Certain new unified integrals associated with the product of generalized Struve function

TL;DR

This paper introduces new generalized integral formulas involving the product of generalized Struve functions, expressed via Lauricella functions, unifying and extending known integral results with special case considerations.

Contribution

The paper derives two new generalized integral formulas involving the product of generalized Struve functions expressed through Lauricella functions, broadening existing integral identities.

Findings

Derived generalized integral formulas involving generalized Struve functions

Expressed integrals in terms of Lauricella functions for broad applicability

Special cases recover known and new integral identities

Abstract

We aim to present two new generalized integral formulae involving product of generalized Struve function $\mathcal{W}_{p,b,c}\left( z\right)$, which are expressed in terms of the generalized Lauricella functions. The main results presented here, being the very general character, reduce to yield known and new integral formulae. Some special cases of our main results are also considered.

Equations

$$x^2{y}^{{}^{\prime \prime }}(x)+x{y}^{{}^{\prime }}\left(x\right)+\left(x^2-p^2\right)y\left(x\right)=\frac{4(x/2{)}^{\alpha +1}}{\sqrt{\pi }\Gamma \left(\alpha +1/2\right)}(\alpha \in \mathbb{C})$$

$${\mathcal{H}}_{\upsilon }\left(z\right)={\left(\frac{z}{2}\right)}^{\upsilon +1}\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k}{\Gamma \left(k+\frac{3}{2}\right)\Gamma \left(k+\upsilon +\frac{1}{2}\right)}{\left(\frac{z}{2}\right)}^{2k}$$

$${\mathcal{L}}_{\upsilon }\left(z\right)={\left(\frac{z}{2}\right)}^{\upsilon +1}\sum _{k=0}^{\infty }\frac{1}{\Gamma \left(k+\frac{3}{2}\right)\Gamma \left(k+\upsilon +\frac{1}{2}\right)}{\left(\frac{z}{2}\right)}^{2k}.$$

$${\mathcal{W}}_{p,b,c}\left(z\right)=\sum _{k=0}^{\infty }\frac{{\left(-c\right)}^k{\left(\frac{z}{2}\right)}^{2k+p+1}}{\Gamma \left(k+\frac{3}{2}\right)\Gamma \left(k+p+\frac{b+2}{2}\right)}(p,b,c\in \mathbb{C}).$$

$${}_p{\Psi }_q\left[\begin{array}{rl}\left({\alpha }_1,A_1\right),\dots ,\left({\alpha }_p,A_p\right)& ;\\ \left({\beta }_1,B_1\right),\dots ,\left({\beta }_q,B_q\right)& ;\end{array}z\right]=\sum _{k=0}^{\infty }\frac{\prod _{j=1}^p\Gamma \left({\alpha }_j+A_{j}k\right)}{\prod _{j=1}^q\Gamma \left({\beta }_j+B_{j}k\right)}\frac{z^k}{k!},$$

$$1+\sum _{j=1}^q{B}_j-\sum _{j=1}^p{A}_j\geqq 0.$$

$${}_p{\Psi }_q\left[\begin{array}{rl}\left({\alpha }_1,1\right),\dots ,\left({\alpha }_p,1\right)& ;\\ \left({\beta }_1,1\right),\dots ,\left({\beta }_q,1\right)& ;\end{array}z\right]=\frac{\prod _{j=1}^p\Gamma \left({\alpha }_j\right)}{\prod _{j=1}^q\Gamma \left({\beta }_j\right)}{}_p{F}_q\left[\begin{array}{rl}{\alpha }_1,\dots ,{\alpha }_p& ;\\ {\beta }_1,\dots ,{\beta }_q& ;\end{array}z\right].$$

$$\displaystyle F_{C:D^{(1)};\cdots;D^{(n)}}^{A:B^{(1)};\cdots;B^{(n)}}\,\left(\begin{array}[]{c}z_{1}\\ \vdots\\ z_{n}\\ \end{array}\right)=F_{C:D^{(1)};\cdots;D^{(n)}}^{A:B^{(1)};\cdots;B^{(n)}}\left(\begin{aligned} &\,\,[(a):\theta^{(1)},\ldots,\theta^{(n)}]:\\ &[(c):\psi^{(1)},\ldots,\psi^{(n)}]:\end{aligned}\right.$$

