Some Results of Extended Beta Function and Hypergeometric Functions by Using Wiman’s Function

Abstract

The main aim of this research paper is to introduce a new extension of the Gauss hypergeometric function and confluent hypergeometric function by using an extended beta function. Some functional relations, summation relations, integral representations, linear transformation formulas, and derivative formulas for these extended functions are derived. We also introduce the logarithmic convexity and some important inequalities for extended beta function.

1. Introduction and Preliminaries

The theory of special functions, especially extensions of beta function, gamma function, hypergeometric functions has been one of the fastest rising investigate a topic in mathematical science because scientific researchers feel it is important to study the behavior of special functions with extended domains. Special functions are naturally generalizations of the elementary functions and they play a vital role in the solution of the differential equations due to which study of the generalizations of the special functions always important. In the past years, many extensions, properties, and applications of special functions have been discussed by many researchers and authors [1,2]. Here, we aim to study some results of the extended beta function and hypergeometric functions by using Wiman’s function.

To obtain our main results, we require prior knowledge of some special functions. First of all, throughout this present paper, there is a need to note the following notations: $\Re \left(z\right)$ represents the real part of any complex number z, C represents a set of complex numbers, and $R^{+}$ represents a set of positive real numbers.

Classical Euler beta function and gamma function are defined as [3]:

$$B(y_1,y_2)={\int }_0^1{t}^{y_1-1}{(1-t)}^{y_2-1}dt,$$

(1)

where $\Re \left(y_1\right)$ and $\Re \left(y_2\right)>0$;

$$\Gamma \left(y_1\right)={\int }_0^{\infty }{t}^{y_1-1}{e}^{-t}dt,$$

(2)

where $\Re \left(y_1\right)>0$.

Gauss hypergeometric function ${}_2{F}_1$ is defined as [4]:

$${}_2{F}_1(p_0,p_1,p_2;z)=F(p_0,p_1,p_2;z)=\sum _{k=0}^{\infty }\frac{{\left(p_0\right)}_k{\left(p_1\right)}_k}{{\left(p_2\right)}_k}\frac{z^k}{k!},$$

(3)

where ${\left(u\right)}_k$ represents the Pochhammer symbol defined below:

$${\left(u\right)}_k:=\frac{\Gamma \left(u+k\right)}{\Gamma \left(u\right)}=\left\{\begin{array}{cc}1\hfill & k=0;u\in \mathbb{C}\backslash \left\{0\right\},\hfill \\ u\left(u+1\right)\cdots \left(u+k-1\right)\hfill & k\in \mathbb{N};u\in \mathbb{C}.\hfill \end{array}\right.$$

Series representation and integral representation of Gauss hypergeometric function ${}_2{F}_1$ is defined as [4]:

$$F(r_0,r_1,r_2;z)=\sum _{n=0}^{\infty }\frac{B(r_1+n,r_2-r_1)}{B(r_1,r_2-r_1)}{\left(r_0\right)}_n\frac{z^n}{n!},$$

(4)

where $\Re \left(r_2\right)>\Re \left(r_1\right)>0$ and $\mid z\mid <1$.

$$F(r_0,r_1,r_2;z)=\frac{1}{B(r_1,r_2-r_1)}{\int }_0^1{t}^{r_1-1}{(1-t)}^{r_2-r_1-1}{(1-zt)}^{-r_0}dt.$$

(5)

Confluent hypergeometric function is defined as [4]:

$${}_1{F}_1(p_1,p_2;z)=\Phi (p_1,p_2;z)=\sum _{k=0}^{\infty }\frac{{\left(p_1\right)}_k}{{\left(p_2\right)}_k}\frac{z^k}{k!}.$$

(6)

Series representation and integral representation of confluent hypergeometric function is defined as [4]:

$$\Phi (r_1,r_2;z)=\sum _{n=0}^{\infty }\frac{B(r_1+n,r_2-r_1)}{B(r_1,r_2-r_1)}\frac{z^n}{n!},$$

(7)

where $\Re \left(r_2\right)>\Re \left(r_1\right)>0$;

$$\Phi (r_1,r_2;z)=\frac{1}{B(r_1,r_2-r_1)}{\int }_0^1{t}^{r_1-1}{(1-t)}^{r_2-r_1-1}{e}^{zt}dt.$$

(8)

One parameter Mittag-Leffler function and two parameter Mittag-Leffler function (Wiman’s function) is defined as [5,6,7]:

$$E_{y_1}\left(z\right)=\sum _{k=0}^{\infty }\frac{z^k}{\Gamma (k{y}_1+1)},$$

(9)

where $\Re \left(y_1\right)\ge 0$ and $z\in C$;

$$E_{y_1,y_2}\left(z\right)=\sum _{k=0}^{\infty }\frac{z^k}{\Gamma (k{y}_1+y_2)},$$

(10)

where $\Re \left(y_1\right)\ge 0$, $\Re \left(y_2\right)\ge 0$ and $z\in C$.

Since 1900, many researchers have been working on extensions of special functions. Chaudhry is one of the mathematician who continuously worked on extensions of beta function and other special functions. In 1997, Chaudhry et al. extended the classical Euler beta function defined in Reference [8], and then, in 2004, he used this extended beta function to generalize the Gauss hypergeometric function and confluent hypergeometric function as [9]:

$$B(y_1,y_2;y)={\int }_0^1{t}^{y_1-1}{(1-t)}^{y_2-1}{e}^{\frac{-y}{t(1-t)}}dt,$$

(11)

where $\Re \left(y_1\right)>0$, $\Re \left(y_2\right)>0$ and $\Re \left(y\right)>0$;

$$F_r(r_0,r_1,r_2;z)=\sum _{n=0}^{\infty }\frac{B(r_1+n,r_2-r_1;r)}{B(r_1,r_2-r_1)}{\left(r_0\right)}_n\frac{z^n}{n!},$$

(12)

where $r\ge 0$, $\Re \left(r_2\right)>\Re \left(r_1\right)>0$, $\mid z\mid <1$;

$${\Phi }_r(r_1,r_2;z)=\sum _{n=0}^{\infty }\frac{B(r_1+n,r_2-r_1;r)}{B(r_1,r_2-r_1)}\frac{z^n}{n!},$$

(13)

where $r\ge 0$ and $\Re \left(r_2\right)>\Re \left(r_1\right)>0$.

Remark 1.

If we set $r=0$ in (12) and (13), we get Gauss hypergeometric function and confluent hypergeometric function given by (4) and (7), respectively:

$$F_0(r_0,r_1,r_2;z)=F(r_0,r_1,r_2;z),$$

(14)

and

$${\Phi }_0(r_1,r_2;z)=\Phi (r_1,r_2;z).$$

(15)

Subsequently, Özergin et al. [10] have used confluent hypergeometric function to extend classic Euler beta function; moreover, the latter function was then exploited to generalize hypergeometric functions as follows:

$$B_u^{(u_1,u_2)}(y_1,y_2)={\int }_0^1{t}^{y_1-1}{(1-t)}^{y_2-1}{}_1{F}_1\left(u_1,u_2;\frac{-u}{t(1-t)}\right)dt,$$

(16)

where $\Re \left(y_1\right),\Re \left(y_2\right)>0$, $\Re \left(u_1\right)>0,\Re \left(u_2\right)>0$ and $\Re \left(u\right)>0$;

$$F_r^{(r_1,r_2)}(p_0,p_1,p_2;z)=\sum _{k=0}^{\infty }\frac{B_r^{(r_1,r_2)}(p_1+k,p_2-p_1)}{B(p_1,p_2-p_1)}{\left(p_0\right)}_k\frac{z^k}{k!},$$

(17)

where $\Re \left(p_2\right)>\Re \left(p_1\right)>0$, $\mid z\mid <1$ and $r\ge 0$;

$${\Phi }_r^{(r_1,r_2)}(p_1,p_2;z)=\sum _{k=0}^{\infty }\frac{B_r^{(r_1,r_2)}(p_1+k,p_2-p_1)}{B(p_1,p_2-p_1)}\frac{z^k}{k!},$$

(18)

where $\Re \left(p_2\right)>\Re \left(p_1\right)>0$ and $r\ge 0$.

