• Title/Summary/Keyword: a triple hypergeometric function

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INTEGRAL REPRESENTATIONS FOR SRIVASTAVA'S HYPERGEOMETRIC FUNCTION HA

  • Choi, June-Sang;Hasanov, Anvar;Turaev, Mamasali
    • Honam Mathematical Journal
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    • v.34 no.1
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    • pp.113-124
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    • 2012
  • While investigating the Lauricella's list of 14 complete second-order hypergeometric series in three variables, Srivastava noticed the existence of three additional complete triple hypergeometric series of the second order, which were denoted by $H_A$, $H_B$ and $H_C$. Each of these three triple hypergeometric functions $H_A$, $H_B$ and $H_C$ has been investigated extensively in many different ways including, for example, in the problem of finding their integral representations of one kind or the other. Here, in this paper, we aim at presenting further integral representations for the Srivatava's triple hypergeometric function $H_A$.

INTEGRAL REPRESENTATIONS FOR SRIVASTAVA'S HYPERGEOMETRIC FUNCTION HB

  • Choi, June-Sang;Hasanov, Anvar;Turaev, Mamasali
    • The Pure and Applied Mathematics
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    • v.19 no.2
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    • pp.137-145
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    • 2012
  • While investigating the Lauricella's list of 14 complete second-order hypergeometric series in three variables, Srivastava noticed the existence of three additional complete triple hypergeometric series of the second order, which were denoted by $H_A$, $H_B$ and $H_C$. Each of these three triple hypergeometric functions $H_A$, $H_B$ and $H_C$ has been investigated extensively in many different ways including, for example, in the problem of finding their integral representations of one kind or the other. Here, in this paper, we aim at presenting further integral representations for the Srivatava's triple hypergeometric function $H_B$.

INTEGRAL REPRESENTATIONS FOR SRIVASTAVA'S HYPERGEOMETRIC FUNCTION HC

  • Choi, Junesang;Hasanov, Anvar;Turaev, Mamasali
    • Honam Mathematical Journal
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    • v.34 no.4
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    • pp.473-482
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    • 2012
  • While investigating the Lauricella's list of 14 complete second-order hypergeometric series in three variables, Srivastava noticed the existence of three additional complete triple hypergeo-metric series of the second order, which were denoted by $H_A$, $H_B$ and $H_C$. Each of these three triple hypergeometric functions $H_A$, $H_B$ and $H_C$ has been investigated extensively in many different ways including, for example, in the problem of finding their integral representations of one kind or the other. Here, in this paper, we aim at presenting further integral representations for the Srivatava's triple hypergeometric function $H_C$.

A REDUCIBILITY OF SRIVASTAVA'S TRIPLE HYPERGEOMETRIC SERIES F(3)[x, y, z]

  • Choi, Junesang;Wang, Xiaoxia;Rathie, Arjun K.
    • Communications of the Korean Mathematical Society
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    • v.28 no.2
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    • pp.297-301
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    • 2013
  • When certain general single or multiple hypergeometric functions were introduced, their reduction formulas have naturally been investigated. Here, in this paper, we aim at presenting a very interesting reduction formula for the Srivastava's triple hypergeometric function $F^{(3)}[x,y,z]$ by applying the so-called Beta integral method to the Henrici's triple product formula for hypergeometric series.

Reduction Formulas for Srivastava's Triple Hypergeometric Series F(3)[x, y, z]

  • CHOI, JUNESANG;WANG, XIAOXIA;RATHIE, ARJUN K.
    • 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.

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.

SUMMATION FORMULAS DERIVED FROM THE SRIVASTAVA'S TRIPLE HYPERGEOMETRIC SERIES HC

  • Kim, Yong-Sup;Rathie, Arjun Kumar;Choi, June-Sang
    • Communications of the Korean Mathematical Society
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    • v.25 no.2
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    • pp.185-191
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    • 2010
  • Srivastava noticed the existence of three additional complete triple hypergeometric functions $H_A$, $H_B$ and $H_C$ of the second order in the course of an extensive investigation of Lauricella's fourteen hypergeometric functions of three variables. In 2004, Rathie and Kim obtained four summation formulas containing a large number of very interesting reducible cases of Srivastava's triple hypergeometric series $H_A$ and $H_C$. Here we are also aiming at presenting two unified summation formulas (actually, including 62 ones) for some reducible cases of Srivastava's $H_C$ with the help of generalized Dixon's theorem and generalized Whipple's theorem on the sum of a $_3F_2$ obtained earlier by Lavoie et al.. Some special cases of our results are also considered.

ON AN EXTENSION FORMULAS FOR THE TRIPLE HYPERGEOMETRIC SERIES X8 DUE TO EXTON

  • Kim, Yong-Sup;Rathie, Arjun K.
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.4
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    • pp.743-751
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    • 2007
  • The aim of this article is to derive twenty five transformation formulas in the form of a single result for the triple hypergeometric series $X_8$ introduced earlier by Exton. The results are derived with the help of generalized Watson#s theorem obtained earlier by Lavoie et al. An interesting special cases are also pointed out.

ANOTHER METHOD FOR PADMANABHAM'S TRANSFORMATION FORMULA FOR EXTON'S TRIPLE HYPERGEOMETRIC SERIES X8

  • Kim, Yong-Sup;Rathie, Arjun Kumar;Choi, June-Sang
    • Communications of the Korean Mathematical Society
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    • v.24 no.4
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    • pp.517-521
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    • 2009
  • The object of this note is to derive Padmanabham's transformation formula for Exton's triple hypergeometric series $X_8$ by using a different method from that of Padmanabham's. An interesting special case is also pointed out.

CERTAIN INTEGRAL REPRESENTATIONS OF EULER TYPE FOR THE EXTON FUNCTION X8

  • Choi, June-Sang;Hasanov, Anvar;Turaev, Mamasali
    • Communications of the Korean Mathematical Society
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    • v.27 no.2
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    • pp.257-264
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    • 2012
  • Exton introduced 20 distinct triple hypergeometric functions whose names are $X_i$ (i = 1, ${\ldots}$, 20) to investigate their twenty Laplace integral representations whose kernels include the confluent hypergeometric functions $_0F_1$, $_1F_1$, a Humbert function ${\Psi}_1$, and a Humbert function ${\Phi}_2$. The object of this paper is to present 18 new integral representations of Euler type for the Exton hypergeometric function $X_8$, whose kernels include the Exton functions ($X_2$, $X_8$) itself, the Horn's function $H_4$, the Gauss hypergeometric function $F$, and Lauricella hypergeometric function $F_C$. We also provide a system of partial differential equations satisfied by $X_8$.