstract

Motivated by the substantial development of the special functions, we contribute to establish some rigorous results on the general series identities with bounded sequences and hypergeometric functions with different arguments, which are generally applicable in nature. For the application purpose, we apply our results to some functions e.g. Trigonometric functions, Elliptic integrals, Dilogarithmic function, Error function, Incomplete gamma function, and many other special functions.

Tables

Table 1. Table 1. Some elementary functions and its hypergeometric representations

Ser. No.	Notation	Hypergeometric Representation
1	$\arcsin \left(x\right)$	$\arcsin \left(x\right)=x{}_2{}^{}F_1^{}\left[\begin{array}{c}\frac{1}{2},\frac{1}{2};\\ \\ \frac{3}{2};\end{array}{x}^2\right]$
2	$\arctan \left(x\right)$	$\arctan \left(x\right)=x{}_2{}^{}F_1^{}\left[\begin{array}{c}1,\frac{1}{2};\\ \\ \frac{3}{2};\end{array}-x^2\right]$
3	$\sin \left(x\right)$	$\sin \left(x\right)=x{}_0{}^{}F_1^{}\left[\begin{array}{c}-;\\ \\ \frac{3}{2};\end{array}-\frac{x^2}{4}\right]$
4	${\left(\arcsin \left(x\right)\right)}^2$	${\left(\arcsin \left(x\right)\right)}^2=x^2{}_3{}^{}F_2^{}\left[\begin{array}{c}1,1,1;\\ \\ 2,\frac{3}{2};\end{array}{x}^2\right]$
5	$\cos \left(x\right)$	$\cos \left(x\right)={}_0{}^{}F_1^{}\left[\begin{array}{c}-;\\ \\ \frac{1}{2};\end{array}-\frac{x^2}{4}\right]$
5	${\left(\frac{2}{1+\sqrt{\left(1-x^2\right)}}\right)}^{2\gamma -1}$	${\left(\frac{2}{1+\sqrt{\left(1-x^2\right)}}\right)}^{2\gamma -1}={}_2{}^{}F_1^{}\left[\begin{array}{c}\gamma ,\gamma -\frac{1}{2};\\ \\ 2\gamma ;\end{array}{x}^2\right]$

Table 2. Table 2. Some special functions and its hypergeometric representations

Ser. No.	Notation	Hypergeometric Representation
1	Complete elliptic integral of first kind: $\text{K}\left(x\right)$	$\text{K}\left(x\right)=\frac{\pi }{2}{}_2{}^{}F_1^{}\left[\begin{array}{c}\frac{1}{2},\frac{1}{2};\\ \\ 1;\end{array}{x}^2\right]$
2	Complete elliptic integral of second kind: $\text{E}\left(x\right)$	$\text{E}\left(x\right)=\frac{\pi }{2}{}_2{}^{}F_1^{}\left[\begin{array}{c}-\frac{1}{2},\frac{1}{2};\\ \\ 1;\end{array}{x}^2\right]$
3	Error function or Probability integral: $\mathrm{erf}\left(x\right)$	$\mathrm{erf}\left(x\right)=\frac{2x}{\sqrt{(}\pi )}{}_1{}^{}F_1^{}\left[\begin{array}{c}\frac{1}{2};\\ \\ \frac{3}{2};\end{array}-x^2\right]$
4	Incomplete gamma function: $\gamma (a,x)$	$\gamma (a,x^2)=\frac{x^{2a}}{a}{}_1{}^{}F_1^{}\left[\begin{array}{c}a;\\ \\ 1+a;\end{array}-x^2\right]$
5	Dilogarithm function: $L{i}_2\left(x\right)$	$L{i}_2\left(x^2\right)=x^2{}_3{}^{}F_2^{}\left[\begin{array}{c}1,1,1;\\ \\ 2,2;\end{array}{x}^2\right]$