Later, in 2014, Choi et al. [11] defined a generalization of extended beta function reported in (11), using it to broaden the definition of extended hypergeometric Gauss function and extended confluent hypergeometric as follows:

$$B(y_1,y_2;u;v)={\int }_0^1{t}^{y_1-1}{(1-t)}^{y_2-1}{e}^{\left(\frac{-u}{t}-\frac{v}{(1-t)}\right)}dt,$$

(19)

where $\Re \left(y_1\right),\Re \left(y_2\right)>0$, $\Re \left(v\right)>0$ and $\Re \left(u\right)>0$;

$$F_{r,s}(r_0,r_1,r_2;z)=\sum _{k=0}^{\infty }\frac{B(r_1+k,r_2-r_1;r,s)}{B(r_1,r_2-r_1)}{\left(r_0\right)}_k\frac{z^k}{k!},$$

(20)

where $\Re \left(r_2\right)>\Re \left(r_1\right)>0$, $r\ge 0$, $s\ge 0$ and $\mid z\mid <1$;

$${\Phi }_{r,s}(r_1,r_2;z)=\sum _{k=0}^{\infty }\frac{B(r_1+k,r_2-r_1;r,s)}{B(r_1,r_2-r_1)}\frac{z^k}{k!},$$

(21)

where $\Re \left(r_2\right)>\Re \left(r_1\right)>0$, $s\ge 0$ and $r\ge 0$.

Remark 2.

If we set $r=0$ and $s=0$ in (20) and (21), we get Gauss hypergeometric function and confluent hypergeometric function given by (4) and (7), respectively:

$$F_{0,0}(r_0,r_1,r_2;z)=F(r_0,r_1,r_2;z),$$

(22)

and

$${\Phi }_{0,0}(r_1,r_2;z)=\Phi (r_1,r_2;z).$$

(23)

In 2018, Shadab et al. [12] extended classic Euler beta function using one parameter Mittag-Leffler function; this generalization was then exploited to expand hypergeometric function of the Gauss and confluent hypergeometric function, thus allowing to study various properties and relationships of these functions:

$$B_u^y(y_1,y_2)={\int }_0^1{t}^{y_1-1}{(1-t)}^{y_2-1}{E}_u\left(\frac{-y}{t(1-t)}\right)dt,$$

(24)

where $\Re \left(y_1\right),\Re \left(y_2\right)>0$, $\Re \left(u\right)>0$ and $\Re \left(y\right)>0$;

$$F_{r,\alpha }(r_0,r_1,r_2;z)=\sum _{k=0}^{\infty }\frac{B_{\alpha }^r(r_1+k,r_2-r_1)}{B(r_1,r_2-r_1)}{\left(r_0\right)}_k\frac{z^k}{k!},$$

(25)

where $\Re \left(r_2\right)>\Re \left(r_1\right)>0$, $\alpha \ge 0$, $r\ge 0$ and $\mid z\mid <1$;

$${\Phi }_{r,\alpha }(r_1,r_2;z)=\sum _{k=0}^{\infty }\frac{B_{\alpha }^r(r_1+k,r_2-r_1)}{B(r_1,r_2-r_1)}\frac{z^k}{k!},$$

(26)

where $\Re \left(r_2\right)>\Re \left(r_1\right)>0$, $\alpha \ge 0$ and $r\ge 0$.

Remark 3.

(i) If we set $\alpha =1$ in (25) and (26), we get extended Gauss hypergeometric function and extended confluent hypergeometric function given by (12) and (13), respectively:

$$F_{r,1}(r_0,r_1,r_2;z)=F_r(r_0,r_1,r_2;z),$$

(27)

and

$${\Phi }_{r,1}(r_1,r_2;z)={\Phi }_r(r_1,r_2;z).$$

(28)

(ii) If we set $\alpha =1$ and $r=0$ in (25) and (26), we get Gauss hypergeometric function and confluent hypergeometric function given by (4) and (7), respectively:

$$F_{0,1}(r_0,r_1,r_2;z)=F(r_0,r_1,r_2;z),$$

(29)

and

$${\Phi }_{0,1}(r_1,r_2;z)=\Phi (r_1,r_2;z).$$

(30)

Very recently, Goyal et al. [13] introduced an extension of the beta function using the Wiman function, thus studying various properties and relationships of that function:

$$B_{(u_1,u_2)}^{\left(u\right)}(y_1,y_2)={\int }_0^1{t}^{y_1-1}{(1-t)}^{y_2-1}{E}_{u_1,u_2}\left(-u{\left(t(1-t)\right)}^{-1}\right)dt,$$

(31)

where $min\left\{\Re \left(y_1\right),\Re \left(y_2\right)\right\}>0$, $\Re \left(u_1\right)>0,\Re \left(u_2\right)>0$, $u\ge 0$ and $E_{u_1,u_2}\left(z\right)$ is 2-parameter Mittag-Leffler function given by (10).

In this paper, we discuss convexity and inequalities related to extended beta function given in (31). Now, we give some definitions about convex functions and show some results.

Definition 1.

(References [14,15]) Let Y be a convex set in a real vector space, and let$g:Y⟶(-\infty ,+\infty )$be a function. Then, g is called convex on Y if the inequality

$$g(\beta x_1+(1-\beta )x_2)\le \beta g\left(x_1\right)+(1-\beta )g\left(x_2\right)$$

(32)

holds for any $x_1,x_2\in Y$ and $\beta \in [0,1]$.

The function g is concave if the function $-g$ is convex.

The function g is called logarithmically convex (or logarithmically concave, respectively) on Y if $g>0$ and $logg$ ( $-logg$, respectively) is convex (or concave, respectively) on Y.

Lemma 1.

(References [16,17,18]) (Chebyshev’s integral inequality)

Let $f_1,f_2:[c,d]\subseteq (-\infty ,+\infty )⟶(-\infty ,+\infty )$ be integrable functions. Assume that:

$$[f_1\left(x\right)-f_1\left(y\right)][f_2\left(x\right)-f_2\left(y\right)]⋛0,\forall x,y\in [c,d].$$

Let $h\left(x\right):[c,d]\subseteq (-\infty ,+\infty )⟶(-\infty ,+\infty )$ be a positive integrable function. Then:

$${\int }_c^{d}h\left(x\right)f_1\left(x\right)dx·{\int }_c^{d}h\left(x\right)f_2\left(x\right)dx⋛{\int }_c^{d}h\left(x\right)dx·{\int }_c^{d}h\left(x\right)f_1\left(x\right)f_2\left(x\right)dx.$$

(33)

Lemma 2.

(References [19,20]) (Hölder Inequality) Let ${\theta }_1$ and ${\theta }_2$ be positive numbers such that

$$\frac{1}{{\theta }_1}+\frac{1}{{\theta }_2}=1.$$

Let $f_1,f_2:[c,d]⟶(-\infty ,+\infty )$ be integrable functions. Then

$${\int }_c^d{f}_1\left(x\right)f_2\left(x\right)dx\le {\left({\int }_c^d{f_1\left(x\right)}^{{\theta }_1}dx\right)}^{\frac{1}{{\theta }_1}}·{\left({\int }_c^d{f_2\left(x\right)}^{{\theta }_2}dx\right)}^{\frac{1}{{\theta }_2}}.$$

(34)

2. Inequalities of Extended Beta function ${\mathit{B}}_{({\mathit{u}}_{\mathbf{1}},{\mathit{u}}_{\mathbf{2}})}^{\left(\mathit{u}\right)}({\mathit{y}}_{\mathbf{1}},{\mathit{y}}_{\mathbf{2}})$

Theorem 1.

Assume that:

• $x,y,x_1,y_1$ are non-zero and non-negative numbers such that $(x-x_1)(y-y_1)\ge 0$,
• $u_1\in [0,1]$ and $u_2\in [0,1]$.

Then,

$$B_{(u_1,u_2)}^{\left(u\right)}(x,y_1)·B_{(u_1,u_2)}^{\left(u\right)}(x_1,y)\le B_{(u_1,u_2)}^{\left(u\right)}(x,y)B_{(u_1,u_2)}^{\left(u\right)}(x_1,y_1).$$

(35)

Proof. 

Let $f_1,f_2,f_3:[0,1]⟶$${\mathbb{R}}^{+}\cup 0$ be functions such that

$$f_1\left(t\right)=t^{(x-x_1)},f_2\left(t\right)={(1-t)}^{(y-y_1)},$$

$$f_3\left(t\right)=t^{(x_1-1)}{(1-t)}^{(y_1-1)}{E}_{u_1,u_2}\left(-u{\left(t(1-t)\right)}^{-1}\right)$$

Since

$$f_1^{\prime }\left(t\right)=(x-x_1)t^{(x-x_1-1)},\mathrm{and}{f}_2^{\prime }\left(t\right)=(y-y_1){(1-t)}^{(y-y_1-1)},$$

functions $f_1$ and $f_2$ have the same monotonicity (increasing or decreasing) on $[0,1]$ as $(x-x_1)(y-y_1)\ge 0$.

