On an identity for H-function

TL;DR

This paper presents a new identity for the H-function that generalizes previous identities, contributing to the mathematical understanding of special functions.

Contribution

It introduces a generalized identity for the H-function, extending earlier results by Rathie and Rathie et al.

Findings

Provides a new mathematical identity for the H-function

Generalizes previous identities involving H-function

Enhances theoretical understanding of special functions

Abstract

The main objective of this research note is to provide an identity for the H-function, which generalizes two identities involving H-function obtained earlier by Rathie and Rathie et al.

Equations

$$H_{p+1,q+1}^{m,n}$$

$$\displaystyle=\frac{1}{2\pi i}\left\{e^{i\pi\alpha}H_{\;p,\;q}^{\;m,\;n}\left[ze^{-i\pi\lambda}|\begin{array}[]{c}\;{}_{1}(a_{j},e_{j})_{p}\\ \;{}_{1}(b_{j},f_{j})_{q}\end{array}\right]\right.$$

$$\displaystyle\qquad\qquad-\left.e^{-i\pi\alpha}H_{\;p,\;q}^{\;m,\;n}\left[ze^{i\pi\lambda}|\begin{array}[]{c}\;{}_{1}(a_{j},e_{j})_{p}\\ \;{}_{1}(b_{j},f_{j})_{q}\end{array}\right]\right\}$$

$$H_{p+1,q+1}^{m+1,n+1}$$

$$\displaystyle=e^{i\pi\alpha}\;H_{\;p+1,\;q+1}^{\;m+1,\;n+1}\left[ze^{-i\pi\lambda}|\begin{array}[]{c}(2\alpha,2\lambda),\;{}_{1}(a_{j},e_{j})_{p}\\ (2\alpha,2\lambda),\;{}_{1}(b_{j},f_{j})_{q}\end{array}\right]$$

$$\displaystyle\qquad+e^{-i\pi\alpha}\;H_{\;p+1,\;q+1}^{\;m+1,\;n+1}\left[ze^{i\pi\lambda}|\begin{array}[]{c}(2\alpha,2\lambda),\;{}_{1}(a_{j},e_{j})_{p}\\ (2\alpha,2\lambda),\;{}_{1}(b_{j},f_{j})_{q}\end{array}\right]$$

$$2\pi i$$

$$\displaystyle=e^{i\pi(\alpha+\beta)}\;H_{\;p+1,\;q+1}^{\;m+1,\;n+1}\left[ze^{-i\pi(\lambda+\delta)}|\begin{array}[]{c}(2\beta,2\delta),\;{}_{1}(a_{j},e_{j})_{p}\\ (2\beta,2\delta),\;{}_{1}(b_{j},f_{j})_{q}\end{array}\right]$$

$$\displaystyle+e^{i\pi(\alpha-\beta)}\;H_{\;p+1,\;q+1}^{\;m+1,\;n+1}\left[ze^{-i\pi(\lambda-\delta)}|\begin{array}[]{c}(2\beta,2\delta),\;{}_{1}(a_{j},e_{j})_{p}\\ (2\beta,2\delta),\;{}_{1}(b_{j},f_{j})_{q}\end{array}\right]$$

$$\displaystyle-e^{-i\pi(\alpha-\beta)}\;H_{\;p+1,\;q+1}^{\;m+1,\;n+1}\left[ze^{i\pi(\lambda-\delta)}|\begin{array}[]{c}(2\beta,2\delta),\;{}_{1}(a_{j},e_{j})_{p}\\ (2\beta,2\delta),\;{}_{1}(b_{j},f_{j})_{q}\end{array}\right]$$

$$\displaystyle-e^{-i\pi(\alpha+\beta)}\;H_{\;p+1,\;q+1}^{\;m+1,\;n+1}\left[ze^{i\pi(\lambda+\delta)}|\begin{array}[]{c}(2\beta,2\delta),\;{}_{1}(a_{j},e_{j})_{p}\\ (2\beta,2\delta),\;{}_{1}(b_{j},f_{j})_{q}\end{array}\right]$$

$$I=2\pi i\frac{1}{2\pi i}{\int }_L\theta (s)z^s\frac{\Gamma (\beta -\delta s)\Gamma (1-\beta +\delta s)}{\Gamma (\alpha -\lambda s)\Gamma (1-\alpha +\lambda s)}ds$$

$$\theta (s)=\frac{{\prod }_{j=1}^m\Gamma (b_j-f_{j}s){\prod }_{j=1}^n\Gamma (1-a_j+e_{j}s)}{{\prod }_{j=m+1}^q\Gamma (1-b_j+f_{j}s){\prod }_{j=n+1}^p\Gamma (a_j-e_{j}s)}$$

$$\Gamma (\beta -\delta s)\Gamma (1-\beta +\delta s)=2\pi \frac{\Gamma (2\beta -2\delta s)\Gamma (1-2\beta +2\delta s)}{\Gamma (\frac{1}{2}+\beta -\delta s)\Gamma (\frac{1}{2}-\beta +\delta s)}$$