$$\begin{array}{rl}& [(b{)}^{(1)}:{\varphi }^{(1)}];\dots ;[(b{)}^{(n)}:{\varphi }^{(n)}];\\ & [(d{)}^{(1)}:{\delta }^{(1)}];\dots ;[(d{)}^{(n)}:{\delta }^{(n)}];\end{array}{z}_1,\dots ,z_n)$$

$$=\sum _{k_1,\dots ,k_n=0}^{\infty }\Omega (k_1,\dots ,k_n)\frac{z_1^{k_1}}{k_1!}\cdots \frac{z_n^{k_n}}{k_n!},$$

$$\Omega (k_1,\dots ,k_n)=\frac{\prod _{j=1}^A(a_j{)}_{k_1{\theta }_j^{(1)}+\cdots +k_n{\theta }_j^{(n)}}\prod _{j=1}^{B^{(1)}}(b_j^{(1)}{)}_{k_1{\varphi }_j^{(1)}}\cdots \prod _{j=1}^{B^{(n)}}(b_j^{(n)}{)}_{k_n{\varphi }_j^{(n)}}}{\prod _{j=1}^C(c_j{)}_{k_1{\psi }_j^{(1)}+\cdots +k_n{\psi }_j^{(n)}}\prod _{j=1}^{D^{(1)}}(d_j^{(1)}{)}_{k_1{\delta }_j^{(1)}}\cdots \prod _{j=1}^{D^{(n)}}(d_j^{(n)}{)}_{k_n{\delta }_j^{(n)}}},$$

$$\{\begin{array}{rl}& {\theta }_j^{(m)}(j=1,\dots ,A);{\varphi }_j^{(m)}(j=1,\dots ,B^{(m)});\\ & {\psi }_j^{(m)}(j=1,\dots ,C);{\delta }_j^{(m)}(j=1,\cdots ,D^{(m)});\forall m\in \{1,\dots ,n\}\end{array}$$

$$b_j^{(m)}(j=1,\dots ,B^{(m)});\forall m\in \{1,\dots ,n\},$$

$${\int }_0^{\infty }{x}^{\mu -1}{\left(x+a+\sqrt{x^2+2ax}\right)}^{-\lambda }dx=2\lambda a^{-\lambda }{\left(\frac{a}{2}\right)}^{\mu }\frac{\Gamma \left(2\mu \right)\Gamma \left(\lambda -\mu \right)}{\Gamma \left(1+\lambda +\mu \right)}$$

$$\left(a\in {\mathbb{R}}^{+};0<\Re (\mu )<\Re (\lambda )\right).$$

$$\mathbf{p}:=\sum _{j=1}^n{p}_j.$$

$${\int }_0^{\infty }{x}^{\mu -1}{\left(x+a+\sqrt{x^2+2ax}\right)}^{-\lambda }\prod _{j=1}^n{\mathcal{W}}_{p_j,b,c}\left(\frac{y_j}{x+a+\sqrt{x^2+2ax}}\right)dx$$

$$=\mathcal{A}(a,\mathbf{p},\lambda ,\mu ,p_j,y_j:j=1,\dots ,n)$$

$$\displaystyle\hskip 14.22636pt\times F^{2:1;\cdots;1}_{\,2:2;\cdots;2}\,\bigg{(}\begin{aligned} &[1+\lambda+\mathbf{p}+n:2,\ldots,2],[\lambda+\mathbf{p}+n-\mu:2,\ldots,2]:\\ &[\lambda+\mathbf{p}+n:2,\ldots,2],[1+\lambda+\mathbf{p}+n+\mu:2,\ldots,2]:\end{aligned}$$

$$\displaystyle\hskip 14.22636pt\begin{aligned} &\hskip 184.9429pt[1:1];\cdots;[1:1];\\ &\Big{[}\frac{3}{2}:1\Big{]},\Big{[}p_{1}+\frac{b+2}{2}:1\Big{]};\cdots;\Big{[}\frac{3}{2}:1\Big{]},\Big{[}p_{n}+\frac{b+2}{2}:1\Big{]};\end{aligned}-\frac{c\,y_{1}^{2}}{4\,a^{2}},\cdots,-\frac{c\,y_{n}^{2}}{4\,a^{2}}\bigg{)},$$