Then, applying Chebyshev’s inequality given in (33), to $f_1$, $f_2$, and $f_3$, we have:

$$\begin{array}{c}{\int }_0^1{t}^{x-1}{(1-t)}^{y_1-1}{E}_{u_1,u_2}\left(-u{\left(t(1-t)\right)}^{-1}\right)dt·{\int }_0^1{t}^{x_1-1}{(1-t)}^{y-1}{E}_{u_1,u_2}\left(-u{\left(t(1-t)\right)}^{-1}\right)dt\\ {\int }_0^1{t}^{x-1}{(1-t)}^{y-1}{E}_{u_1,u_2}\left(-u{\left(t(1-t)\right)}^{-1}\right)dt·{\int }_0^1{t}^{x_1-1}{(1-t)}^{y_1-1}{E}_{u_1,u_2}\left(-u{\left(t(1-t)\right)}^{-1}\right)dt.\end{array}$$

(36)

Then, from definition of extended beta function (31), we get our desired result:

$$B_{(u_1,u_2)}^{\left(u\right)}(x,y_1)·B_{(u_1,u_2)}^{\left(u\right)}(x_1,y)\le B_{(u_1,u_2)}^{\left(u\right)}(x,y)·B_{(u_1,u_2)}^{\left(u\right)}(x_1,y_1).$$

□

Corollary 1.

Assuming that $x,y>0$, $u_1\in [0,1]$ and $u_2\in [0,1]$, then:

$${\left[B_{(u_1,u_2)}^{\left(u\right)}(x,y)\right]}^2\ge B_{(u_1,u_2)}^{\left(u\right)}(x,x)·B_{(u_1,u_2)}^{\left(u\right)}(y,y).$$

(37)

Proof. 

Put $y_1=x$ and $x_1=y$ in theorem (1). Using symmetry property of extended beta function $B_{(u_1,u_2)}^{\left(u\right)}(x,y)=B_{(u_1,u_2)}^{\left(u\right)}(y,x)$, we get our desired result:

$${\left[B_{(u_1,u_2)}^{\left(u\right)}(x,y)\right]}^2\ge B_{(u_1,u_2)}^{\left(u\right)}(x,x)·B_{(u_1,u_2)}^{\left(u\right)}(y,y).$$

□

Theorem 2.

The map $(x,y)⟼B_{(u_1,u_2)}^{\left(u\right)}(x,y)$ is logarithmically convex on ${\mathbb{R}}^{+}\times {\mathbb{R}}^{+}$ $\forall u\ge 0$, with $u_1\in [0,1]$ and $u_2\in [0,1]$. Moreover:

$${\left[B_{(u_1,u_2)}^{\left(u\right)}\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\right]}^2\le B_{(u_1,u_2)}^{\left(u\right)}(x_1,y_1)·B_{(u_1,u_2)}^{\left(u\right)}(x_2,y_2).$$

(38)

Proof. 

Let $(x_1,y_1),(x_2,y_2)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^{+}$ and let $a,b\ge 0$ with $a+b=1$.

After some algebraic manipulation on the left-hand side terms of the extended beta function (31), we get

$$B_{(u_1,u_2)}^{\left(u\right)}\left(a(x_1,y_1)+b(x_2,y_2)\right)=B_{(u_1,u_2)}^{\left(u\right)}(a{x}_1+b{x}_2,a{y}_1+b{y}_2)$$

Moreover, using definition of the extended beta function (31), we can write

$$B_{(u_1,u_2)}^{\left(u\right)}(a{x}_1+b{x}_2,a{y}_1+b{y}_2)={\int }_0^1{t}^{a{x}_1+b{x}_2-1}{(1-t)}^{a{y}_1+b{y}_2-1}{E}_{u_1,u_2}\left(-u{\left(t(1-t)\right)}^{-1}\right)dt;$$

$$\Downarrow$$

$$B_{(u_1,u_2)}^{\left(r\right)}(a{x}_1+b{x}_2,a{y}_1+b{y}_2)={\int }_0^1{t}^{a(x_1-1)}{t}^{b(x_2-1)}{(1-t)}^{a(y_1-1)}{(1-t)}^{b(y_2-1)}·$$

$$·{\left[E_{u_1,u_2}\left(-u{\left(t(1-t)\right)}^{-1}\right)\right]}^a·{\left[E_{u_1,u_2}\left(-u{\left(t(1-t)\right)}^{-1}\right)\right]}^{b}dt;$$

$$\Downarrow$$

$$B_{(u_1,u_2)}^{\left(u\right)}(a{x}_1+b{x}_2,a{y}_1+b{y}_2)={\int }_0^1{\left[t^{x_1-1}{(1-t)}^{y_1-1}{E}_{u_1,u_2}\left(-u{\left(t(1-t)\right)}^{-1}\right)\right]}^a·$$

$$·{\left[t^{x_2-1}{(1-t)}^{y_2-1}{E}_{u_1,u_2}\left(-u{\left(t(1-t)\right)}^{-1}\right)\right]}^{b}dt.$$

Putting ${\theta }_1=\frac{1}{a}$ and ${\theta }_2=\frac{1}{b}$, Hölder inequality (34) gives:

$$B_{(u_1,u_2)}^{\left(u\right)}(a{x}_1+b{x}_2,a{y}_1+b{y}_2)\le {\left[{\int }_0^1{t}^{x_1-1}{(1-t)}^{y_1-1}{E}_{u_1,u_2}\left(-u{\left(t(1-t)\right)}^{-1}\right)dt\right]}^a·$$

$$·{\left[{\int }_0^1{t}^{x_2-1}{(1-t)}^{y_2-1}{E}_{u_1,u_2}\left(-u{\left(t(1-t)\right)}^{-1}\right)dt\right]}^b={\left[B_{(u_1,u_2)}^{\left(u\right)}(x_1,y_1)\right]}^a·{\left[B_{(u_1,u_2)}^{\left(u\right)}(x_2,y_2)\right]}^b.$$

Then, from Definition (1), the function $B_{(u_1,u_2)}^{\left(u\right)}(x,y)$ is logarithmically convex on ${\mathbb{R}}^{+}\times {\mathbb{R}}^{+}$.

If $a=b=\frac{1}{2}$, we get our desired result:

$${\left[B_{(u_1,u_2)}^{\left(u\right)}\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\right]}^2\le B_{(u_1,u_2)}^{\left(u\right)}(x_1,y_1)·B_{(u_1,u_2)}^{\left(u\right)}(x_2,y_2).$$

□

3. Extension of Gauss Hypergeometric Function and Confluent Hypergeometric Function

In this section, we introduce a new extension of Gauss hypergeometric function and confluent hypergeometric function by using extended beta function given in (31). In integral representation of Gauss hypergeometric function and confluent hypergeometric function, we introduce Wiman’s function as a kernel.

Definition 2.

A new extended Gauss hypergeometric function $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$ is defined as follows:

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\sum _{n=0}^{\infty }\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+n,p_2-p_1)}{B(p_1,p_2-p_1)}{\left(p_0\right)}_n\frac{z^n}{n!},$$

(39)

where $\Re \left(p_2\right)>\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$, $r\ge 0$, $\mid z\mid <1$ and $B_{(r_1,r_2)}^{\left(r\right)}(x_1,x_2)$ extended beta function.

After substitution of beta function value in terms of gamma function

$$B(x,y)=\frac{\Gamma \left(x\right)\Gamma \left(y\right)}{\Gamma (x+y)},$$

in the above definition, we have another representation of a new extended Gauss hypergeometric function (39):

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\frac{\Gamma \left(p_2\right)}{\Gamma \left(p_1\right)\Gamma (p_2-p_1)}\sum _{n=0}^{\infty }{\left(p_0\right)}_n{B}_{(r_1,r_2)}^{\left(r\right)}(p_1+n,p_2-p_1)\frac{z^n}{n!}.$$

(40)

Definition 3.

A new extended confluent hypergeometric function ${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)$ is defined as follows:

$${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)=\sum _{n=0}^{\infty }\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+n,p_2-p_1)}{B(p_1,p_2-p_1)}\frac{z^n}{n!},$$

(41)

where $\Re \left(p_2\right)>\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$, $r\ge 0$ and $B_{(r_1,r_2)}^{\left(r\right)}(x_1,x_2)$ extended beta function.

