Euler-type integrals for the generalized hypergeometric matrix function

TL;DR

This paper explores Euler-type integral representations for the generalized hypergeometric matrix function, providing new transformations and specific cases for unit and half arguments, advancing the mathematical understanding of these functions.

Contribution

It introduces novel Euler-type integral representations and transformations for the generalized hypergeometric matrix function, including special cases with unit and half arguments.

Findings

Derived new Euler-type integral representations.

Developed transformations in terms of hypergeometric matrix functions.

Provided specific cases for unit and half arguments.

Abstract

In this paper, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and develop some transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have been provided for several particular cases.

Equations

$$a(P)=\max \left\{\Re (z):z\in \sigma (P)\right\},b(P)=\min \left\{\Re (z):z\in \sigma (P)\right\},$$

$$||P||=\underset{x\ne 0}{\sup }\frac{||Px|{|}_2}{||x|{|}_2}=\max \left\{\sqrt{\lambda }:\lambda \in \sigma (P^{\ast }P)\right\},$$

$$\Vert e^{Pt}\Vert \le e^{ta(P)}\sum _{u=0}^{r-1}\frac{{\left(\Vert P\Vert r^{1/2}t\right)}^u}{u!}(t\ge 0),$$

$$\Vert t^P\Vert \le \Vert e^{P\ln t}\Vert \le t^{a(P)}\sum _{u=0}^{r-1}\frac{{\left(\Vert P\Vert r^{1/2}\ln t\right)}^u}{u!}(t\ge 1).$$

$$f(P)g(P)=g(P)f(P).$$

$$f(P)g(Q)=g(Q)f(P).$$

$$\mu (P)=\underset{k\to 0}{\lim }\frac{\Vert I+kP\Vert -1}{k}=\max \left\{z|z\in \sigma \left(\frac{P+P^{\ast }}{2}\right)\right\}$$

$$\tilde{\mu }(P)=-\mu (-P)=\min \left\{z|z\in \sigma \left(\frac{P+P^{\ast }}{2}\right)\right\}.$$

$${\Gamma }^{-1}(P)=P(P+I)\dots (P+(n-1)I){\Gamma }^{-1}(P+nI),n\ge 1.$$

$$(P{)}_m=\{\begin{array}{l}I,\text{if}m=0\\ P(P+I)\dots (P+(m-1)I),\text{if}m\ge 1,\end{array}$$

$$(P{)}_m={\Gamma }^{-1}(P)\Gamma (P+mI),m\ge 1.$$

$$\Gamma (P)=\underset{m\to \infty }{\lim }(m-1)!(P{)}_m^{-1}{m}^P.$$

$$\Gamma (P)={\int }_0^{\infty }{e}^{-t}{t}^{P-1}dt;t^{P-1}=\exp ((P-I)\ln t),$$

$$B(P,Q)={\int }_0^1{t}^{P-1}(1-t{)}^{Q-1}dt.$$

$$B(P,Q)=\Gamma (P)\Gamma (Q){\left[\Gamma (P+Q)\right]}^{-1}.$$

$${}_p{F}_q\left(P_1,\dots ,P_p;Q_1,\dots ,Q_q;z\right)=\sum _{m=0}^{\infty }\frac{(P_1{)}_m\dots (P_p{)}_m(Q_1{)}_m^{-1}\dots (Q_q{)}_m^{-1}}{m!}{z}^m,$$

$${}_3{F}_2\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};z\right)=\frac{\Gamma (R)}{\Gamma (Q)\Gamma (R-Q)}{\int }_0^1{u}^{Q-I}(1-u{)}^{R-Q-I}(1-z{u}^2{)}^{-P}du.$$

$${}_3{F}_2\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};z\right)=\sum _{m=0}^{\infty }\frac{{\left(P\right)}_m{\left(\frac{Q}{2}\right)}_m{\left(\frac{Q+I}{2}\right)}_m}{{\left(\frac{R}{2}\right)}_m{\left(\frac{R+I}{2}\right)}_m}\frac{z^m}{m!}.$$

$${\left(P\right)}_{2m}=2^{2m}{\left(\frac{P}{2}\right)}_m{\left(\frac{P+I}{2}\right)}_m,$$

$${}_3{F}_2\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};z\right)$$

$$=\frac{\Gamma (R)}{\Gamma (Q)\Gamma (R-Q)}{\int }_0^1{u}^{Q-I}(1-u{)}^{R-Q-I}\sum _{m=0}^{\infty }\frac{{\left(P\right)}_m(z{u}^2{)}^m}{m!}du$$