$$\mathcal{A}(a,\mathbf{p},\lambda ,\mu ,p_j,y_j:j=1,\dots ,n)$$

$$:=\frac{(\lambda +\mathbf{p}+n)2^{1-\mu -\mathbf{p}-n}{a}^{\mu -\lambda -\mathbf{p}-n}\Gamma (2\mu )\Gamma (\lambda +\mathbf{p}+n-\mu )\prod _{j=1}^n{y_j}^{p_j+1}}{{\left\{\Gamma \left(\frac{3}{2}\right)\right\}}^n\Gamma (1+\lambda +\mathbf{p}+n+\mu )\prod _{j=1}^n\Gamma \left(p_j+\frac{b+2}{2}\right)}.$$

$$\mathcal{L}=$$

$$\times {\int }_0^{\infty }{x}^{\mu -1}{\left(x+a+\sqrt{x^2+2ax}\right)}^{-\lambda -(p_1+\cdots +p_n)-2(k_1+\cdots +k_n)-n}dx.$$

$$(\lambda {)}_{\nu }:=\frac{\Gamma (\lambda +\nu )}{\Gamma (\lambda )}=\{\begin{array}{rl}& 1(\nu =0;\lambda \in \mathbb{C}\setminus \{0\})\\ & \lambda (\lambda +1)\cdots (\lambda +n-1)(\nu =n\in \mathbb{N};\lambda \in \mathbb{C})\end{array}$$

$$\lambda +k=\frac{\lambda \cdot (1+\lambda {)}_k}{(\lambda {)}_k}\left(k\in {\mathbb{N}}_0\right),$$

$$\mathcal{L}=$$

$$\times \sum _{k_1,\dots ,k_n=0}^{\infty }\frac{(-c{)}^{k_1}{\left(\frac{y_1}{2}\right)}^{2k_1}\cdots (-c{)}^{k_n}{\left(\frac{y_n}{2}\right)}^{2k_n}{a}^{-2(k_1+\cdots +k_n)}}{{\left(\frac{3}{2}\right)}_{k_1}{\left(p_1+\frac{b+2}{2}\right)}_{k_1}\cdots {\left(\frac{3}{2}\right)}_{k_n}{\left(p_n+\frac{b+2}{2}\right)}_{k_n}}$$

$$\times \frac{(1+\lambda +\mathbf{p}+n{)}_{2(k_1+\cdots +k_n)}(\lambda +\mathbf{p}+n-\mu {)}_{2(k_1+\cdots +k_n)}}{(\lambda +\mathbf{p}+n{)}_{2(k_1+\cdots +k_n)}(1+\lambda +\mathbf{p}+n+\mu {)}_{2(k_1+\cdots +k_n)}},$$

$${\int }_0^{\infty }{x}^{\mu -1}{\left(x+a+\sqrt{x^2+2ax}\right)}^{-\lambda }\prod _{j=1}^n{\mathcal{W}}_{p_j,b,c}\left(\frac{x{y}_j}{x+a+\sqrt{x^2+2ax}}\right)dx$$

$$=\mathcal{B}(a,\mathbf{p},\lambda ,\mu ,p_j,y_j:j=1,\dots ,n)$$

$$\displaystyle\hskip 14.22636pt\times F^{2:1;\cdots;1}_{\,2:2;\cdots;2}\,\bigg{(}\begin{aligned} &\hskip 14.22636pt[2\mu+2\mathbf{p}+2n:4,\ldots,4],[1+\lambda+\mathbf{p}+n:2,\ldots,2]:\\ &[1+\lambda+\mu+2\mathbf{p}+2n:4,\ldots,4],[\lambda+\mathbf{p}+n:2,\ldots,2]:\end{aligned}$$

$$\displaystyle\hskip 14.22636pt\begin{aligned} &\hskip 184.9429pt[1:1];\cdots;[1:1];\\ &\Big{[}\frac{3}{2}:1\Big{]},\Big{[}p_{1}+\frac{b+2}{2}:1\Big{]};\cdots;\Big{[}\frac{3}{2}:1\Big{]},\Big{[}p_{n}+\frac{b+2}{2}:1\Big{]};\end{aligned}-\frac{c\,y_{1}^{2}}{16},\cdots,-\frac{c\,y_{n}^{2}}{16}\bigg{)},$$

