- Open Access
Certain unified integrals associated with Bessel functions
© Choi and Agarwal; licensee Springer. 2013
- Received: 14 January 2013
- Accepted: 5 April 2013
- Published: 18 April 2013
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 . Using Ali’s method, in this paper, we present two generalized integral formulas involving the Bessel function of the first kind , 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.
- Gamma function
- hypergeometric function
- generalized hypergeometric function
- generalized (Wright) hypergeometric functions
- Bessel function of the first kind
- Oberhettinger’s integral formula
- Garg and Mittal’s integral formula
Integrals involving products of Gamma functions along vertical lines were first studied by Pincherle in 1888 and an extensive theory was developed by Barnes  and Mellin . Cahen  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 [, p.98]. The work of Pincherle provided an impetus for the subsequent investigations of Barnes  and Mellin  on the integral representations of solutions of generalized hypergeometric series (see [, Chapter 16 and the comment on p.225]). A detailed commentary on Pincherle’s work  set against a historical backdrop is available in .
Indeed, a remarkably large number of integral formulas involving a variety of special functions have been developed by many authors (see, for example, ; for a very recent work, see also ). Recently, Garg and Mittal  obtained an interesting unified integral involving Fox H-function. Motivated by the work of Garg and Mittal , very recently, Ali  gave three interesting unified integrals involving the hypergeometric function . Also, many integral formulas involving the Bessel function (1.1) have been presented (see, e.g., [, pp.196-204]; see also [, pp.373-476]). Here, by using Ali’s method , 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.
where is a confluent hypergeometric series of in (1.5), ℂ denotes the set of complex numbers and is the familiar Gamma function (see [, Section 1.1]).
and denotes the set of nonpositive integers.
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).
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).
provided and . Again, let us try to reduce to in the integrand of (2.5) by using the principle of confluence (2.4). Replacing y by in the of (2.5) and letting in the resulting identity, we easily see that both sides reduce to zero. On the other hand, in view of the last expression of in (1.1), we also see that cannot directly generate in the integrals of Ali’s main results [, p.152]. Even though, here, the authors make use of the method of Ali’s work  (see also ), we may carefully conclude that those results in both  and this paper do not seem to yield the other ones.
Now we are ready to state the following two corollaries.
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 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). □
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.
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.
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.
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.
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  applied their integral involving the product of several Bessel functions to give an explicit expression of a generalized random walk.
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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