• Honam Mathematical Journal
• /
• v.35 no.4
• /
• pp.701-706
• /
• 2013

Summation theorems for hypergeometric series $_2F_1$ and generalized hypergeometric series $_pF_q$ play important roles in themselves and their diverse applications. Some summation theorems for $_2F_1$ and $_pF_q$ have been established in several or many ways. Here we give a proof of Watson's classical summation theorem for the series $_3F_2$(1) by following the same lines used by Rakha [7] except for the last step in which we applied an integral formula introduced by Choi et al. [3].

• Communications of the Korean Mathematical Society
• /
• v.30 no.3
• /
• pp.227-237
• /
• 2015

By employing certain extended classical summation theorems, several surprising ${\pi}$ and other formulae are displayed.

• Honam Mathematical Journal
• /
• v.39 no.1
• /
• pp.73-82
• /
• 2017

In this paper we aim to demonstrate how one can obtain so far unknown Laplace transforms for the generalized hypergeometric functions $_pF_p$ for p = 2, 3, 4, and 5 by employing known summation theorems available in the literature.

• Communications of the Korean Mathematical Society
• /
• v.32 no.4
• /
• pp.865-874
• /
• 2017

In the theory of hypergeometric and generalized hypergeometric series, classical summation theorems have been found interesting applications in obtaining various series identities for ${\Pi}$, ${\Pi}^2$ and ${\frac{1}{\Pi}}$. The aim of this research paper is to provide twelve general formulas for ${\frac{1}{\Pi}}$. On specializing the parameters, a large number of very interesting series identities for ${\frac{1}{\Pi}}$ not previously appeared in the literature have been obtained. Also, several other results for multiples of ${\Pi}$, ${\Pi}^2$, ${\frac{1}{{\Pi}^2}}$, ${\frac{1}{{\Pi}^3}}$ and ${\frac{1}{\sqrt{\Pi}}}$ have been obtained. The results are established with the help of the extensions of classical Gauss's summation theorem available in the literature.

• Communications of the Korean Mathematical Society
• /
• v.32 no.4
• /
• pp.979-990
• /
• 2017

Inspired by the results obtained by Brychkov ([2]), the authors evaluate a large number of new and interesting double definite integrals. The results are obtained with the use of classical hypergeometric summation theorems and a well-known double finite integral due to Edwards ([3]). The results are given in terms of Psi and Hurwitz zeta functions suitable for numerical computations.

• Kyungpook Mathematical Journal
• /
• v.47 no.3
• /
• pp.449-454
• /
• 2007

InIn this paper we consider generalizations of the classical Dixon's theorem and the classical Whipple's theorem on the sum of a $_3F_2$. The results are derived with the help of generalized Watson's theorem obtained earlier by Mitra. A large number of results contiguous to Dixon's and Whipple's theorems obtained earlier by Lavoie, Grondin and Rathie, and Lavoie, Grondin, Rathie and Arora follow special cases of our main findings.

• The Pure and Applied Mathematics
• /
• v.25 no.1
• /
• pp.7-16
• /
• 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
• /
• v.21 no.1
• /
• pp.49-56
• /
• 1999

The object of this note is to provide various methods for proving two results contiguous to Kummer's second theorem by using the classical Gauss's summation theorem and contiguous relations.

• Honam Mathematical Journal
• /
• v.39 no.3
• /
• pp.453-463
• /
• 2017

The aim of this paper is to provide a class of six definite general integrals in terms of gamma function. The results are established with the help of generalized summation formulas obtained earlier by Rakha and Rathie. The results established in this paper are simple, interesting, easily established and may be useful potentially.

• Honam Mathematical Journal
• /
• v.38 no.1
• /
• pp.25-37
• /
• 2016

In 1972, Andrews derived the basic analogue of Gauss's second summation theorem and Bailey's theorem by implementing basic analogue of Kummer's theorem into identity due to Jackson. Recently Lavoie et.al. derived many results closely related to Kummer's theorem, Gauss's second summation theorem and Bailey's theorem and also Rakha et. al. derive the basic analogues of results closely related Kummer's theorem. The aim of this paper is to derive basic analogues of results closely related Gauss's second summation theorem and Bailey's theorem. Some applications and limiting cases are also considered.