After substitution of beta function value in terms of gamma function, in the above definition, we have another representation of a new extended confluent hypergeometric function (41):

$${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)=\frac{\Gamma \left(p_2\right)}{\Gamma \left(p_1\right)\Gamma (p_2-p_1)}\sum _{n=0}^{\infty }{B}_{(r_1,r_2)}^{\left(r\right)}(p_1+n,p_2-p_1)\frac{z^n}{n!},$$

(42)

where $\Re \left(p_2\right)>\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$ and $r\ge 0.$

Definition 4.

(New) A new form of beta function is defined as:

$$\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+n,p_2-p_1)}{B(p_1,p_2-p_1)}={\widehat{\mathcal{B}}}_{(r_1,r_2)}^{\left(r\right)}(p_1+n,p_2-p_1).$$

(43)

Using the above relation, we have a new form of extended Gauss hypergeometric function $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$ and extended confluent hypergeometric function ${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)$:

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\sum _{n=0}^{\infty }{\widehat{\mathcal{B}}}_{(r_1,r_2)}^{\left(r\right)}(p_1+n,p_2-p_1){\left(p_0\right)}_n\frac{z^n}{n!},$$

(44)

and

$${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)=\sum _{n=0}^{\infty }{\widehat{\mathcal{B}}}_{(r_1,r_2)}^{\left(r\right)}(p_1+n,p_2-p_1)\frac{z^n}{n!}.$$

(45)

Remark 4.

We know that Gauss’s hypergeometric function does not change if the $p_0$ and $p_1$ parameters are swapped while keeping $p_2$ fixed. This symmetric property with respect to the parameters $p_0$ and $p_1$ can also be deduced from the new extended Gauss hypergeometric function (39):

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=F_{(r_1,r_2)}^{\left(r\right)}(p_1,p_0,p_2;z).$$

(46)

Then, replacing $p_0$ with $p_1$ in (39), from above property (46), we have

$$\begin{array}{ccc}\hfill F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)& =& \sum _{n=0}^{\infty }\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+n,p_2-p_1)}{B(p_1,p_2-p_1)}{\left(p_0\right)}_n\frac{z^n}{n!}=\hfill \\ & =& \sum _{n=0}^{\infty }\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_0+n,p_2-p_0)}{B(p_0,p_2-p_0)}{\left(p_1\right)}_n\frac{z^n}{n!}.\hfill \end{array}$$

(47)

The beta function (43) can be rewritten in a similar form:

$$\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_0+n,p_2-p_0)}{B(p_0,p_2-p_0)}={\tilde{\mathcal{B}}}_{(r_1,r_2)}^{\left(r\right)}(p_0+n,p_2-p_0).$$

(48)

Using the above relation (48), we can rewrite extended Gauss hypergeometric function$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$in a similar form:

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\sum _{n=0}^{\infty }{\tilde{\mathcal{B}}}_{(r_1,r_2)}^{\left(r\right)}(p_0+n,p_2-p_0){\left(p_1\right)}_n\frac{z^n}{n!}.$$

(49)

Then, from above definitions and relation (44), (46) and (49), we get

$$\begin{array}{ccc}\hfill F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)& =& \sum _{n=0}^{\infty }{\tilde{\mathcal{B}}}_{(r_1,r_2)}^{\left(r\right)}(p_0+n,p_2-p_0){\left(p_1\right)}_n\frac{z^n}{n!}=\hfill \\ & =& \sum _{n=0}^{\infty }{\widehat{\mathcal{B}}}_{(r_1,r_2)}^{\left(r\right)}(p_1+n,p_2-p_1)\frac{z^n}{n!}.\hfill \end{array}$$

(50)

Remark 5.

(i) If we set$r_2=1$in (39) and (41), then, we get a new extension of Gauss hypergeometric function and confluent hypergeometric function given by (25) and (26), respectively:

$$F_{(r_1,1)}^{\left(r\right)}(p_0,p_1,p_2;z)=F_{r,r_1}(p_0,p_1,p_2;z);$$

(51)

and

$${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)={\Phi }_{r,r_1}(p_1,p_2;z).$$

(52)

(ii) If we set $r_1=r_2=1$ in (39) and (41), then we get extended Gauss hypergeometric function and extended confluent hypergeometric function given by (12) and (13), respectively:

$$F_{(1,1)}^{\left(r\right)}(p_0,p_1,p_2;z)=F_r(p_0,p_1,p_2;z);$$

(53)

and

$${\Phi }_{(1,1)}^{\left(r\right)}(p_1,p_2;z)={\Phi }_r(p_1,p_2;z).$$

(54)

(iii) If we set $r_1=r_2=1$ and $r=0$ in (39) and (41), then we get Gauss hypergeometric function and confluent hypergeometric function given by (3) and (6), respectively:

$$F_{(1,1)}^{\left(0\right)}(p_0,p_1,p_2;z)=F(p_0,p_1,p_2;z);$$

(55)

and

$${\Phi }_{(1,1)}^{\left(0\right)}(p_1,p_2;z)=\Phi (p_1,p_2;z).$$

(56)

Theorem 3.

Consider $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$ and ${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)$ functions. Then, the following functional relations hold:

(1)

$$\begin{array}{ccc}\hfill F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)& =& \frac{p_1}{p_2}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1+1,p_2+1;z)\hfill \\ & +& \frac{(p_2-p_1)}{p_1}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2+1;z),\hfill \end{array}$$

(57)

where $\Re \left(p_2\right)>0,\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$, $r\ge 0$ and $\mid z\mid <1$.

(2)

$$\begin{array}{ccc}\hfill {\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)& =& \frac{p_1}{p_2}{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1+1,p_2+1;z)\hfill \\ & +& \frac{(p_2-p_1)}{p_1}{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2+1;z),\hfill \end{array}$$

(58)

where $\Re \left(p_2\right)>0,\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$ and $r\ge 0$.

Proof. 

Using the following known relation from Reference [13]:

$$B_{(u_1,u_2)}^{\left(u\right)}(y_1,y_2)=B_{(u_1,u_2)}^{\left(u\right)}(y_1,y_2+1)+B_{(u_1,u_2)}^{\left(u\right)}(y_1+1,y_2),$$

(59)

in (39), we get

$$\begin{array}{ccc}& & F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\hfill \\ & =& \sum _{n=0}^{\infty }{\left(p_0\right)}_n\left(\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+n+1,p_2-p_1)+B_{(r_1,r_2)}^{\left(r\right)}(p_1+n,p_2-p_1+1)}{B(p_1,p_2-p_1)}\right)\frac{z^n}{n!}=\hfill \\ & =& \sum _{n=0}^{\infty }{\left(p_0\right)}_n\left(\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+n+1,p_2-p_1)}{B(p_1,p_2-p_1)}\right)\frac{z^n}{n!}+\hfill \\ & +& \sum _{n=0}^{\infty }{\left(p_0\right)}_n\left(\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+n,p_2-p_1+1)}{B(p_1,p_2-p_1)}\right)\frac{z^n}{n!}=\hfill \\ & =& \frac{B(p_1+1,p_2-p_1)}{B(p_1,p_2-p_1)}\sum _{n=0}^{\infty }{\left(p_0\right)}_n\left(\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+n+1,p_2-p_1)}{B(p_1+1,p_2-p_1)}\right)\frac{z^n}{n!}+\hfill \\ & +& \frac{B(p_1,p_2-p_1+1)}{B(p_1,p_2-p_1)}\sum _{n=0}^{\infty }{\left(p_0\right)}_n\left(\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+n,p_2-p_1+1)}{B(p_1,p_2-p_1+1)}\right)\frac{z^n}{n!}\hfill \end{array}$$

Then, using the value of beta function in terms of gamma function together with (39), allow us to get the desired result.

Similarly, using (59) in (41) and following the same rule that led to the result (57), we obtain the desired statement (58), and the Theorem is fully proved. □

Remark 6.

(i) From (46), we observe that, for$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$, another functional relation that also holds is:

$$\begin{array}{ccc}\hfill F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)& =& \frac{p_0}{p_2}{F}_{(r_1,r_2)}^{\left(r\right)}(p_1,p_0+1,p_2+1;z)\hfill \\ & +& \frac{(p_2-p_0)}{p_0}{F}_{(r_1,r_2)}^{\left(r\right)}(p_1,p_0,p_2+1;z).\hfill \end{array}$$

(60)

(ii) If we used (46) on the right-hand side of (60),

we obtain another functional relation for $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$:

$$\begin{array}{ccc}\hfill F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)& =& \frac{p_0}{p_2}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0+1,p_1,p_2+1;z)\hfill \\ & +& \frac{(p_2-p_0)}{p_0}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2+1;z).\hfill \end{array}$$

(61)

Theorem 4.