$$=\frac{\Gamma (R)}{\Gamma (Q)\Gamma (R-Q)}{\int }_0^1{u}^{Q-I}(1-u{)}^{R-Q-I}(1-z{u}^2{)}^{-P}du.$$

$${}_3{F}_2\left(-kI,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};z\right)=\frac{\Gamma (R)}{\Gamma (Q)\Gamma (R-Q)}{\int }_0^1{u}^{Q-I}(1-u{)}^{R-Q-I}(1-z{u}^2{)}^{kI}du.$$

$${}_3{F}_2\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};1\right)=\frac{\Gamma (R)\Gamma (R-Q-P)}{\Gamma (R-P)\Gamma (R-Q)}{}_2{F}_1\left(P,Q;R-P;-1\right).$$

$${}_3{F}_2\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};1\right)$$

$$=\frac{\Gamma (R)}{\Gamma (Q)\Gamma (R-Q)}\sum _{m=0}^{\infty }\left(\genfrac{}{}{0ex}{}{-P}{m}\right){\int }_0^1{u}^{Q+(m-1)I}(1-u{)}^{R-Q-P-I}du$$

$$=\frac{\Gamma (R)}{\Gamma (Q)\Gamma (R-Q)}\sum _{m=0}^{\infty }\left(\genfrac{}{}{0ex}{}{-P}{m}\right)\frac{\Gamma (Q+mI)\Gamma (R-Q-P)}{\Gamma (R-P+mI)}$$

$$=\frac{\Gamma (R)\Gamma (R-Q-P)}{\Gamma (R-P)\Gamma (R-Q)}\sum _{m=0}^{\infty }\frac{(P{)}_m(Q{)}_m}{(R-P{)}_m}\frac{(-1{)}^m}{m!}$$

$$=\frac{\Gamma (R)\Gamma (R-Q-P)}{\Gamma (R-P)\Gamma (R-Q)}{}_2{F}_1\left(P,Q;R-P;-1\right),$$

$${}_3{F}_2\left(-nI,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};1\right)=\frac{{\left(R-Q\right)}_n}{{\left(R\right)}_n}{}_2{F}_1\left(-nI,Q;R+nI;-1\right).$$

$${}_3{F}_2\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};\frac{1}{2}\right)=2^P\sum _{m=0}^{\infty }\left(\genfrac{}{}{0ex}{}{-P}{m}\right)\frac{(R-Q{)}_m}{(R{)}_m}\times {}_2{F}_1\left(-mI,Q;R+mI;-1\right).$$

$${}_3{F}_2\left(P,\frac{Q}{2},\frac{Q+I}{2};\frac{R}{2},\frac{R+I}{2};\frac{1}{2}\right)$$

$$=\frac{\Gamma (R)}{\Gamma (Q)\Gamma (R-Q)}{\int }_0^1{u}^{Q-I}(1-u{)}^{R-Q-I}{\left(1-\frac{1}{2}{u}^2\right)}^{-P}du$$

$$=\frac{2^P\Gamma (R)}{\Gamma (Q)\Gamma (R-Q)}{\int }_0^1{u}^{Q-I}(1-u{)}^{R-Q-I}{\left(2-u^2\right)}^{-P}du$$

Full text

**Euler-type integrals for the generalized hypergeometric matrix function **

**Ankit Pala, Kiran Kumarib

**ankit.pal@vitbhopal.ac.in, kiran.kumariprajapati@gmail.com,

aDepartment of Mathematics, School of Advanced Sciences & Languages

VIT Bhopal University, Sehore-466 114, Madhya Pradesh, India.

bDepartment of Mathematics and Humanities,

Sardar Vallabhbhai National Institute of Technology, Surat-395 007, Gujarat, India.

Abstract: In this paper, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and develop some transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have been provided for several particular cases.

Keywords: Generalized hypergeometric matrix function, Matrix functional calculus, Gamma and Beta matrix function.

AMS Subject Classification (2010): 15A15, 33C05, 33C20, 44A99.

