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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Certain unified integrals associated with Bessel functions

Junesang Choi1* and Praveen Agarwal2

Author Affiliations

1 Department of Mathematics, Dongguk University, Gyeongju, Korea

2 Department of Mathematics, Anand International College of Engineering, Jaipur, 303012, India

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Boundary Value Problems 2013, 2013:95  doi:10.1186/1687-2770-2013-95

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/95


Received:14 January 2013
Accepted:5 April 2013
Published:18 April 2013

© 2013 Choi and Agarwal; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A remarkably large number of integral formulas involving a variety of special functions have been developed by many authors. Very recently, Ali gave three interesting unified integrals involving the hypergeometric function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M3">View MathML</a>. Using Ali’s method, in this paper, we present two generalized integral formulas involving the Bessel function of the first kind <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M2">View MathML</a>, which are expressed in terms of the generalized (Wright) hypergeometric functions. Some interesting special cases of our main results are also considered.

MSC: 33B20, 33C20, 33B15, 33C05.

Keywords:
Gamma function; hypergeometric function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M3">View MathML</a>; generalized hypergeometric function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M4">View MathML</a>; generalized (Wright) hypergeometric functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M5">View MathML</a>; Bessel function of the first kind; Oberhettinger’s integral formula; Garg and Mittal’s integral formula

1 Introduction and preliminaries

Integrals involving products of Gamma functions along vertical lines were first studied by Pincherle in 1888 and an extensive theory was developed by Barnes [1] and Mellin [2]. Cahen [3] employed some of these integrals in the study of the Riemann Zeta function and other Dirichlet series. In a spirit of Mellin’s theory, some of Ramanujan’s formulas were generalized by Hardy [[4], p.98]. The work of Pincherle provided an impetus for the subsequent investigations of Barnes [1] and Mellin [2] on the integral representations of solutions of generalized hypergeometric series (see [[5], Chapter 16 and the comment on p.225]). A detailed commentary on Pincherle’s work [6] set against a historical backdrop is available in [7].

Indeed, a remarkably large number of integral formulas involving a variety of special functions have been developed by many authors (see, for example, [8]; for a very recent work, see also [9]). Recently, Garg and Mittal [10] obtained an interesting unified integral involving Fox H-function. Motivated by the work of Garg and Mittal [10], very recently, Ali [11] gave three interesting unified integrals involving the hypergeometric function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M3">View MathML</a>. Also, many integral formulas involving the Bessel function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M2">View MathML</a> (1.1) have been presented (see, e.g., [[8], pp.196-204]; see also [[12], pp.373-476]). Here, by using Ali’s method [11], we aim at presenting two generalized integral formulas involving the Bessel function of the first kind (1.1), which are expressed in terms of the generalized (Wright) hypergeometric functions (1.4). Some interesting special cases of our main results are also considered.

For our purpose, we begin by recalling some known functions and earlier works. The Bessel function of the first kind <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M2">View MathML</a> is defined for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M10">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M11">View MathML</a> by the following series (see, e.g., [[13], p.217, Entry 10.2.2] and [[12], p.40, Eq. (8)]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M12">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M13">View MathML</a> is a confluent hypergeometric series of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M4">View MathML</a> in (1.5), ℂ denotes the set of complex numbers and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M15">View MathML</a> is the familiar Gamma function (see [[14], Section 1.1]).

An interesting further generalization of the generalized hypergeometric series <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M4">View MathML</a> (1.5) is due to Fox [15] and Wright [16-18] who studied the asymptotic expansion of the generalized (Wright) hypergeometric function defined by (see [[19], p.21]; see also [20])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M17">View MathML</a>

(1.2)

where the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M19">View MathML</a> are positive real numbers such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M20">View MathML</a>

(1.3)

A special case of (1.2) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M21">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M4">View MathML</a> is the generalized hypergeometric series defined by (see [[14], Section 1.5])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M23">View MathML</a>

(1.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M24">View MathML</a> is the Pochhammer symbol defined (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M25">View MathML</a>) by (see [[14], p.2 and pp.4-6]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M26">View MathML</a>

(1.6)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M27">View MathML</a> denotes the set of nonpositive integers.

For our present investigation, we also need to recall the following Oberhettinger’s integral formula [21]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M28">View MathML</a>

(1.7)

provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M29">View MathML</a>.

2 Main results

We establish two generalized integral formulas, which are expressed in terms of the generalized (Wright) hypergeometric functions (1.4), by inserting the Bessel function of the first kind (1.1) with suitable arguments into the integrand of (1.7).

Theorem 1The following integral formula holds true: For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M30">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M32">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M33">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M34">View MathML</a>

(2.1)

Theorem 2The following integral formula holds true: For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M30">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M32">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M33">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M39">View MathML</a>

(2.2)

Proof By applying (1.1) to the integrand of (2.1) and then interchanging the order of integral sign and summation, which is verified by uniform convergence of the involved series under the given conditions, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M40">View MathML</a>

(2.3)

In view of the conditions given in Theorem 1, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M41">View MathML</a>

we can apply the integral formula (1.7) to the integral in (2.3) and obtain the following expression:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M42">View MathML</a>

which, upon using (1.2), yields (2.1). This completes the proof of Theorem 2. □

It is easy to see that a similar argument as in the proof of Theorem 2 will establish the integral formula (2.2).