Consider $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$ and ${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)$ functions. The following Sum relations hold:

(1)

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=(p_2-p_1)\sum _{k=0}^{\infty }\frac{{\left(p_1\right)}_k}{{\left(p_2\right)}_{k+1}}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1+1,p_2+k+1;z),$$

(62)

where $\Re \left(p_2\right)>0,\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$, $r\ge 0$ and $\mid z\mid <1$.

(2)

$${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)=(p_2-p_1)\sum _{k=0}^{\infty }\frac{{\left(p_1\right)}_k}{{\left(p_2\right)}_{k+1}}{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1+k,p_2+k+1;z),$$

(63)

where $\Re \left(p_2\right)>0,\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$ and $r\ge 0$.

Proof. 

Using the following known relation from Reference [13]:

$$B_{(u_1,u_2)}^{\left(u\right)}(y_1,y_2)=\sum _{k=0}^{\infty }{B}_{(u_1,u_2)}^{\left(u\right)}(y_1+k,y_2+1),$$

(64)

in (39), we have

$$\begin{array}{ccc}& & F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\hfill \\ & =& \sum _{n=0}^{\infty }\frac{{\sum }_{k=0}^{\infty }{B}_{(r_1,r_2)}^{\left(r\right)}(p_1+n+k,p_2-p_1+1)}{B(p_1,p_2-p_1)}{\left(p_0\right)}_n\frac{z^n}{n!}=\hfill \\ & =& \sum _{k=0}^{\infty }\left(\sum _{n=0}^{\infty }\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+n+k,p_2-p_1+1)}{B(p_1,p_2-p_1)}{\left(p_0\right)}_n\frac{z^n}{n!}\right)=\hfill \\ & =& \sum _{k=0}^{\infty }\frac{B(p_1+k,p_2-p_1+1)}{B(p_1,p_2-p_1)}\sum _{n=0}^{\infty }\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+k+n,p_2-p_1+1)}{B(p_1+k,p_2-p_1+1)}{\left(p_0\right)}_n\frac{z^n}{n!}.\hfill \end{array}$$

Moreover, from (39)

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\sum _{k=0}^{\infty }\frac{B(p_1+k,p_2-p_1+1)}{B(p_1,p_2-p_1)}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1+k,p_2+k+1;z).$$

Using value of beta function in terms of gamma function and value of gamma function in terms of Pochhammer symbol in the above equation, we get our desired statement:

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=(p_2-p_1)\sum _{k=0}^{\infty }\frac{{\left(p_1\right)}_k}{{\left(p_2\right)}_{k+1}}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1+1,p_2+k+1;z).$$

Similarly, using (64) in (41) and following the same rule getting to result (62), we obtain the desired statement (63), and the Theorem is fully proved. □

Remark 7.

(i) From (46), we observe that, for $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$, another summation relation that also holds is:

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=(p_2-p_0)\sum _{k=0}^{\infty }\frac{{\left(p_0\right)}_k}{{\left(p_2\right)}_{k+1}}{F}_{(r_1,r_2)}^{\left(r\right)}(p_1,p_0+1,p_2+k+1;z).$$

(65)

(ii) If we used (46) on the right-hand side of (65), we obtain another summation relation for $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$:

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=(p_2-p_0)\sum _{k=0}^{\infty }\frac{{\left(p_0\right)}_k}{{\left(p_2\right)}_{k+1}}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0+1,p_1,p_2+k+1;z).$$

(66)

Theorem 5.

Consider $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$ and ${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)$ functions. The following Sum relations hold:

(1)

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\sum _{k=0}^{\infty }\frac{{(p_1-p_2+1)}_{k}B(p_1+k,1)}{k!B(p_1,p_2-p_1)}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1+k,p_1+k+1;z),$$

(67)

where $\Re \left(p_2\right)>0$, $\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0$, $\Re \left(r_2\right)>0$, $r\ge 0$ and $\mid z\mid <1$.

(2)

$${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)=\sum _{k=0}^{\infty }\frac{{(p_1-p_2+1)}_{k}B(p_1+k,1)}{k!B(p_1,p_2-p_1)}{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1+k,p_1+k+1;z),$$

(68)

where $\Re \left(p_2\right)>0$, $\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0$, $\Re \left(r_2\right)>0$ and $r\ge 0$.

Proof. 

Using the following known relation from Reference [13]:

$$B_{(u_1,u_2)}^{\left(u\right)}(y_1,1-y_2)=\sum _{k=0}^{\infty }\frac{{\left(y_2\right)}_k}{k!}{B}_{(u_1,u_2)}^{\left(u\right)}(y_1+k,1),$$

(69)

in (39), we get

$$\begin{array}{ccc}& & F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\hfill \\ & =& \sum _{n=0}^{\infty }\frac{{\left(p_0\right)}_n}{B(p_1,p_2-p_1)}\left(\sum _{k=0}^{\infty }\frac{{(1+p_1-p_2)}_k}{k!}{B}_{(r_1,r_2)}^{\left(r\right)}(p_1+n+k,1)\right)\frac{z^n}{n!}=\hfill \\ & =& \sum _{k=0}^{\infty }\frac{B(p_1+k,1){(1+p_1-p_2)}_k}{B(p_1,p_2-p_1)k!}\sum _{n=0}^{\infty }\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+k+n,1)}{B(p_1+k,1)}{\left(p_0\right)}_n\frac{z^n}{n!}=\hfill \\ & =& \sum _{k=0}^{\infty }\frac{{(p_1-p_2+1)}_{k}B(p_1+k,1)}{k!B(p_1,p_2-p_1)}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1+k,p_1+k+1;z).\hfill \end{array}$$

Similarly, using (69) in (41) and following the same rule getting to result (67), we obtain the desired statement (68), and the Theorem is fully proved. □

Remark 8.

(i) From (46), we observe that, for function$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$, another summation relation that also holds is:

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\sum _{k=0}^{\infty }\frac{{(p_0-p_2+1)}_{k}B(p_0+k,1)}{k!B(p_0,p_2-p_0)}{F}_{(r_1,r_2)}^{\left(r\right)}(p_1,p_0+k,p_0+k+1;z).$$

(70)

(ii) If we used (46) on the right-hand side of (70),

we have another summation relation for $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$:

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\sum _{k=0}^{\infty }\frac{{(p_0-p_2+1)}_{k}B(p_0+k,1)}{k!B(p_0,p_2-p_0)}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0+k,p_1,p_0+k+1;z).$$

(71)

Theorem 6.

Consider $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$ and ${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)$ functions. The following results hold:

(1)

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\sum _{l=0}^k\left(\genfrac{}{}{0pt}{}{k}{l}\right)\frac{B(p_1+l,p_2-p_1-l+k)}{B(p_1,p_2-p_1)}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1+l,p_2+k;z),$$

(72)

where $\Re \left(p_2\right)>0,\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$, $r\ge 0$,$k\in N$ and $\mid z\mid <1$.

(2)

$${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)=\sum _{l=0}^k\left(\genfrac{}{}{0pt}{}{k}{l}\right)\frac{B(p_1+l,p_2-p_1-l+k)}{B(p_1,p_2-p_1)}{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1+l,p_2+k;z),$$

(73)

where $\Re \left(p_2\right)>0,\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$, $r\ge 0$ and $k\in N$.

Proof. 

Using the following known relation from Reference [13]:

$$B_{(u_1,u_2)}^{\left(u\right)}(y_1,y_2)=\sum _{l=0}^k\left(\genfrac{}{}{0pt}{}{k}{l}\right)B_{(u_1,u_2)}^{\left(u\right)}(y_1+l,y_2+k-l),$$

(74)

in (39), we have

$$\begin{array}{ccc}& & F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\hfill \\ & =& \sum _{n=0}^{\infty }\frac{{\left(p_0\right)}_n}{B(p_1,p_2-p_1)}\left(\sum _{l=0}^k\left(\genfrac{}{}{0pt}{}{k}{l}\right)B_{(r_1,r_2)}^{\left(r\right)}(p_1+n+l,p_2-p_1+k-l)\right)\frac{z^n}{n!}=\hfill \\ & =& \sum _{l=0}^k\left(\genfrac{}{}{0pt}{}{k}{l}\right)\frac{1}{B(p_1,p_2-p_1)}·\sum _{n=0}^{\infty }\left(B_{(r_1,r_2)}^{\left(r\right)}(p_1+n+l,p_2-p_1+k-l){\left(p_0\right)}_n\frac{z^n}{n!}\right)=\hfill \\ & =& \sum _{l=0}^k\left(\genfrac{}{}{0pt}{}{k}{l}\right)\frac{B(p_1+l,p_2-p_1-l+k)}{B(p_1,p_2-p_1)}\sum _{n=0}^{\infty }{\left(p_0\right)}_n\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+n+l,p_2-p_1+k-l)}{B(p_1+l,p_2-p_1-l+k)}\frac{z^n}{n!}.\hfill \end{array}$$

From (39), we get our desired result:

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\sum _{l=0}^k\left(\genfrac{}{}{0pt}{}{k}{l}\right)\frac{B(p_1+l,p_2-p_1-l+k)}{B(p_1,p_2-p_1)}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1+l,p_2+k;z).$$

Similarly, using (74) in (41) and following the same rule getting to result (72), we obtain the desired statement (73), and the Theorem is fully proved. □

Remark 9.