$$sin\pi z=\frac{\pi }{\Gamma (z)\Gamma (1-z)}=\frac{e^{i\pi z}-e^{-i\pi z}}{2i}$$

$$cos\pi z=\frac{\pi }{\Gamma \left(\frac{1}{2}-z\right)\Gamma \left(\frac{1}{2}+z\right)}=\frac{e^{i\pi z}+e^{-i\pi z}}{2}$$

$$I$$

$$.\left(e^{i\pi (\alpha -\lambda s)}-e^{-i\pi (\alpha -\lambda s)}\right)\left(e^{i\pi (\beta -\delta s)}+e^{-i\pi (\beta -\delta s)}\right)ds$$

$$=\frac{1}{2\pi i}{\int }_L\theta (s)z^s\Gamma (2\beta -2\delta s)\Gamma (1-2\beta +2\delta s)$$

$$.\{e^{i\pi (\alpha +\beta -\lambda s-\delta s)}+e^{i\pi (\alpha -\beta -\lambda s+\delta s)}$$

$$-e^{-i\pi (\alpha -\beta -\lambda s+\delta s)}-e^{-i\pi (\alpha +\beta -\lambda s-\delta s)}\}ds$$

$$\displaystyle H_{\;p+2,\;q+2}^{\;m+1,\;n+1}\left[z|\begin{array}[]{c}(\beta,\delta),\;{}_{1}(a_{j},e_{j})_{p},(\alpha,\lambda)\\ (\beta,\delta),\;{}_{1}(b_{j},f_{j})_{q},(\alpha,\lambda)\end{array}\right]$$

$$\displaystyle\quad=\;H_{\;p+4,\;q+4}^{\;m+2,\;n+2}\left[z|\begin{array}[]{c}(2\beta,2\delta),\left(\frac{1}{2}+\alpha,\lambda\right),\;{}_{1}(a_{j},e_{j})_{p},(2\alpha,2\lambda),\left(\frac{1}{2}+\beta,\delta\right)\\ (2\beta,2\delta),\left(\frac{1}{2}+\alpha,\lambda\right),\;{}_{1}(b_{j},f_{j})_{q},(2\alpha,2\lambda),\left(\frac{1}{2}+\beta,\delta\right)\end{array}\right]$$

$$H_{p+1,q+1}^{m,n}$$

$$\displaystyle=\frac{1}{2\pi}\;H_{\;p+2,\;q+2}^{\;m+1,\;n+1}\left[z|\begin{array}[]{c}\left(\frac{1}{2}+\alpha,\lambda\right),\;{}_{1}(a_{j},e_{j})_{p},(2\alpha,2\lambda)\\ \left(\frac{1}{2}+\alpha,\lambda\right),\;{}_{1}(b_{j},f_{j})_{q},(2\alpha,2\lambda),\end{array}\right]$$

$$H_{p+1,q+1}^{m,n}$$

$$\displaystyle=2\pi\;H_{\;p+2,\;q+2}^{\;m+1,\;n+1}\left[z|\begin{array}[]{c}(2\beta,2\delta),\;{}_{1}(a_{j},e_{j})_{p},\left(\frac{1}{2}+\beta,\delta\right)\\ (2\beta,2\delta),\;{}_{1}(b_{j},f_{j})_{q},\left(\frac{1}{2}+\beta,\delta\right)\end{array}\right]$$

Full text

Keywords : H-function, Identity

2000 Mathematics Subject Classification : 33C60, 33C20, 33C70

1. Introduction

In 1981, Rathie [2] established the following identity for the H-function viz.

[TABLE]

Very recently, Rathie et al.[3] established another identity for the H-function viz.

[TABLE]

For interesting applications of the identities (1.1) and (1.2), we refer the recent paper by Rathie et al.[3]

The aim of this short note is to provide a natural generalization of (1.1) and (1.2).

2. Main Result

The identity for the H-function to be established in this note is the following.

[TABLE]

where $H_{\;p,\;q}^{\;m,\;n}[z]$ is the well known H-function[1].

**Proof : ** In order to establish the identity (2.1), we proceed as follows.

Denoting the left-hand of H-function by I, expressing the H-function with the help of its definition we have,

[TABLE]

where $\theta(s)$ is given by

[TABLE]

Using the results

[TABLE]

[TABLE]

and

[TABLE]

and after some algebra, we have

[TABLE]

Now, breaking in to four parts and after some simplification, using the definition of H-function, we easily arrive at the right-hand side of (2.1).

This completes the proof of the identity (2.1).

3. Special Cases

1. (a)

In (2.1), if we take $\delta=0$, we get, after some simplification, the identity obtained earlier by Rathie[2]. 2. (b)

In (2.1), if we take $\lambda=0$, we get, after some simplification, the identity obtained very recently by Rathie et al.[3].

4. Concluding Remarks

If we use the result (2.4), we get the following identity for the H-function.

[TABLE]

Further in this, if we take $\delta=0$ and $\lambda=0$, we respectively get

[TABLE]

and

[TABLE]