1 Introduction and Preliminaries

In recent years, mathematical analysis has paid a lot of attention to the theory of special matrix functions and polynomials. The matrix analogue of hypergeometric functions play a significant role in the area of applied and pure analysis. These functions appears in the study of statistics [3], probability theory [13] and Lie theory [10]. The matrix analogue of the Gauss hypergeometric function was introduced by Jódar and Cortés [11]. They studied the integral representation and matrix differential equation. Later, Abdalla [1, 2], Dwivedi and Sahai [6, 7] studied the various properties like integral and differential relations, finite sum formulas, generating functions of special matrix functions. Particularly, these functions play a vital role in solving numerous problems of mathematical physics, engineering and mathematical sciences.

Throughout this work, we consider the complex space $\mathbf{C}^{R\times R}$ of complex matrices of common order $R$. For any matrix $P\in\mathbf{C}^{R\times R}$, $\sigma(P)$ is the spectrum of $P$ and

[TABLE]

where $a(P)$ is the spectral abscissa of $P$ and $b(P)=-a(-P)$. A Hermitian matrix $P$ in $\mathbf{C}^{R\times R}$ is a positive stable matrix if $\Re(\lambda)>0$ for all $\lambda\in\sigma(P)$, where $\sigma(P)$ is the set of all eigenvalues of $P$ or spectrum of $A$ and its two-norm is given by

[TABLE]

where for any vector $x\in\mathbf{C}^{r}$, $||x||_{2}=(x^{*}x)^{1/2}$ is the Euclidean norm of $x$ and $P^{*}$ denotes the conjugate transpose of $P$. $I$ and $\mathbf{0}$ stands for the identity matrix and null matrix in $\mathbf{C}^{R\times R}$, respectively. Taking into account the Schür decomposition of a matrix $P$ [8], we have

[TABLE]

which yields

[TABLE]

If $f(z)$ and $g(z)$ are holomorphic functions of the complex variable $z$, which are defined in an open set $\Omega$ of the complex plane, and $P$ is a matrix in $\mathbf{C}^{R\times R}$ with $\sigma(P)\subset\Omega$, then from the properties of the matrix functional calculus [5], it follows that

[TABLE]

Furthermore, if $Q\in\mathbf{C}^{R\times R}$ with $\sigma(Q)\subset\Omega$, and if $PQ=QP$, then

[TABLE]

The logarithmic norm of a matrix $P\in\mathbf{C}^{R\times R}$ is defined as (see [4, 9]),

[TABLE]

Let the number $\tilde{\mu}(P)$ such that

[TABLE]

The reciprocal gamma function $\Gamma^{-1}(z)=1/\Gamma(z)$ is an entire function of the complex variable $z$. The image of $\Gamma^{-1}(z)$ acting on $P$, denoted by $\Gamma^{-1}(P)$, is a well defined matrix. If $P+nI$ is invertible for all integers $n\geq 0$, then the reciprocal gamma function is defined as (see [12]),

[TABLE]

By applications of the matrix functional calculus, the Pochhammer symbol [12] for $P\in\mathbf{C}^{R\times R}$ is given by

[TABLE]

which gives

[TABLE]

If $P\in\mathbf{C}^{R\times R}$ is a positive stable matrix and $m\geq 1$ is an integer, then the gamma matrix function can be represented in the following limit form as [11]:

[TABLE]

Let $P$ and $Q$ be two positive stable matrices in $\mathbf{C}^{R\times R}$. The gamma matrix function $\Gamma(P)$ and the beta matrix function $B(P,Q)$ have been defined in [11, 12], as follows

[TABLE]

and

[TABLE]

Let P and Q be commuting matrices in $\mathbf{C}^{R\times R}$ such that the matrices $P+nI,Q+nI$ and $P+Q+nI$ are invertible for every integer $n\geq 0$, then according to [12], we have

[TABLE]

2 Main results

Dwivedi and Sahai [6] introduced a natural generalization of hypergeometric matrix function called generalized hypergeometric matrix function and defined by

[TABLE]

where $P_{i},Q_{j}\in\mathbf{C}^{R\times R},1\leq i\leq p,1\leq j\leq q$, such that $Q_{j}+mI,1\leq j\leq q$ are invertible for all integers $m\geq 0$.

In the present work, we consider an Euler-type integral and present some integral representations of ${}_{3}F_{2}(\cdot)$ matrix function using the suitable adjustment of matrix parameters.

Theorem 2.1**.**

Let $P,Q$ and $R$ be commuting matrices in $\mathbf{C}^{R\times R}$ such that $Q,R$ and $R-Q$ are positive stable. Then, for $|z|<1$, we have the following integral representation:

[TABLE]

Proof.