Equations

$$(\lambda)_{\nu}:=\frac{\Gamma(\lambda+\nu)}{\Gamma(\lambda)}=\begin{cases}$1$&;(\nu=0;~{}\lambda\in\mathbb{C}\backslash\{0\})\\ \lambda(\lambda+1)\ldots(\lambda+n-1)&;~{}(\nu=n\in\mathbb{N};~{}\lambda\in\mathbb{C}\backslash{\mathbb{Z}_{0}^{-}})\\ \frac{(-1)^{n}k!}{(k-n)!}&;~{}(\lambda=-k;~{}\nu=n;~{}n,k\in\mathbb{N}_{0};~{}{0}\leq{n}\leq{k})\\ $0$&;~{}(\lambda=-k;~{}\nu=n;~{}n,k\in\mathbb{N}_{0};~{}{n}>{k})\\ \frac{(-1)^{n}}{(1-\lambda)_{n}}&;~{}(\nu=-n;n\in\mathbb{N};~{}\lambda\neq 0,\pm{1},\pm{2},\ldots),\end{cases}$$

$${{}_{p}F_{q}}\left[\begin{array}[]{c}\alpha_{1},\ldots,\alpha_{p};\\ \beta_{1},\ldots,\beta_{q};\end{array}z\right]=\sum_{n=0}^{\infty}\frac{(\alpha_{1})_{n}\ldots(\alpha_{p})_{n}}{(\beta_{1})_{n}\ldots(\beta_{q})_{n}}\frac{z^{n}}{n!},$$

$${\beta }_j\ne 0,-1,-2,\dots ;j=1,\dots ,q.$$

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

$$\displaystyle{{}_{p}\Psi_{q}}\left[\begin{array}[]{c}(\alpha_{1},A_{1}),\ldots,(\alpha_{p},A_{p});\\ (\beta_{1},B_{1}),\ldots,(\beta_{q},B_{q});\end{array}z\right]$$

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

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

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

$${\sigma }^{\ast }=(1+A_1+...+A_p)-(B_1+...+B_q)=1-{\Delta }^{\ast }.$$

$$\alpha =\exp \left(\frac{2\pi i}{5}\right),i=\sqrt{(-1)}$$

$$\displaystyle 1+\alpha^{2r}+\alpha^{4r}+\alpha^{6r}+\alpha^{8r}=\begin{cases}\begin{array}[]{c}5;\\ {}\hfil\\ 0;\end{array}&\begin{array}[]{c}r\in\{0,5,10,15,20,\dots\}\\ {}\hfil\\ r\in\{1,2,3,4,6,7,8,9,11,12,13,14,\dots\}.\end{array}\end{cases}$$

$$\alpha =\exp \left(\frac{2\pi i}{5}\right),i=\sqrt{(-1)}$$

$$1+{\alpha }^{2r+1}+{\alpha }^{4r+2}+{\alpha }^{6r+3}+{\alpha }^{8r+4}$$

$$\displaystyle\hskip 42.67912pt=\begin{cases}\begin{array}[]{c}5;\\ {}\hfil\\ 0;\end{array}&\begin{array}[]{c}r\in\{2,7,12,17,\dots\}\\ {}\hfil\\ r\in\{0,1,3,4,5,6,8,9,10,11,13,14,15,16,\dots\}.\end{array}\end{cases}$$

$$\displaystyle(b)_{mn}=m^{mn}\displaystyle\prod_{j=1}^{m}\Big{(}\frac{b+j-1}{m}\Big{)}_{n}.$$

$$\alpha =\exp \left(\frac{2\pi i}{5}\right),i=\sqrt{(-1)}$$

$$\sum _{r=0}^{\infty }\varphi (r)\frac{c^r(x{)}^{2r}}{r!}+\sum _{r=0}^{\infty }\varphi (r)\frac{c^r(x\alpha {)}^{2r}}{r!}+\sum _{r=0}^{\infty }\varphi (r)\frac{c^r(x{\alpha }^2{)}^{2r}}{r!}+\sum _{r=0}^{\infty }\varphi (r)\frac{c^r(x{\alpha }^3{)}^{2r}}{r!}$$

$$+\sum _{r=0}^{\infty }\varphi (r)\frac{c^r(x{\alpha }^4{)}^{2r}}{r!}=5\sum _{r=0}^{\infty }\varphi (5r)\frac{c^{5r}{x}^{10r}}{(5r)!},$$

$$S=\sum _{r=0}^{\infty }\varphi (r)\frac{c^r{x}^{2r}}{r!}\{1+{\alpha }^{2r}+{\alpha }^{4r}+{\alpha }^{6r}+{\alpha }^{8r}\}.$$