(i) From (46), we observe that, for $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$, the following result holds:

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\sum _{l=0}^k\left(\genfrac{}{}{0pt}{}{k}{l}\right)\frac{B(p_0+l,p_2-p_0+k)}{B(p_0,p_2-p_0)}{F}_{(r_1,r_2)}^{\left(r\right)}(p_1,p_0+l,p_2+k;z).$$

(75)

(ii) If we used (46) on the right-hand side of (75), we have another result for $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$:

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\sum _{l=0}^k\left(\genfrac{}{}{0pt}{}{k}{l}\right)\frac{B(p_0+l,p_2-p_0+k)}{B(p_0,p_2-p_0)}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0+l,p_1,p_2+k;z).$$

(76)

Theorem 7.

Consider $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$ and ${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)$ functions. The following integral representations hold:

(1)

$$\begin{array}{ccc}\hfill & & F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\hfill \\ & =& \frac{1}{B(p_1,p_2-p_1)}{\int }_0^1{t}^{p_1-1}{(1-t)}^{p_2-p_1-1}{(1-zt)}^{-p_0}{E}_{r_1,r_2}\left(-r{\left(t(1-t)\right)}^{-1}\right)dt,\hfill \end{array}$$

(77)

where $\Re \left(p_2\right)>\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$, $r\ge 0$ and $\mid z\mid <1$.

(2)

$$\begin{array}{ccc}\hfill & & {\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)=\hfill \\ & =& \frac{1}{B(p_1,p_2-p_1)}{\int }_0^1{t}^{p_1-1}{(1-t)}^{p_2-p_1-1}{e}^{zt}{E}_{r_1,r_2}\left(-r{\left(t(1-t)\right)}^{-1}\right)dt,\hfill \end{array}$$

(78)

where $\Re \left(p_2\right)>\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$ and $r\ge 0$.

Proof. 

Using the following known relation from Reference [13]:

$$B_{(u_1,u_2)}^{\left(u\right)}(y_1,y_2)={\int }_0^1{t}^{y_1-1}{(1-t)}^{y_2-1}{E}_{u_1,u_2}\left(-u{\left(t(1-t)\right)}^{-1}\right)dt$$

(79)

in (39), we have:

$$\begin{array}{ccc}& & F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\hfill \\ & =& \frac{1}{B(p_1,p_2-p_1)}\sum _{n=0}^{\infty }{\left(p_0\right)}_n·\left({\int }_0^1{t}^{p_1+n-1}{(1-t)}^{p_2-p_1-1}{E}_{r_1,r_2}\left(-r{\left(t(1-t)\right)}^{-1}\right)dt\right)\frac{z^n}{n!}\hfill \\ & =& \frac{1}{B(p_1,p_2-p_1)}\sum _{n=0}^{\infty }{\left(p_0\right)}_n·\left({\int }_0^1{t}^{p_1-1}{(1-t)}^{p_2-p_1-1}{E}_{r_1,r_2}\left(-r{\left(t(1-t)\right)}^{-1}\right)t^{n}dt\right)\frac{z^n}{n!}\hfill \end{array}$$

Changing the order of integration and summation, we get:

$$\begin{array}{ccc}& & F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\hfill \\ & =& \frac{1}{B(p_1,p_2-p_1)}{\int }_0^1{t}^{p_1-1}{(1-t)}^{p_2-p_1-1}{E}_{r_1,r_2}\left(-r{\left(t(1-t)\right)}^{-1}\right)dt\sum _{n=0}^{\infty }{\left(p_0\right)}_n\frac{{\left(zt\right)}^n}{n!}\hfill \end{array}$$

Since

$${(1-zt)}^{-p_0}=\sum _{n=0}^{\infty }\frac{{\left(p_0\right)}_n}{n!}{\left(zt\right)}^n,\mid t\mid <1,$$

(80)

the last expression becomes the searched result:

$$\begin{array}{ccc}& & F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\hfill \\ & =& \frac{1}{B(p_1,p_2-p_1)}{\int }_0^1{t}^{p_1-1}{(1-t)}^{p_2-p_1-1}{(1-zt)}^{-p_0}{E}_{r_1,r_2}\left(-r{\left(t(1-t)\right)}^{-1}\right)dt.\hfill \end{array}$$

Similarly, using (79) in (41) and following the same rule getting to result (77), we obtain the desired statement (78), and the Theorem is fully proved. □

Remark 10.

From (46), we observe that, for $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$, the following integral representation holds:

$$\begin{array}{ccc}\hfill & & F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\hfill \\ & =& \frac{1}{B(p_0,p_2-p_0)}{\int }_0^1{t}^{p_0-1}{(1-t)}^{p_2-p_0-1}{(1-zt)}^{-p_1}{E}_{r_1,r_2}\left(-r{\left(t(1-t)\right)}^{-1}\right)dt.\hfill \end{array}$$

(81)

Corollary 2.

Consider $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$ and ${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)$ functions. The following integral representations hold:

(1)

$$\begin{array}{ccc}\hfill F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)& =& \frac{2}{B(p_1,p_2-p_1)}{\int }_0^{\frac{\pi }{2}}{\left(cos\left(x\right)\right)}^{2p_1-1}{\left(sin\left(x\right)\right)}^{2p_2-2p_1-1}·\hfill \\ & ·& E_{r_1,r_2}\left(\frac{-r}{{(cos\left(x\right))}^2{(sin\left(x\right))}^2}\right)\left({(1-z{(cos\left(x\right))}^2)}^{-p_0}\right)dx.\hfill \end{array}$$

(82)

(2)

$$\begin{array}{ccc}\hfill {\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)& =& \frac{2}{B(p_1,p_2-p_1)}{\int }_0^{\frac{\pi }{2}}{\left(cos\left(x\right)\right)}^{2p_1-1}{\left(sin\left(x\right)\right)}^{2p_2-2p_1-1}·\hfill \\ & ·& E_{r_1,r_2}\left(\frac{-r}{{(cos\left(x\right))}^2{(sin\left(x\right))}^2}\right)e^{z{(cos\left(x\right))}^2}dx.\hfill \end{array}$$

(83)

Proof. 

The corollary is easily proved by choosing $t={cos}^2\left(x\right)$ in (77) and (78). □

Corollary 3.

Consider $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$ and ${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2,z)$ functions. The following integral representations hold:

(1)

$$\begin{array}{ccc}\hfill F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)& =& \frac{2}{B(p_1,p_2-p_1)}{\int }_0^{\frac{\pi }{2}}{\left(sin\left(x\right)\right)}^{2p_1-1}{\left(cos\left(x\right)\right)}^{2p_2-2p_1-1}·\hfill \\ & ·& E_{r_1,r_2}\left(\frac{-r}{{(cos\left(x\right))}^2{(sin\left(x\right))}^2}\right)\left(1-z{\left({(sin\left(x\right))}^2\right)}^{-p_0}\right)dx.\hfill \end{array}$$

(84)

(2)

$$\begin{array}{ccc}\hfill {\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)& =& \frac{2}{B(p_1,p_2-p_1)}{\int }_0^{\frac{\pi }{2}}{\left(sin\left(x\right)\right)}^{2p_1-1}{\left(cos\left(x\right)\right)}^{2p_2-2p_1-1}·\hfill \\ & ·& E_{r_1,r_2}\left(\frac{-r}{{(cos\left(x\right))}^2{(sin\left(x\right))}^2}\right)e^{z{(sin\left(x\right))}^2}dx.\hfill \end{array}$$

(85)

Proof. 

The corollary is easily proved by choosing $t={sin}^2\left(x\right)$ in (77) and (78). □

Theorem 8.

