• 제목/요약/키워드: Whipple's theorem

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GENERALIZATION OF WHIPPLE'S THEOREM FOR DOUBLE SERIES

  • RATHIE, ARJUN K.;GAUR, VIMAL K.;KIM, YONG SUP;PARK, CHAN BONG
    • 호남수학학술지
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    • 제26권1호
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    • pp.119-132
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    • 2004
  • In 1965, Bhatt and Pandey have obtained an analogue of the Whipple's theorem for double series by using Watson's theorem on the sum of a $_3F_2$. The aim of this paper is to derive twenty five results for double series closely related to the analogue of the Whipple's theorem for double series obtained by Bhatt and Pandey. The results are derived with the help of twenty five summation formulas closely related to the Watson's theorem on the sum of a $_3F_2$ obtained recently by Lavoie, Grondin, and Rathie.

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Generalizations of Dixon's and Whipple's Theorems on the Sum of a 3F2

  • Choi, Junesang;Malani, Shaloo;Rathie, Arjun K.
    • Kyungpook Mathematical Journal
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    • 제47권3호
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    • pp.449-454
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    • 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.

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

  • Kim, Yong-Sup;Rathie, Arjun Kumar;Choi, June-Sang
    • 대한수학회논문집
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    • 제25권2호
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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.

APPLICATIONS OF GENERALIZED KUMMER'S SUMMATION THEOREM FOR THE SERIES 2F1

  • Kim, Yong-Sup;Rathie, Arjun K.
    • 대한수학회보
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    • 제46권6호
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    • pp.1201-1211
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    • 2009
  • The aim of this research paper is to establish generalizations of classical Dixon's theorem for the series $_3F_2$, a result due to Bailey involving product of generalized hypergeometric series and certain very interesting summations due to Ramanujan. The results are derived with the help of generalized Kummer's summation theorem for the series $_2F_1$ obtained earlier by Lavoie, Grondin, and Rathie.

FURTHER SUMMATION FORMULAS FOR THE APPELL'S FUNCTION $F_1$

  • CHOI JUNESANG;HARSH HARSHVARDHAN;RATHIE ARJUN K.
    • 한국수학교육학회지시리즈B:순수및응용수학
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    • 제12권3호
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    • pp.223-228
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    • 2005
  • In 2001, Choi, Harsh & Rathie [Some summation formulas for the Appell's function $F_1$. East Asian Math. J. 17 (2001), 233-237] have obtained 11 results for the Appell's function $F_1$ with the help of Gauss's summation theorem and generalized Kummer's summation theorem. We aim at presenting 22 more results for $F_1$ with the help of the generalized Gauss's second summation theorem and generalized Bailey's theorem obtained by Lavoie, Grondin & Rathie [Generalizations of Whipple's theorem on the sum of a $_3F_2$. J. Comput. Appl. Math. 72 (1996), 293-300]. Two interesting (presumably) new special cases of our results for $F_1$ are also explicitly pointed out.

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NEW SERIES IDENTITIES FOR ${\frac{1}{\Pi}}$

  • Awad, Mohammed M.;Mohammed, Asmaa O.;Rakha, Medhat A.;Rathie, Arjun K.
    • 대한수학회논문집
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    • 제32권4호
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    • pp.865-874
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    • 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.

NEW LAPLACE TRANSFORMS FOR THE GENERALIZED HYPERGEOMETRIC FUNCTION 2F2

  • KIM, YONG SUP;RATHIE, ARJUN K.;LEE, CHANG HYUN
    • 호남수학학술지
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    • 제37권2호
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    • pp.245-252
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    • 2015
  • This paper is in continuation of the paper very recently published [New Laplace transforms of Kummer's confluent hypergeometric functions, Math. Comp. Modelling, 55 (2012), 1068-1071]. In this paper, our main objective is to show one can obtain so far unknown Laplace transforms of three rather general cases of generalized hypergeometric function $_2F_2(x)$ by employing generalized Watson's, Dixon's and Whipple's summation theorems for the series $_3F_2$ obtained earlier in a series of three research papers by Lavoie et al. [5, 6, 7]. The results established in this paper may be useful in theoretical physics, engineering and mathematics.