$$S=5\varphi (0)\frac{c^0{x}^0}{(0)!}+5\varphi (5)\frac{c^5{x}^{10}}{(5)!}+5\varphi (10)\frac{c^{10}{x}^{20}}{(10)!}+5\varphi (15)\frac{c^{15}{x}^{30}}{(15)!}+\dots$$

$$=5\sum _{r=0}^{\infty }\varphi (5r)\frac{c^{5r}{x}^{10r}}{(5r)!}.$$

$$\alpha =\exp \left(\frac{2\pi i}{5}\right),i=\sqrt{(-1)}$$

$$\sum _{r=0}^{\infty }\varphi (r)\frac{c^r(x{)}^{2r}}{r!}+\alpha \sum _{r=0}^{\infty }\varphi (r)\frac{c^r(x\alpha {)}^{2r}}{r!}+{\alpha }^2\sum _{r=0}^{\infty }\varphi (r)\frac{c^r(x{\alpha }^2{)}^{2r}}{r!}+{\alpha }^3\sum _{r=0}^{\infty }\varphi (r)\frac{c^r(x{\alpha }^3{)}^{2r}}{r!}$$

$$+{\alpha }^4\sum _{r=0}^{\infty }\varphi (r)\frac{c^r(x{\alpha }^4{)}^{2r}}{r!}=5\sum _{r=0}^{\infty }\varphi (5r+2)\frac{c^{(5r+2)}{x}^{(10r+4)}}{(5r+2)!},$$

$$T=\sum _{r=0}^{\infty }\varphi (r)\frac{c^r{x}^{2r}}{r!}\{1+{\alpha }^{2r+1}+{\alpha }^{4r+2}+{\alpha }^{6r+3}+{\alpha }^{8r+4}\}.$$

$$T=5\varphi (2)\frac{c^2{x}^4}{(2)!}+5\varphi (7)\frac{c^7{x}^{14}}{(7)!}+5\varphi (12)\frac{c^{12}{x}^{24}}{(12)!}+5\varphi (17)\frac{c^{17}{x}^{34}}{(17)!}+\dots$$

$$=5\sum _{r=1}^{\infty }\varphi (5r-3)\frac{c^{(5r-3)}{x}^{(10r-6)}}{(5r-3)!}.$$

$$T=5\sum _{r=0}^{\infty }\varphi (5r+2)\frac{c^{(5r+2)}{x}^{(10r+4)}}{(5r+2)!}.$$

$$\displaystyle{{}_{p}\Psi_{q}}\left[\begin{array}[]{r}(a_{1},A_{1}),\,(a_{2},A_{2}),\,\dots\,,(a_{p},A_{p});\\ {}\hfil\\ (b_{1},B_{1}),\,(b_{2},B_{2}),\,\dots\,,(b_{q},B_{q});\end{array}c(x)^{2}\right]$$

$$\displaystyle+{{}_{p}\Psi_{q}}\left[\begin{array}[]{r}(a_{1},A_{1}),\,(a_{2},A_{2}),\,\dots\,,(a_{p},A_{p});\\ {}\hfil\\ (b_{1},B_{1}),\,(b_{2},B_{2}),\,\dots\,,(b_{q},B_{q});\end{array}c(x\alpha)^{2}\right]$$

$$\displaystyle+{{}_{p}\Psi_{q}}\left[\begin{array}[]{r}(a_{1},A_{1}),\,(a_{2},A_{2}),\,\dots\,,(a_{p},A_{p});\\ {}\hfil\\ (b_{1},B_{1}),\,(b_{2},B_{2}),\,\dots\,,(b_{q},B_{q});\end{array}c(x\alpha^{2})^{2}\right]$$

$$\displaystyle+{{}_{p}\Psi_{q}}\left[\begin{array}[]{r}(a_{1},A_{1}),\,(a_{2},A_{2}),\,\dots\,,(a_{p},A_{p});\\ {}\hfil\\ (b_{1},B_{1}),\,(b_{2},B_{2}),\,\dots\,,(b_{q},B_{q});\end{array}c(x\alpha^{3})^{2}\right]$$

$$\displaystyle{{}_{p}\Psi_{q}}\left[\begin{array}[]{r}(a_{1},A_{1}),\,(a_{2},A_{2}),\,\dots\,,(a_{p},A_{p});\\ {}\hfil\\ (b_{1},B_{1}),\,(b_{2},B_{2}),\,\dots\,,(b_{q},B_{q});\end{array}cx^{2}\right]$$