The Mellin Transformations for $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$ and ${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)$ are given by:

(1)

$$\mathit{M}\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z);s\}=\frac{{\Gamma }_0^{(r_1,r_2)}\left(s\right)B(p_1+s,p_2-p_1+s)}{B(p_1,p_2-p_1)}F(p_0,p_1+s,p_2+2s;z),$$

(86)

where $\Re \left(p_2\right)>\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$, $s\ge 0$, $r\ge 0$ and $\mid z\mid <1$.

(2)

$$\mathit{M}\{{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z);s\}=\frac{{\Gamma }_0^{(r_1,r_2)}\left(s\right)B(p_1+s,p_2-p_1+s)}{B(p_1,p_2-p_1)}\Phi (p_1+s,p_2+2s;z),$$

(87)

where $\Re \left(p_2\right)>\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$, $s\ge 0$ and $r\ge 0$.

Proof. 

From the definition of Mellin Transformation, we have:

$$\mathbf{M}\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z);s\}={\int }_0^{\infty }{r}^{s-1}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)dr.$$

Using (77) on the right-hand side, we have

$$\begin{array}{ccc}\hfill \mathbf{M}\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z);s\}& =& {\int }_0^{\infty }\{\frac{1}{B(p_1,p_2-p_1)}{\int }_0^1{t}^{p_1-1}{(1-t)}^{p_2-p_1-1}{(1-zt)}^{-p_0}·\hfill \\ & ·& E_{r_1,r_2}\left(-r{\left(t(1-t)\right)}^{-1}\right)dt\}{r}^{s-1}dr\hfill \end{array}$$

Interchanging the order of integrations, the last expression reads as:

$$\begin{array}{ccc}\hfill \mathbf{M}\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z);s\}& =& \frac{1}{B(p_1,p_2-p_1)}{\int }_0^1{t}^{p_1-1}{(1-t)}^{p_2-p_1-1}{(1-zt)}^{-p_0}·\hfill \\ & ·& \left[{\int }_0^{\infty }{r}^{s-1}{E}_{r_1,r_2}\left(-r{\left(t(1-t)\right)}^{-1}\right)dr\right]dt.\hfill \end{array}$$

Let $v=\frac{r}{t(1-t)}$. Then:

$$\begin{array}{ccc}\hfill \mathbf{M}\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z);s\}& =& \frac{1}{B(p_1,p_2-p_1)}{\int }_0^1{t}^{p_1+s-1}{(1-t)}^{p_2-p_1+s-1}{(1-zt)}^{-p_0}·\hfill \\ & ·& \left[{\int }_0^{\infty }{v}^{s-1}{E}_{r_1,r_2}(-v)dv\right]dt\hfill \end{array}$$

From the known result in Reference [21]:

$${\Gamma }_r^{(r_1,r_2)}\left(x_1\right)={\int }_0^{\infty }{t}^{x_1-1}{E}_{r_1,r_2}\left(-t-r{t}^{-1}\right)dt,$$

(88)

and, using it with $r=0$ in the above equation, we get

$$\begin{array}{ccc}& & \mathbf{M}\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z);s\}=\hfill \\ \\ & =& \frac{1}{B(p_1,p_2-p_1)}{\int }_0^1{t}^{p_1+s-1}{(1-t)}^{p_2-p_1+s-1}{(1-zt)}^{-p_0}\left[{\Gamma }_0^{(r_1,r_2)}\left(s\right)\right]dt=\hfill \\ \\ & =& \frac{B(p_1+s,p_2-p_1+s)}{B(p_1,p_2-p_1)}{\int }_0^1\frac{t^{p_1+s-1}{(1-t)}^{p_2-p_1+s-1}{(1-zt)}^{-p_0}}{B(p_1+s,p_2-p_1+s)}\left[{\Gamma }_0^{(r_1,r_2)}\left(s\right)\right]dt=\hfill \\ \\ & =& \frac{B(p_1+s,p_2-p_1+s){\Gamma }_0^{(r_1,r_2)}\left(s\right)}{B(p_1,p_2-p_1)}{\int }_0^1\frac{t^{p_1+s-1}{(1-t)}^{p_2-p_1+s-1}{(1-zt)}^{-p_0}}{B(p_1+s,p_2-p_1+s)}dt.\hfill \end{array}$$

Finally, from integral representation of Gauss hypergeometric function (5), we get our desired result:

$$\mathbf{M}\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z);s\}=\frac{{\Gamma }_0^{(r_1,r_2)}\left(s\right)B(p_1+s,p_2-p_1+s)}{B(p_1,p_2-p_1)}F(p_0,p_1+s,p_2+2s;z).$$

Similarly, using Mellin transformation, Equation (78), and following the same rule getting to result (86), we obtain the desired statement (87), and the Theorem is fully proved. □

Remark 11.

(i) From (46), we observe that, for $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$, the following Mellin representation also holds:

$$\mathit{M}\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z);s\}=\frac{{\Gamma }_0^{(r_1,r_2)}\left(s\right)B(p_0+s,p_2-p_0+s)}{B(p_0,p_2-p_0)}F(p_1,p_0+s,p_2+2s;z).$$

(89)

(ii) Using symmetric property of Gauss hypergeometric function $F(p_o,p_1,p_2;z)=F(p_1,p_0,p_2;z)$ on the right-hand side of (89), we have another Mellin representation for $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$:

$$\mathit{M}\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z);s\}=\frac{{\Gamma }_0^{(r_1,r_2)}\left(s\right)B(p_0+s,p_2-p_0+s)}{B(p_0,p_2-p_0)}F(p_0+s,p_1,p_2+2s;z).$$

(90)

Theorem 9.

The Pfaff’s transformations for $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$ is given by:

(1)

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)={(1-z)}^{-p_0}{F}_{(r_1,r_2)}^{\left(r\right)}\left(p_0,p_2-p_1,p_2;\frac{z}{z-1}\right),$$

(91)

where $\Re \left(p_2\right)>\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$, $r\ge 0$ and $\mid \frac{z}{z-1}\mid <1$.

(2)

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)={(1-z)}^{-p_1}{F}_{(r_1,r_2)}^{\left(r\right)}\left(p_2-p_0,p_1,p_2;\frac{z}{z-1}\right),$$

(92)

where $\Re \left(p_2\right)>\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$, $r\ge 0$ and $\mid \frac{z}{z-1}\mid <1$.

Proof. 

From (77), we deduce:

$$\begin{array}{ccc}& & F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)=\hfill \\ & =& \frac{1}{B(p_1,p_2-p_1)}·{\int }_0^1{v}^{p_1-1}{(1-v)}^{p_2-p_1-1}{(1-zv)}^{-p_0}{E}_{r_1,r_2}\left(-r{\left(v(1-v)\right)}^{-1}\right)dv.\hfill \end{array}$$

Let $v=1-t$. After some algebraic manipulation, the last expression reads as:

$${[1-z(1-t)]}^{-p_0}={(1-z)}^{-p_0}{\left(1+\frac{z}{1-z}t\right)}^{-p_0}.$$

Thus, we get our desired result (91):

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)={(1-z)}^{-p_0}{F}_{(r_1,r_2)}^{\left(r\right)}\left(p_0,p_2-p_1,p_2;\frac{z}{z-1}\right).$$

From (46) and (91), we deduce that:

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)={(1-z)}^{-p_1}{F}_{(r_1,r_2)}^{\left(r\right)}\left(p_1,p_2-p_0,p_2;\frac{z}{z-1}\right).$$

Moreover, using same property (46) on the right-hand side of the above equation, we get our statement (92):

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)={(1-z)}^{-p_1}{F}_{(r_1,r_2)}^{\left(r\right)}\left(p_2-p_0,p_1,p_2;\frac{z}{z-1}\right).$$

□

Theorem 10.

The Euler’s transformation for $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$ is given by:

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)={(1-z)}^{p_2-p_0-p_1}{F}_{(r_1,r_2)}^{\left(r\right)}(p_2-p_0,p_2-p_1,p_2;z),$$

(93)

where $\Re \left(p_2\right)>\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$, $r\ge 0$ and $\mid z\mid <1$.

Proof. 

From (91) and (92), we obtain:

$${(1-z)}^{-p_0}{F}_{(r_1,r_2)}^{\left(r\right)}\left(p_0,p_2-p_1,p_2;\frac{z}{z-1}\right)={(1-z)}^{-p_1}{F}_{(r_1,r_2)}^{\left(r\right)}\left(p_2-p_0,p_1,p_2;\frac{z}{z-1}\right).$$

Now, replacing $p_0$ with $p_2-p_0$ and z with $\frac{z}{z-1}$, we complete the theorem’s proof:

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)={(1-z)}^{p_2-p_0-p_1}{F}_{(r_1,r_2)}^{\left(r\right)}(p_2-p_0,p_2-p_1,p_2;z).$$

□

Remark 12.