$$\mathcal{B}(a,\mathbf{p},\lambda ,\mu ,p_j,y_j:j=1,\dots ,n)$$

$$:=\frac{(\lambda +\mathbf{p}+n)2^{1-\mu -2\mathbf{p}-2n}{a}^{\mu -\lambda }\Gamma (\lambda -\mu )\Gamma (2\mu +2\mathbf{p}+2n)\prod _{j=1}^n{y}_j^{p_j+1}}{{\left\{\Gamma \left(\frac{3}{2}\right)\right\}}^n\Gamma (1+\lambda +\mu +2\mathbf{p}+2n)\prod _{j=1}^n\Gamma \left(p_j+\frac{b+2}{2}\right)},$$

$${\int }_0^{\infty }{x}^{\mu -1}{\left(x+a+\sqrt{x^2+2ax}\right)}^{-\lambda }{\mathcal{W}}_{p,b,c}\left(\frac{y}{x+a+\sqrt{x^2+2ax}}\right)dx$$

$$=(1+\lambda +p)2^{-\mu -p}{a}^{\mu -1-\lambda -p}{y}^{p+1}\frac{\Gamma (2\mu )\Gamma (1+\lambda +p-\mu )}{\Gamma \left(\frac{3}{2}\right)\Gamma (2+\lambda +p+\mu )\Gamma \left(1+\frac{b}{2}+p\right)}$$

Full text

1. Introduction and Preliminaries

A solution of the following non-homogeneous Bessel’s differential equation

[TABLE]

is known as the Struve function $\mathcal{H}_{\alpha}$ of order $\alpha$, where $\Gamma$ is the familiar gamma function (see, e.g., [18, Section 1.1]). Here and in the following, let $\mathbb{C}$, $\mathbb{R}^{+}$, and $\mathbb{N}$ be the sets of complex numbers, positive real numbers, and positive integers, respectively, and let $\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}$. The Struve functions occur in many areas such as water wave and surface-wave problems (see [2, 9]), problems on unsteady aerodynamics [16], particle quantum dynamical studies of spin decoherence [15] and nanotubes [14]. The Struve functions $\mathcal{H}_{v}\left(z\right)$ and $\mathcal{L}_{v}\left(z\right)$ are defined as the following infinite series

[TABLE]

and

[TABLE]

The generalized Struve function is given as follows (see, e.g., [11]):

[TABLE]

An interesting further generalization of the generalized hypergeometric series ${}_{p}F_{q}$ (see, e.g., [18, Section 1.5]) is due to Fox [7] and Wright [21, 22] who studied the asymptotic expansion of the generalized (Wright) hypergeometric function defined by (see [20, p. 21])

[TABLE]

where the coefficients $A_{1},\,\ldots,\,A_{p}\in\mathbb{R}^{+}$ and $B_{1},\,\ldots,\,B_{q}\in\mathbb{R}^{+}$ such that

[TABLE]

A special case of (1.5) is

[TABLE]

An interesting further several-variable-generalization of the generalized Lauricella series (see, for example, [20, p. 36, Eq. (19)]) is defined by (cf. Srivastava and Daoust [19, p. 454]; see also [20, p. 37])

[TABLE]

where, for convenience,

[TABLE]

the coefficients

[TABLE]

are real and positive, and $(a)$ abbreviates the array of $A$ parameters $a_{1},\ldots,a_{A},$ $(b^{(m)})$ abbreviates the array of $B^{(m)}$ parameters

[TABLE]

with similar interpretations for $(c)$ and $(d^{(m)})\,\,(m=1,\ldots,n)$; et cetera.

For more details about the generalized Lauricella function (1.8), the reader may be referred, for example, to [17, 20, 19, 8, 6].

We also need the following integral formula due to Oberhettinger [13]

[TABLE]

[TABLE]

Many integral formulas involving Bessel functions, generalized Bessel functions, and modified Bessel functions, and so on, have recently been established (see, e.g., [1, 3, 4, 5, 10, 12]). In this sequel, we aim to present two new integral formulas involving a finite product of the generalized Struve functions (1.4), which are expressed in terms of the generalized Lauricella function (1.8). The main formulas presented here, being very general, reduce to yield known and new integral formulas. Some interesting special cases are also considered.