Full text

*Corresponding author

1. Introduction, Preliminaries and Notations

In present paper, we shall use the following standard notations:

$\mathbb{N}:=\{1,2,3,\ldots\}$,

$\mathbb{N}_{0}:=\{0,1,2,3,\ldots\}=\mathbb{N}\cup\{0\}$,

$\mathbb{Z}_{0}^{-}:=\{0,-1,-2,-3,\ldots\}$,

$\mathbb{Z}^{-}:=\{-1,-2,-3,\ldots\}=\mathbb{Z}_{0}^{-}\backslash\{0\},$ and

$\mathbb{Z}=\mathbb{Z}_{0}^{-}\cup\mathbb{N}$.

Here, as usual, $\mathbb{Z}$ denotes the set of integers, $\mathbb{R}$ denotes the set of real numbers, $\mathbb{R^{+}}$ denotes the set of positive real numbers and $\mathbb{C}$ denotes the set of complex numbers.

The Pochhammer symbol (or the shifted factorial) $(\lambda)_{\nu}$ $(\lambda,\nu\in\mathbb{C})$ is defined in terms of the familiar Gamma function, by

[TABLE]

it being understood conventionally that $(0)_{0}=1$, and assumed tacitly that the Gamma quotient exists.

In the Gaussian hypergeometric series ${}_{2}F_{1}(a,b;c;z)$, there are two numerator parameters $a$, $b$ and one denominator parameter $c$. A natural generalization of this series is accomplished by introducing any arbitrary number of numerator and denominator parameters. The non-terminating hypergeometric series [9, p.42-43]

[TABLE]

is known as the generalized Gauss and Kummer series, or simply, the generalized hypergeometric series. Here $p$ and $q$ are positive integers or zero (interpreting an empty product as unity), and we assume that the variable $z$, the numerator parameters ${\alpha_{1},\ldots,\alpha_{p}}$ and the denominator parameters ${\beta_{1},\ldots,\beta_{q}}$ take on complex values, provided that

[TABLE]

Convergence conditions [9, p.43] for generalized hypergeometric function are as follows:

Suppose that none of the numerator parameters is zero or a negative integer (otherwise the question of convergence will not arise), and with the usual restriction (1.3), the ${{}_{p}F_{q}}$ series in the definition (1.2)

(i) converges for ${|z|<\infty}$, if $p~{}{\leqq}~{}q$,

(ii) converges for ${|z|<1}$, if $p~{}=~{}q+1$.

Furthermore, if we denote

[TABLE]

it is known that the ${{}_{p}F_{q}}$ series, with $p=q+1$, is

(a) absolutely convergent for $|z|=1$, if $\Re(\omega)>0$,

(b) conditionally convergent for $|z|=1$, $z\neq 1$, if $-1<\Re(\omega)\leqq 0$.

The Fox-Wright psi function of one variable ([3, p.389]; see also [4, 10, 11]) is given by

[TABLE]

[TABLE]

[TABLE]

where $\rho^{2}=-1$, $z\in\mathbb{C}$; parameters $\alpha_{i},\beta_{j}\in\mathbb{C}$; coefficients $A_{i},B_{j}\in\mathbb{R}=(-\infty,+\infty)$ in case of series (1) (or $A_{i},B_{j}\in\mathbb{R}_{+}=(0,+\infty)$ in case of contour integral (1.16)), $A_{i}\neq 0~{}(i=1,2,...,p),B_{j}\neq 0~{}(j=1,2,...,q)$. In equation (1), 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 [1, 2].