Using (46) on the right-hand side of (93), we deduce another Euler’s transformation for $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$

$$F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)={(1-z)}^{p_2-p_0-p_1}{F}_{(r_1,r_2)}^{\left(r\right)}(p_2-p_1,p_2-p_0,p_2;z).$$

(94)

Theorem 11.

For ${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)$, we obtain the extension of Kummer transformation:

$${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)=e^z{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_2-p_1,p_2;-z),$$

(95)

where $\Re \left(p_2\right)>0,\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$ and $r\ge 0$.

Proof. 

From (78), we get:

$${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)=\frac{1}{B(p_1,p_2-p_1)}{\int }_0^1{u}^{p_1-1}{(1-u)}^{p_2-p_1-1}{e}^{zu}{E}_{r_1,r_2}\left(-r{\left(u(1-u)\right)}^{-1}\right)du.$$

Let $u=1-t$. After some algebraic manipulation, the last expression becomes:

$${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)=e^z\frac{1}{B(p_1,p_2-p_1)}{\int }_0^1{t}^{p_2-p_1-1}{(1-t)}^{p_1-1}{e}^{-zt}{E}_{r_1,r_2}\left(-r{\left(t(1-t)\right)}^{-1}\right)du.$$

Then, from (78), we obtain our statement (95). □

Theorem 12.

Consider ${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)$ function. The following relation holds:

$${\int }_0^{\infty }{z}^{p_0-1}{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;-z)dz=\Gamma \left(p_0\right)F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_2-p_1,p_2;1),$$

(96)

where $\Re \left(p_2\right)>0,\Re \left(p_1\right)>0$, $\Re \left(p_0\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$ and $r\ge 0$.

Proof. 

Replacing z by $-z$ in (95), we can write

$${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;-z)=e^{-z}{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_2-p_1,p_2;z).$$

Multiplying by $z^{p_0-1}$ and integrating the above equation with respect to z from $z=0$ to $z=\infty$, we get:

$${\int }_0^{\infty }{z}^{p_0-1}{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;-z)dz={\int }_0^{\infty }{z}^{p_0-1}\left\{e^{-z}{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_2-p_1,p_2;z)\right\}dz.$$

Using (41) on the right-hand side of last expression, we obtain:

$${\int }_0^{\infty }{z}^{p_0-1}{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;-z)dz={\int }_0^{\infty }{z}^{p_0-1}{e}^{-z}\sum _{n=0}^{\infty }\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_2-p_1+n,p_1)}{B(p_2-p_1,p_1)}\frac{z^n}{n!}dz.$$

Then, changing order of integration and summation, we can read:

$${\int }_0^{\infty }{z}^{p_0-1}{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;-z)dz=\sum _{n=0}^{\infty }\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_2-p_1+n,p_1)}{B(p_2-p_1,p_1)n!}{\int }_0^{\infty }{z}^{n+p_0-1}{e}^{-z}dz.$$

From definition of gamma function and from its relation with Pochhammer symbol

$$\Gamma (n+p_0)={\left(p_0\right)}_n\Gamma \left(p_0\right),$$

we obtain:

$${\int }_0^{\infty }{z}^{p_0-1}{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;-z)dz=\sum _{n=0}^{\infty }\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_2-p_1+n,p_1)}{B(p_2-p_1,p_1)n!}{\left(p_0\right)}_n\Gamma \left(p_0\right).$$

Finally, from (39), we get our desired result:

$${\int }_0^{\infty }{z}^{p_0-1}{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;-z)dz=\Gamma \left(p_0\right)F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_2-p_1,p_2;1).$$

□

Remark 13.

Using (46) on the right-hand side of Equation (96), we get another relation for ${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)$:

$${\int }_0^{\infty }{z}^{p_0-1}{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;-z)dz=\Gamma \left(p_0\right)F_{(r_1,r_2)}^{\left(r\right)}(p_2-p_1,p_0,p_2;1).$$

(97)

Theorem 13.

Consider $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$ and ${\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)$ functions. The following differentiation formulas hold:

(1)

$$\frac{d^m}{d{z}^m}\left\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)\right\}=\frac{{\left(p_0\right)}_m{\left(p_1\right)}_m}{{\left(p_2\right)}_m}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0+m,p_1+m,p_2+m;z),$$

(98)

where $\Re \left(p_2\right)>\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$, $r\ge 0$,$m\ge 0$ and $\mid z\mid <1$.

(2)

$$\frac{d^m}{d{z}^m}\left\{{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)\right\}=\frac{{\left(p_1\right)}_m}{{\left(p_2\right)}_m}{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1+m,p_2+m;z),$$

(99)

where $\Re \left(p_2\right)>0,\Re \left(p_1\right)>0$, $\Re \left(r_1\right)>0,\Re \left(r_2\right)>0$,$m\ge 0$ and $r\ge 0$.

Proof. 

Let $\frac{d^m}{d{z}^m}$ be the m-th derivative with respect to the variable z. Differentiating (39), we obtain

$$\frac{d^m}{d{z}^m}\left\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)\right\}=\frac{d^m}{d{z}^m}\sum _{n=0}^{\infty }\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+n,p_2-p_1)}{B(p_1,p_2-p_1)}{\left(p_0\right)}_n\frac{z^n}{n!},$$

that is:

$$\frac{d^m}{d{z}^m}\left\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)\right\}=\sum _{n=m}^{\infty }\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+n,p_2-p_1)}{B(p_1,p_2-p_1)}{\left(p_0\right)}_n\frac{z^{n-m}n!}{n!(n-m)!}.$$

Replacing $n-m$ with m, we can read:

$$\frac{d^m}{d{z}^m}\left\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)\right\}=\sum _{m=0}^{\infty }\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+m+m,p_2-p_1)}{B(p_1,p_2-p_1)}{\left(p_0\right)}_{2m}\frac{z^m}{m!}.$$

Using the property of Pochhammer symbol ${\left(p_0\right)}_{m+k}={\left(p_0\right)}_m{(p_0+k)}_m$, and after manipulation, we obtain:

$$\frac{d^m}{d{z}^m}\left\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)\right\}=\frac{{\left(p_1\right)}_m}{{\left(p_2\right)}_m}\left\{\sum _{m=0}^{\infty }\frac{B_{(r_1,r_2)}^{\left(r\right)}(p_1+m+m,p_2-p_1)}{B(p_1+m,p_2-p_1)}{(p_0+m)}_m\frac{z^m}{m!}\right\}{\left(p_0\right)}_m.$$

By exploiting (39), we get our statement:

$$\frac{d^m}{d{z}^m}\left\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)\right\}=\frac{{\left(p_0\right)}_m{\left(p_1\right)}_m}{{\left(p_2\right)}_m}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0+m,p_1+m,p_2+m;z).$$

Similarly, following the same rule getting to result (98) applied in (41), we obtain the desired statement (99), and the Theorem is fully proved. □

Corollary 4.

If we consider $m=1$ in (98) and (99), we get following derivative formulas:

$$\frac{d}{dz}\left\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)\right\}=\frac{p_0{p}_1}{p_2}{F}_{(r_1,r_2)}^{\left(r\right)}(p_0+1,p_1+1,p_2+1;z);$$

(100)

$$\frac{d}{dz}\left\{{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1,p_2;z)\right\}=\frac{p_1}{p_2}{\Phi }_{(r_1,r_2)}^{\left(r\right)}(p_1+1,p_2+1;z).$$

(101)

Remark 14.

Using (46) on the right-hand side of the (98), we can write another derivative formula for $F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)$

$$\frac{d^m}{d{z}^m}\left\{F_{(r_1,r_2)}^{\left(r\right)}(p_0,p_1,p_2;z)\right\}=\frac{{\left(p_0\right)}_m{\left(p_1\right)}_m}{{\left(p_2\right)}_m}{F}_{(r_1,r_2)}^{\left(r\right)}(p_1+m,p_0+m,p_2+m;z).$$

(102)

4. Conclusions

We conclude our analysis by mentioning that the results obtained in this paper are new and potentially useful. First, we have investigated some inequalities of the extended beta function. Then, we have introduced a new extension of Gauss hypergeometric and confluent hypergeometric function and investigated some properties of these extended functions. We are also trying to find certain applications of the results obtained here in some significant research areas, such as statistics, physics, engineering, applied mathematics, and computer algebra. In the future, we will work on matrix forms of these extended functions and also try to develop algorithms for these extended functions by using maple computer software.