2. Main results

Here, we establish two integral formulas involving a finite product of the generalized Struve functions (1.4), which are asserted by Theorems 1 and 2.

Theorem 1**.**

Let $n\in\mathbb{N}$ be fixed and $a,\,y_{j}\in\mathbb{R}^{+}$ $(j=1,\ldots,n)$. Also, let $b,\,c,\,p_{j},\,\lambda,\,\mu\in\mathbb{C}$ $(j=1,\ldots,n)$ with $0<\Re(\mu)<\Re\left(\lambda+\mathbf{p}\right)+n$, where

[TABLE]

Then

[TABLE]

where $1;\cdots;1$ and $2;\cdots;2$ are $n$ arrays, respectively, and

[TABLE]

Proof.

Let $\mathcal{L}$ be the left-hand side of (2.2). By using the generalized Struve function (1.4) and changing the order of integration and summations, which is verified under the given conditions of this theorem, we have

[TABLE]

By using (1.11) to evaluate the integral in (2.3) and interpreting the resulting expression in terms of the Pochhammer symbol defined (for $\lambda,\,\nu\in\mathbb{C}$) by

[TABLE]

together with the following easily-derivable identity:

[TABLE]

we obtain

[TABLE]

which, upon the multiple summations being expressed in terms of (1.8), leads to the right-hand side of (2.2).

∎

Theorem 2**.**

Let $n\in\mathbb{N}$ be fixed and $a,\,y_{j}\in\mathbb{R}^{+}$ $(j=1,\ldots,n)$. Also, let $b,\,c,\,p_{j},\,\lambda,\,\mu\in\mathbb{C}$ $(j=1,\ldots,n)$ with $\Re(\mu+\mathbf{p})>-n$ and $\Re(\lambda)>\Re(\mu)$. Then

[TABLE]

where $1;\cdots;1$ and $2;\cdots;2$ are $n$ arrays, respectively, and

[TABLE]

and $\mathbf{p}$ is the same as in (2.1).

Proof.

The proof runs parallel to that of Theorem 1. We omit the details. ∎

3. Special cases

Here, we consider some special cases of the main results.

Setting $n=1$ in the result in Theorem 1, we have an integral formula involving the generalized Struve function (1.4), which is given in Corollary 3.1.

Corollary 3.1**.**

Let $a,\,y\in\mathbb{R}^{+}$. Also, let $b,\,c,\,p,\,\lambda,\,\mu\in\mathbb{C}$ with $0<\Re(\mu)<\Re\left(\lambda+p\right)+1$. Then

[TABLE]

Setting $n=1$ in the result in Theorem 2, we have another integral formula involving the generalized Struve function (1.4), which is given in Corollary 3.2.

Corollary 3.2**.**

Let $a,\,y\in\mathbb{R}^{+}$. Also, let $b,\,c,\,p,\,\lambda,\,\mu\in\mathbb{C}$ with $\Re(\mu+p)>-1$ and $\Re(\lambda)>\Re(\mu)$. Then

[TABLE]

It is easy to see from (1.2) and (1.4) that

[TABLE]

Considering the relation (3.3) and setting $n=1$, $b=-1$, and $c=1$ in Theorems 1 and 2 (or setting $b=-1$ and $c=1$ in Corollaries 3.1 and 3.2, we obtain two integral formulas involving the Struve function $\mathcal{H}_{p}$ of order $p$, which are given, respectively, in Corollaries 3.3 and 3.4.

Corollary 3.3**.**

Let $a,\,y\in\mathbb{R}^{+}$. Also, let $p,\,\lambda,\,\mu\in\mathbb{C}$ with $0<\Re(\mu)<\Re\left(\lambda+p\right)+1$. Then

[TABLE]

Corollary 3.4**.**

Let $a,\,y\in\mathbb{R}^{+}$. Also, let $p,\,\lambda,\,\mu\in\mathbb{C}$ with $\Re(\mu+p)>-1$ and $\Re(\lambda)>\Re(\mu)$. Then

[TABLE]