[TABLE]

and

[TABLE]

Case(I): When contour (L) is a left loop beginning and ending at $-\infty$, then ${{}_{p}}\Psi_{q}$ given by (1) or (1.16) holds the following convergence conditions

i) When $\Delta^{*}>-1,0<|z|<\infty,z\neq 0,$

ii) When $\Delta^{*}=-1,0<|z|<\delta^{*},$

iii)When $\Delta^{*}=-1,|z|=\delta^{*},\text{ and }\Re(\mu^{*})>\frac{1}{2}.$

Case(II): When contour (L) is a right loop beginning and ending at $+\infty$, then ${{}_{p}}\Psi_{q}$ given by (1) or (1.16) holds the following convergence conditions

i) When $\Delta^{*}<-1,0<|z|<\infty,z\neq 0,$

ii) When $\Delta^{*}=-1,|z|>\delta^{*},$

iii)When $\Delta^{*}=-1,|z|=\delta^{*},\text{ and }\Re(\mu^{*})>\frac{1}{2}.$

Case(III): When contour (L) is starting from $\gamma-i\infty$ and ending at $\gamma+i\infty$, where $\gamma\in\mathbb{R}=(-\infty,+\infty)$, then ${{}_{p}}\Psi_{q}$ is also convergent under the following conditions

i) When $\sigma^{*}>0,|\arg(-z)|<\frac{\pi}{2}\sigma^{*},0<|z|<\infty,z\neq 0,$

ii) When $\sigma^{*}=0,\arg(-z)=0,0<|z|<\infty,z\neq 0\text{ such that }-\gamma\Delta^{*}+\Re(\mu^{*})>\frac{1}{2}+\gamma,$

iii)When $\gamma=0,\sigma^{*}=0,\arg(-z)=0,0<|z|<\infty,z\neq 0\text{ such that },\Re(\mu^{*})>\frac{1}{2}.$

Next we collect some results that we will need in the sequel.

Identity 1. Let

[TABLE]

and $r$ being non-negative integer, then

[TABLE]

Identity 2. Let

[TABLE]

and $r$ being non-negative integer, then

[TABLE]

Above identities can be verified with the help of De Moivre’s theorem and some trigonometrical identities.

Gauss multiplication formula. Let m being positive integer and n being non-negative integer, then

[TABLE]

Now, we are recalling some functions in the hypergeometric notations [7, pp. 71, 115] (see also, [5]), which we will use in the applications.

2. General Series Identities

Theorem 1**.**

Suppose $\{\phi(r)\}_{r=0}^{\infty}$ is a bounded sequence of arbitrary real and complex numbers and

[TABLE]

then

[TABLE]

provided that each of the series involved is absolutely convergent.

Proof.

Suppose LHS of equation (1) is denoted by $S$, then

[TABLE]

Now, we apply Identity 1 in equation (2.2), we get

[TABLE]

∎

Theorem 2**.**

Suppose $\{\phi(r)\}_{r=0}^{\infty}$ is a bounded sequence of arbitrary real and complex numbers and

[TABLE]

then

[TABLE]

provided that each of the series involved is absolutely convergent.

Proof.

Suppose LHS of equation (2) is denoted by $T$, then

[TABLE]

Now, we apply Identity 2 in equation (2.5), we get

[TABLE]

Replacing $r$ by $r+1$ in equation (2), we get

[TABLE]

∎

3. Hypergeometric Representations

Any values of parameters and variables leading to the results, which do not make sense, are tacitly excluded.

Theorem 3**.**

Following sum of Fox-Wright hypergeometric functions with different arguments holds true:

[TABLE]

[TABLE]

where $\alpha=\exp{\left(\frac{2\pi i}{5}\right)}.$

Theorem 4**.**

Following sum of Fox-Wright hypergeometric functions with different arguments holds true:

[TABLE]

where $\alpha=\exp{\left(\frac{2\pi i}{5}\right)}$.

Proof.

On setting $\phi(r)=\frac{\displaystyle\prod_{j=1}^{p}\Gamma{(a_{j}+A_{j}r)}}{\displaystyle\prod_{j=1}^{q}\Gamma{(b_{j}+B_{j}r)}}$ in general series identities (1) and (2), and applying the definition of Fox-Wright hypergeometric function ${}_{p}\Psi_{q}$, we get equations (3.24), (3.29) respectively. ∎

Theorem 5**.**

Following sum of special case of Fox-Wright hypergeometric functions with different arguments holds true:

[TABLE]

where $\alpha=\exp{\left(\frac{2\pi i}{5}\right)}.$

Theorem 6**.**

Following sum of special case of Fox-Wright hypergeometric functions with different arguments holds true:

[TABLE]

[TABLE]

where $\alpha=\exp{\left(\frac{2\pi i}{5}\right)}.$

Proof.