From the left-hand side of (2.2), we find that

[TABLE]

Using the relation

[TABLE]

equation (2.3) becomes

[TABLE]

This completes the proof of Theorem 2.1. ∎

Corollary 2.1**.**

Under the conditions stated in Theorem 2.1, the following integral relation holds true:

[TABLE]

Theorem 2.2**.**

Let $P,Q$ and $R$ be commuting matrices in $\mathbf{C}^{R\times R}$ such that $R,R-P,R-Q$ and $R-Q-P$ are positive stable. Then the following integral representation holds true:

[TABLE]

Proof.

Substituting $z=1$ in (2.2), it becomes

[TABLE]

which completes the proof of Theorem 2.2. ∎

Corollary 2.2**.**

Under the conditions stated in Theorem 2.2, the following integral relation holds true:

[TABLE]

Theorem 2.3**.**

Let $P,Q$ and $R$ be commuting matrices in $\mathbf{C}^{R\times R}$ such that $Q,R$ and $R-Q$ are positive stable. Then the following integral representation holds true:

[TABLE]

Proof.

Substituting $z=\frac{1}{2}$ in Theorem 2.1, we obtain

[TABLE]

Using the transformation $(R)_{k+m}=(R)_{m}(R+m)_{k}$, we arrive at

[TABLE]

This completes the proof of Theorem 2.3. ∎

Corollary 2.3**.**

Under the conditions stated in Theorem 2.3, the following integral relation holds true:

[TABLE]

Now we generalize the Theorem 2.1 in the following form using the suitable adjustment of argument in the ${}_{3}F_{2}(\cdot)$ matrix function.

Theorem 2.4**.**

Let $P,Q$ and $R$ be commuting matrices in $\mathbf{C}^{R\times R}$ such that $Q,R$ and $R-Q$ are positive stable and $w\in\mathbf{R}\backslash\left\{0,-1\right\}$. Then the following integral representation holds true:

[TABLE]

Proof.

Substituting $z=\frac{1}{w+1}$ in Theorem 2.1, we have

[TABLE]

Using Corollary 2.2, this yields the right hand side of Theorem 2.4. ∎

Remark 2.1**.**

It is interesting to observe that for $w=1$, Theorem 2.4 reduces to Theorem 2.3 and Corollary 2.3 and for $w=-2$, it reduces to the following result asserted by Corollary 2.4.

Corollary 2.4**.**

Under the conditions stated in Theorem 2.4, the following integral relation holds true:

[TABLE]

Next we generalize the Theorem 2.1 in the obvious way using the suitable adjustment of matrix parameters by introducing the sequence of $q$ parameters in the ${}_{3}F_{2}(\cdot)$ matrix function.

Theorem 2.5**.**

Let $P,Q$ and $R$ be commuting matrices in $\mathbf{C}^{R\times R}$ such that $Q,R$ and $R-Q$ are positive stable. Then, for $|z|<1$, the following integral representation holds true:

[TABLE]

Proof.

Using the following relation:

[TABLE]

we can easily proceed for the proof similar to Theorem 2.1. ∎

Now for $q=3$ and $z=1$, Theorem 2.5 leads to the following result asserted by Theorem 2.6.

Theorem 2.6**.**

Let $P,Q$ and $R$ be commuting matrices in $\mathbf{C}^{R\times R}$ such that $R,R-P,R-Q$ and $R-Q-P$ are positive stable. Then the following integral representation holds true:

[TABLE]

Proof.

Substituting $q=3$ and $z=1$ in Theorem 2.5, we have

[TABLE]

This completes the proof. ∎

Theorem 2.7**.**

Let $P,Q$ and $R$ be commuting matrices in $\mathbf{C}^{R\times R}$ such that $Q,R$ and $R-Q$ are positive stable. Then, for $|z|<1$, the following integral representation holds true:

[TABLE]

Proof.

From Theorem 2.5, we have

[TABLE]

Substitution $s=u^{q}$ in the integral on the right hand side of above equation yields

[TABLE]

Now using the Pochhammer matrix symbols (1.7) yields the right hand side of Theorem 2.7. ∎

Statements & Declarations

Funding

Not applicable.

Conflicts of interest/Competing interests

The authors declare that they have no competing interests.

Authors contributions

Both the authors have equally contributed of reading and writing the manuscript.