• Title/Summary/Keyword: Kummer′s second theorem

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ON BASIC ANALOGUE OF CLASSICAL SUMMATION THEOREMS DUE TO ANDREWS

  • Harsh, Harsh Vardhan;Rathie, Arjun K.;Purohit, Sunil Dutt
    • Honam Mathematical Journal
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    • v.38 no.1
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    • pp.25-37
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    • 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.

ANOTHER PROOF OF KUMMER'S SECOND THEOREM

  • Arjun K. Rathie;Choi, June-Sang
    • Communications of the Korean Mathematical Society
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    • v.13 no.4
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    • pp.933-936
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    • 1998
  • We aim at giving another method of proving the well-known and useful Kummer's second theorem without changing its original form.

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OTHER PROOFS OF KUMMER'S SECOND THEOREM

  • Malani, Shaloo;Choi, June-Sang
    • East Asian mathematical journal
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    • v.17 no.1
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    • pp.129-133
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    • 2001
  • The aim of this research note is to derive the well known Kummer's second theorem by transforming the integrals which represent some generalized hypergeometric functions. This theorem can also be shown by combining two known Bailey's and Preece's identities for the product of generalized hypergeometric series.

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A SUMMATION FORMULA FOR THE SERIES 3F2 DUE TO FOX AND ITS GENERALIZATIONS

  • Choi, Junesang;Rathie, Arjun K.
    • Communications of the Korean Mathematical Society
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    • v.30 no.2
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    • pp.103-108
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    • 2015
  • Fox [2] presented an interesting identity for $_pF_q$ which is expressed in terms of a finite summation of $_pF_q$'s whose involved numerator and denominator parameters are different from those in the starting one. Moreover Fox [2] found a very interesting and general summation formula for $_3F_2(1/2)$ as a special case of his above-mentioned general identity with the help of Kummer's second summation theorem for $_2F_1(1/2)$. Here, in this paper, we show how two general summation formulas for $$_3F_2\[\array{\hspace{110}{\alpha},{\beta},{\gamma};\\{\alpha}-m,\;\frac{1}{2}({\beta}+{\gamma}+i+1);}\;{\frac{1}{2}}\]$$, m being a nonnegative integer and i any integer, can be easily established by suitably specializing the above-mentioned Fox's general identity with, here, the aid of generalizations of Kummer's second summation theorem for $_2F_1(1/2)$ obtained recently by Rakha and Rathie [7]. Several known results are also seen to be certain special cases of our main identities.

TWO RESULTS FOR THE TERMINATING 3F2(2) WITH APPLICATIONS

  • Kim, Yong-Sup;Choi, June-Sang;Rathie, Arjun K.
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.3
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    • pp.621-633
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    • 2012
  • By establishing a new summation formula for the series $_3F_2(\frac{1}{2})$, recently Rathie and Pogany have obtained an interesting result known as Kummer type II transformation for the generalized hypergeometric function $_2F_2$. Here we aim at deriving their result by using a very elementary method and presenting two elegant results for certain terminating series $_3F_2(2)$. Furthermore two interesting applications of our new results are demonstrated.

ON CERTAIN REDUCIBILITY OF KAMPE DE FERIET FUNCTION

  • Kim, Yong-Sup
    • Honam Mathematical Journal
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    • v.31 no.2
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    • pp.167-176
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    • 2009
  • The aim of this paper is to obtain three interesting results for reducibility of Kamp$\'{e}$ de $\'{e}$riet function. The results are derived with the help of contiguous Gauss's second summation formulas obtained earlier by Lavoie et al. The results obtained by Bailey, Rathie and Nagar follow special cases of our main findings.

ON THE REDUCIBILITY OF KAMPÉ DE FÉRIET FUNCTION

  • Choi, Junesang;Rathie, Arjun K.
    • Honam Mathematical Journal
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    • v.36 no.2
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    • pp.345-355
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    • 2014
  • The main objective of this paper is to obtain a formula containing eleven interesting results for the reducibility of Kamp$\acute{e}$ de F$\acute{e}$riet function. The results are derived with the help of two general results for the series $_2F_1(2)$ very recently presented by Kim et al. Well known Kummer's second theorem and its contiguous results proved earlier by Rathie and Nagar, and Kim et al. follow special cases of our main findings.

APPELL'S FUNCTION F1 AND EXTON'S TRIPLE HYPERGEOMETRIC FUNCTION X9

  • Choi, Junesang;Rathie, Arjun K.
    • The Pure and Applied Mathematics
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    • v.20 no.1
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    • pp.37-50
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    • 2013
  • In the theory of hypergeometric functions of one or several variables, a remarkable amount of mathematicians's concern has been given to develop their transformation formulas and summation identities. Here we aim at presenting explicit expressions (in a single form) of the following weighted Appell's function $F_1$: $$(1+2x)^{-a}(1+2z)^{-b}F_1\;\(c,\;a,\;b;\;2c+j;\;\frac{4x}{1+2x},\;\frac{4z}{1+2z}\)\;(j=0,\;{\pm}1,\;{\ldots},\;{\pm}5)$$ in terms of Exton's triple hypergeometric $X_9$. The results are derived with the help of generalizations of Kummer's second theorem very recently provided by Kim et al. A large number of very interesting special cases including Exton's result are also given.