On setting $\phi(r)=\frac{(a_{1})_{rA_{1}}(a_{2})_{rA_{2}}\dots(a_{p})_{rA_{p}}}{(b_{1})_{rB_{1}}(b_{2})_{rB_{2}}\dots(b_{q})_{rB_{q}}}$ in general series identities (1) and (2), and applying the definition of Fox-Wright hypergeometric function ${}_{p}\Psi^{*}_{q}$, we get equations (3.57), (3.90) respectively. ∎

Theorem 7**.**

Following sum of generalized hypergeometric functions with different arguments holds true:

[TABLE]

where \alpha=\exp{\left(\frac{2\pi i}{5}\right)};5p\leq 5q+4,\Big{|}\left(\frac{cx^{2}}{5^{(1+q-p)}}\right)^{5}\Big{|}<\infty;p=q+1,\Big{|}\left(\frac{cx^{2}}{5^{(1+q-p)}}\right)^{5}\Big{|}<1.

Theorem 8**.**

Following sum of generalized hypergeometric functions with different arguments holds true:

[TABLE]

[TABLE]

where \alpha=\exp{\left(\frac{2\pi i}{5}\right)};5p\leq 5q+4,\Big{|}\left(\frac{cx^{2}}{5^{(1+q-p)}}\right)^{5}\Big{|}<\infty;p=q+1,\Big{|}\left(\frac{cx^{2}}{5^{(1+q-p)}}\right)^{5}\Big{|}<1.

Proof.

On setting $\phi(r)=\frac{(a_{1})_{r}(a_{2})_{r}\dots(a_{p})_{r}}{(b_{1})_{r}(b_{2})_{r}\dots(b_{q})_{r}}$ in general series identities (1) and (2), and applying the definition of generalized hypergeometric function ${}_{p}F_{q}$, we get equations (3.116), (3.145) respectively. ∎

4. Applications

As the direct application of our theorems 7 & 8, we obtain following results on the sum of special functions and elementary functions with different arguments for $\alpha=\exp{\left(\frac{2\pi i}{5}\right)}$:

On setting $p=0,q=1,c=\frac{-1}{4}$ and $b_{1}=\frac{3}{2}$ in equation (3.116), we obtain

[TABLE]

On setting $p=0,q=1,c=\frac{-1}{4}$ and $b_{1}=\frac{3}{2}$ in equation (3.145), we obtain

[TABLE]

On setting $p=0,q=1,c=\frac{-1}{4}$ and $b_{1}=\frac{1}{2}$ in equation (3.116), we obtain

[TABLE]

On setting $p=0,q=1,c=\frac{-1}{4}$ and $b_{1}=\frac{1}{2}$ in equation (3.145), we obtain

[TABLE]

On setting $p=2,q=1,c=-1$ and $a_{1}=\frac{1}{2},a_{2}=1,b_{1}=\frac{3}{2}$ in equation (3.116), we obtain

[TABLE]

On setting $p=2,q=1,c=-1$ and $a_{1}=\frac{1}{2},a_{2}=1,b_{1}=\frac{3}{2}$ in equation (3.145), we obtain

[TABLE]

On setting $p=2,q=1,c=1$ and $a_{1}=\frac{1}{2},a_{2}=-\frac{1}{2},b_{1}=1$ in equation (3.116), we obtain

[TABLE]

On setting $p=2,q=1,c=1$ and $a_{1}=\frac{1}{2},a_{2}=-\frac{1}{2},b_{1}=1$ in equation (3.145), we obtain

[TABLE]

On setting $p=1,q=1,c=-1$ and $a_{1}=\frac{1}{2},b_{1}=\frac{3}{2}$ in equation (3.116), we obtain

[TABLE]

On setting $p=1,q=1,c=-1$ and $a_{1}=\frac{1}{2},b_{1}=\frac{3}{2}$ in equation (3.145), we obtain