Remark We begin by stating the principle of confluence involved in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M4">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M44">View MathML</a>

(2.4)

In view of this principle of confluence (2.4), for example, replacing y by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M45">View MathML</a> at the second integral of Ali’s work [[11], Eq. (2.2)] and taking the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M46">View MathML</a> on each side of the resulting identity, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M47">View MathML</a>

(2.5)

provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M48">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M49">View MathML</a>. Again, let us try to reduce <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M50">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M13">View MathML</a> in the integrand of (2.5) by using the principle of confluence (2.4). Replacing y by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M52">View MathML</a> in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M50">View MathML</a> of (2.5) and letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M54">View MathML</a> in the resulting identity, we easily see that both sides reduce to zero. On the other hand, in view of the last expression of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M2">View MathML</a> in (1.1), we also see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M13">View MathML</a> cannot directly generate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M3">View MathML</a> in the integrals of Ali’s main results [[11], p.152]. Even though, here, the authors make use of the method of Ali’s work [11] (see also [10]), we may carefully conclude that those results in both [11] and this paper do not seem to yield the other ones.

Next, we consider other variations of Theorem 1 and Theorem 2. In fact, we establish some integral formulas for the Bessel function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M2">View MathML</a> expressed in terms of the generalized hypergeometric function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M4">View MathML</a>. To do this, we recall the well-known Legendre duplication formula for the Gamma function Γ:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M60">View MathML</a>

(2.6)

which is equivalently written in terms of the Pochhammer symbol (1.6) as follows (see, for example, [[14], p.6]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M61">View MathML</a>

(2.7)

Now we are ready to state the following two corollaries.

Corollary 1Let the condition of Theorem 1 be satisfied and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M62">View MathML</a>. Then the following integral formula holds true:

(2.8)

Corollary 2Let the condition of Theorem 2 be satisfied and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M64">View MathML</a>. Then the following integral formula holds true:

(2.9)

Proof By writing the right-hand side of Eq. (2.1) in the original summation and applying (2.7) to the resulting summation, after a little simplification, we find that, when the last resulting summation is expressed in terms of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M4">View MathML</a> in (1.5), this completes the proof of Corollary 1. A similar argument as in the proof of Corollary 1 will establish the integral formula (2.9). □

3 Special cases

In this section, we derive certain new integral formulas for the cosine and sine functions involving in the integrands of (2.1) and (2.2). To do this, we recall the following known formula (see, for example, [[22], p.79, Eq. (15)]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M67">View MathML</a>

(3.1)

By applying the expression in (3.1) to (2.1), (2.2), (2.8) and (2.9), we obtain four integral formulas in Corollaries 3, 4, 5 and 6, respectively.

Corollary 3The following integral formula holds true: For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M68">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M29">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M33">View MathML</a>,

(3.2)

Corollary 4The following integral formula holds true: For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M68">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M29">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M33">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M75">View MathML</a>

(3.3)

If we employ the same method as in getting (2.8) and (2.9) to (3.2) and (3.3), we obtain the following two corollaries.

Corollary 5Let the condition of Corollary 3 be satisfied and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M76">View MathML</a>. Then the following integral formula holds true:

(3.4)

Corollary 6Let the condition of Corollary 4 be satisfied and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M76">View MathML</a>. Then the following integral formula holds true:

(3.5)

By recalling the following formula (see, for example, [[22], p.79, Eq. (14)]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M80">View MathML</a>

(3.6)

and applying this formula to (2.1), (2.2), (2.8) and (2.9), we obtain four more integral formulas in Corollaries 7, 8, 9 and 10, respectively.

Corollary 7The following integral formula holds true: For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M68">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M29">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M33">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M84">View MathML</a>

(3.7)

Corollary 8The following integral formula holds true: For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M68">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M29">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M33">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M88">View MathML</a>

(3.8)

If we employ the same method as in getting (2.8) and (2.9) to (3.7) and (3.8), we obtain the following two corollaries.

Corollary 9Let the condition of Corollary 7 be satisfied and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M76">View MathML</a>. Then the following integral formula holds true:

(3.9)

Corollary 10Let the condition of Corollary 8 be satisfied and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M91">View MathML</a>. Then the following integral formula holds true:

(3.10)

4 Concluding remark

In this section, we briefly consider another variation of the results derived in the preceding sections. The Fox H-function due to Charles Fox [15] can be regarded as an extreme generalization of the generalized hypergeometric functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M4">View MathML</a>, beyond the Meijer G-functions. Like the Meijer G-functions, the Fox H-functions turn out to be related to the Mellin-Barnes integrals and to the Mellin transforms, but in a more general way. Its asymptotic behavior and other properties of this function can be seen from the works of [23,24] and [25]. Further, it can be easily seen that the Bessel function of the first kind in (1.1) is a special case of the Fox H-function as follows (see [[25], p.2, Eq. (1.1)]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/95/mathml/M94">View MathML</a>

(4.1)

Therefore, the results presented in this paper are easily converted in terms of the Fox H-function after some suitable parametric replacement. We are also trying to find certain possible applications of those results presented here to some other research areas, for example, Srivastava and Exton [26] applied their integral involving the product of several Bessel functions to give an explicit expression of a generalized random walk.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors have equal contributions to each part of this paper. All authors have read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors should express their deepest thanks for the referees’ valuable comments and essential suggestions to improve this paper as in the present form. The first-named author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (2010-0011005).

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