• Title/Summary/Keyword: Gauss's 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.

GAUSS SUMMATION THEOREM AND ITS APPLICATIONS

  • Lee, Hyung-Jae;Cho, Young-Joon;Choi, June-Sang
    • East Asian mathematical journal
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    • v.17 no.1
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    • pp.147-158
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    • 2001
  • The Gauss summation theorem plays a key role in the theory of(generalized) hypergeometric series. The authors study several proofs of the theorem and consider some applications of it.

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ALTERNATIVE DERIVATIONS OF CERTAIN SUMMATION FORMULAS CONTIGUOUS TO DIXON'S SUMMATION THEOREM FOR A HYPERGEOMETRIC $_3F_2$ SERIES

  • Choi, June-Sang;Rathie Arjun K.;Malani Shaloo;Mathur Rachana
    • The Pure and Applied Mathematics
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    • v.13 no.4 s.34
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    • pp.255-259
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    • 2006
  • In 1994, Lavoie et al. have obtained twenty tree interesting results closely related to the classical Dixon's theorem on the sum of a $_3F_2$ by making a systematic use of some known relations among contiguous functions. We aim at showing that these results can be derived by using the same technique developed by Bailey with the help of Gauss's summation theorem and generalized Kummer's theorem obtained by Lavoie et al..

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ON A GENERALIZATION OF FENCHEL`S THEOREM

  • Chai, Y.D.;Kim, Moon-Jeong
    • Communications of the Korean Mathematical Society
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    • v.15 no.1
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    • pp.103-109
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    • 2000
  • In this paper, we present the proof of generalized Fenchel's theorem by estimating the Gauss-Kronecker curvature of the tube of a nondegenerate closed curve in R$^{n}$ .

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BOUR'S THEOREM IN 4-DIMENSIONAL EUCLIDEAN SPACE

  • Hieu, Doan The;Thang, Nguyen Ngoc
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.6
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    • pp.2081-2089
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    • 2017
  • In this paper we generalize 3-dimensional Bour's Theorem to the case of 4-dimension. We proved that a helicoidal surface in $\mathbb{R}^4$ is isometric to a family of surfaces of revolution in $\mathbb{R}^4$ in such a way that helices on the helicoidal surface correspond to parallel circles on the surfaces of revolution. Moreover, if the surfaces are required further to have the same Gauss map, then they are hyperplanar and minimal. Parametrizations for such minimal surfaces are given explicitly.

THREE-TERM CONTIGUOUS FUNCTIONAL RELATIONS FOR BASIC HYPERGEOMETRIC SERIES 2φ1

  • KIM, YONG-SUP;RATHIE ARJUN K.;CHOI, JUNE-SANG
    • Communications of the Korean Mathematical Society
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    • v.20 no.2
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    • pp.395-403
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    • 2005
  • The authors aim mainly at giving fifteen three-term contiguous relations for the basic hypergeometric series $series\;_2{\phi}_1$ corresponding to Gauss's contiguous relations for the hypergeometric series $series\;_2F_1$ given in Rainville([6], p.71). They also apply them to obtain two summation formulas closely related to a known q-analogue of Kummer's theorem.

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

  • CHOI JUNESANG;HARSH HARSHVARDHAN;RATHIE ARJUN K.
    • The Pure and Applied Mathematics
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    • v.12 no.3 s.29
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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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NOTE ON THE CLASSICAL WATSON'S THEOREM FOR THE SERIES 3F2

  • Choi, Junesang;Agarwal, P.
    • Honam Mathematical Journal
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    • v.35 no.4
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    • pp.701-706
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    • 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].