[TABLE]

On setting $p=3,q=2,c=1$ and $a_{1}=a_{2}=a_{3}=1$, $b_{1}=2,b_{2}=\frac{3}{2}$ in equation (3.116), we obtain

[TABLE]

On setting $p=3,q=2,c=1$ and $a_{1}=a_{2}=a_{3}=1$, $b_{1}=2,b_{2}=\frac{3}{2}$ in equation (3.145), we obtain

[TABLE]

On setting $p=2,q=1,c=1$ and $a_{1}=a_{2}=\frac{1}{2},b_{1}=1$ in equation (3.116), we obtain

[TABLE]

On setting $p=2,q=1,c=1$ and $a_{1}=a_{2}=\frac{1}{2},b_{1}=1$ in equation (3.145), we obtain

[TABLE]

On setting $p=3,q=2,c=1$ and $a_{1}=a_{2}=a_{3}=1,b_{1}=b_{2}=2$ in equation (3.116), we obtain

[TABLE]

On setting $p=3,q=2,c=1$ and $a_{1}=a_{2}=a_{3}=1,b_{1}=b_{2}=2$ in equation (3.145), we obtain

[TABLE]

On setting $p=1,q=1,c=-1$ and $a_{1}=a,b_{1}=1+a$ in equation (3.116), we obtain

[TABLE]

On setting $p=1,q=1,c=-1$ and $a_{1}=a,b_{1}=1+a$ in equation (3.145), we obtain

[TABLE]

On setting $p=2,q=1,c=1$ and $a_{1}=\gamma,a_{2}=\gamma-\frac{1}{2},b_{1}=2\gamma$ in equation (3.116), we obtain

[TABLE]

On setting $p=2,q=1,c=1$ and $a_{1}=\gamma,a_{2}=\gamma-\frac{1}{2},b_{1}=2\gamma$ in equation (3.145), we obtain

[TABLE]

Remark: Making suitable adjustments of parameters and variables in Theorems 3,4,5 and 6, we can derive some more results involving generalised Bessel functions $\Phi(\alpha,\beta;z)\text{ or }J^{\mu}_{\nu}(z)$, Mittag-Leffler functions $E_{\alpha}(z)$ and its generalizations $E_{\alpha,\beta}(z)$. Since Wright’s generalized function ${}_{p}\Psi_{q}$ of one variable is the particular case of Fox H-function of one variable. Therefore, for more special cases of ${}_{p}\Psi_{q}$, we refer two monographs of Mathai-Saxena [6] and Srivastava, Gupta and Goyal [8].

5. Conclusion

Here, we have established some results on the sum of hypergeometric functions with different arguments. We applied our results to Trigonometric functions, Elliptic integrals, Dilogarithmic function, Error function, and Incomplete gamma function.

One can also establish the some results on the sum of these special functions for example: ordinary Bessel function Jν(x), modified Bessel function Iν(x), complete elliptic integrals B(x), C(x), D(x), Lerch’s transcendent $\Phi(x,q,a)$, Fresnel’s integrals S(x), S1(x), S2(x), C*∗(x), C1*(x), C2(x), Sine integral Si(x), hyperbolic sine integral Shi(x), Polylogarithm function $Li_{q}$(x), Sturve function Hν(x), Modified Sturve functions Lν(x), hμ,ν(x), Lommel function sμ,ν(x), Kelvin’s functions $\text{ ber}(x),\text{ bei(x)}$, Incomplete beta function B${}_{x}(\alpha,\beta)$, Hyperbessel function of Humbert Jm,n(x), Modified hyperbessel function of Delerue Im,n(x), Arctangents function Ti2(x), $\sinh(x),\cosh(x),\tanh^{-1}(x),\sinh^{-1}(x)$,

$\frac{\sinh^{-1}(x)}{\sqrt{\left(1+x^{2}\right)}}$, $\left(\sinh^{-1}x\right)^{2}$, $\sin{\left(a\sin^{-1}x\right)}$, $\cos{\left(a\sin^{-1}x\right)}$, $\log_{e}(1\pm x)$, $\frac{\arcsin(x)}{\sqrt{(1-x^{2})}}$ and $\exp{(a\sin^{-1}x)}$ etc.