• Kyungpook Mathematical Journal
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• v.60 no.4
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• pp.781-795
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• 2020

In this paper, we obtain an analytical solution for an unsolved definite integral RC (m, n) from a 1915 paper of Srinivasa Ramanujan. We obtain our solution using the hypergeometric approach and an infinite series decomposition identity. Also, we give some generalizations of Ramanujan's integral RC (m, n) defined in terms of the ordinary hypergeometric function 2F3 with suitable convergence conditions. Moreover as applications of our result we obtain nine new infinite summation formulas associated with the hypergeometric functions 0F1, 1F2 and 2F3.

• Kyungpook Mathematical Journal
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• v.55 no.2
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• pp.439-447
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• 2015

Very recently the authors have obtained a very interesting reduction formula for the Srivastava's triple hypergeometric series $F^{(3)}$(x, y, z) by applying the so-called Beta integral method to the Henrici's triple product formula for the hypergeometric series. In this sequel, we also present three more interesting reduction formulas for the function $F^{(3)}$(x, y, z) by using the well known identities due to Bailey and Ramanujan. The results established here are simple, easily derived and (potentially) useful.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.423-436
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• 2018

We aim to establish certain Saigo hypergeometric fractional integral formulas for a finite product of the generalized k-Bessel functions, which are also used to present image formulas of several integral transforms including beta transform, Laplace transform, and Whittaker transform. The results presented here are potentially useful, and, being very general, can yield a large number of special cases, only two of which are explicitly demonstrated.

• Communications of the Korean Mathematical Society
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• v.29 no.3
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• pp.429-437
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• 2014

Motivated by the recent investigations of several authors, in this paper we present a generalization of a result obtained recently by Choi et al. ([3]) involving hypergeometric identities. The result is obtained by suitably applying fractional calculus method to a generalization of the hypergeometric transformation formula due to Kummer.

• Communications of the Korean Mathematical Society
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• v.29 no.1
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• pp.65-74
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• 2014

The well known quadratic transformation formula due to Gauss: $$(1-x)^{-2a}{_2F_1}\[{{a,b;}\\\hfill{21}{2b;}}\;-\frac{4x}{(1-x)^2}\]={_2F_1}\[{{a,a-b+\frac{1}{2};}\\\hfill{65}{b+\frac{1}{2};}}\;x^2\]$$ plays an important role in the theory of (generalized) hypergeometric series. In 2001, Rathie and Kim have obtained two results closely related to the above quadratic transformation for $_2F_1$. Our main objective of this paper is to deduce some interesting known or new results for the series $_2F_1(x)$ by using the above Gauss's quadratic transformation and its contiguous relations and then apply our results to provide a list of a large number of integrals involving confluent hypergeometric functions, some of which are (presumably) new. The results established here are (potentially) useful in mathematics, physics, statistics, engineering, and so on.

• Honam Mathematical Journal
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• v.37 no.3
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• pp.339-352
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• 2015

We aim at giving explicit expressions of $${\sum_{m,n=0}^{{\infty}}}{\frac{{\Delta}_{m+n}(-1)^nx^{m+n}}{({\rho})_m({\rho}+i)_nm!n!}$$, where i = 0, ${\pm}1$, ${\ldots}$, ${\pm}9$ and $\{{\Delta}_n\}$ is a bounded sequence of complex numbers. The main result is derived with the help of the generalized Kummer's summation theorem for the series $_2F_1$ obtained earlier by Choi. Further some special cases of the main result considered here are shown to include the results obtained earlier by Kim and Rathie and the identity due to Bailey.

• Kyungpook Mathematical Journal
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• v.58 no.1
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• pp.19-35
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• 2018

Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [30] and the second Appell function [6], we introduce here the incomplete Lauricella functions ${\gamma}^{(n)}_A$ and ${\Gamma}^{(n)}_A$ of n variables. We then systematically investigate several properties of each of these incomplete Lauricella functions including, for example, their various integral representations, finite summation formulas, transformation and derivative formulas, and so on. We provide relevant connections of some of the special cases of the main results presented here with known identities. Several potential areas of application of the incomplete hypergeometric functions in one and more variables are also pointed out.

• The Pure and Applied Mathematics
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• v.25 no.1
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• pp.7-16
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• 2018

The main objective of this paper is to demonstrate how one can obtain very quickly so far unknown Laplace transforms of rather general cases of the generalized hypergeometric function $_3F_3$ by employing generalizations of classical summation theorems for the series $_3F_2$ available in the literature. Several new as well known results obtained earlier by Kim et al. follow special cases of main findings.

• Honam Mathematical Journal
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• v.43 no.2
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• pp.167-181
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• 2021

In this paper, the Saigo's k-fractional integral and derivative operators involving k-hypergeometric function in the kernel are applied to the generalized k-Bessel function; results are expressed in term of k-Wright function, which are used to present image formulas of integral transforms including beta transform. Also special cases related to fractional calculus operators and Bessel functions are considered.

• The Pure and Applied Mathematics
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• v.20 no.4
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• pp.233-242
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• 2013

Certain interesting single (or double) infinite series associated with hypergeometric functions have been expressed in terms of Psi (or Digamma) function ${\psi}(z)$, for example, see Nishimoto and Srivastava [8], Srivastava and Nishimoto [13], Saxena [10], and Chen and Srivastava [5], and so on. In this sequel, with a view to unifying and extending those earlier results, we first establish two relations which some double infinite series involving hypergeometric functions are expressed in a single infinite series involving ${\psi}(z)$. With the help of those series relations we derived, we next present two functional relations which some double infinite series involving $\bar{H}$-functions, which are defined by a generalized Mellin-Barnes type of contour integral, are expressed in a single infinite series involving ${\psi}(z)$. The results obtained here are of general character and only two of their special cases, among numerous ones, are pointed out to reduce to some known results.