• Title/Summary/Keyword: Gauss's contiguous functions relations

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A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS

  • Harsh, Harsh Vardhan;Kim, Yong Sup;Rakha, Medhat Ahmed;Rathie, Arjun Kumar
    • Communications of the Korean Mathematical Society
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    • v.31 no.1
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    • pp.65-94
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    • 2016
  • In 1812, Gauss obtained fifteen contiguous functions relations. Later on, 1847, Henie gave their q-analogue. Recently, good progress has been done in finding more contiguous functions relations by employing results due to Gauss. In 1999, Cho et al. have obtained 24 new and interesting contiguous functions relations with the help of Gauss's 15 contiguous relations. In fact, such type of 72 relations exists and therefore the rest 48 contiguous functions relations have very recently been obtained by Rakha et al.. Thus, the paper is in continuation of the paper [16] published in Computer & Mathematics with Applications 61 (2011), 620.629. In this paper, first we obtained 15 q-contiguous functions relations due to Henie by following a different method and then with the help of these 15 q-contiguous functions relations, we obtain 72 new and interesting q-contiguous functions relations. These q-contiguous functions relations have wide applications.

SOME PROPERTIES OF EXTENDED τ-HYPERGEOMETRIC FUNCTION

  • Jana, Ranjan Kumar;Maheshwari, Bhumika;Shukla, Ajay Kumar
    • Communications of the Korean Mathematical Society
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    • v.33 no.4
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    • pp.1159-1170
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    • 2018
  • Recently, Parmar [5] introduced a new extension of the ${\tau}$-Gauss hypergeometric function $_2R^{\tau}_1(z)$. The main object of this paper is to study this extended ${\tau}$-Gauss hypergeometric function and obtain its properties including connection with modified Bessel function of third kind and extended generalized hypergeometric function, several contiguous relations, differential relations, integral transforms and elementary integrals. Various special cases of our results are also discussed.

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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NEW RESULTS FOR THE SERIES 2F2(x) WITH AN APPLICATION

  • Choi, Junesang;Rathie, Arjun Kumar
    • Communications of the Korean Mathematical Society
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    • v.29 no.1
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    • pp.65-74
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    • 2014
  • The well known quadratic transformation formula due to Gauss: $$(1-x)^{-2a}{_2F_1}\[{{a,b;}\\\hfill{21}{2b;}}\;-\frac{4x}{(1-x)^2}\]={_2F_1}\[{{a,a-b+\frac{1}{2};}\\\hfill{65}{b+\frac{1}{2};}}\;x^2\]$$ plays an important role in the theory of (generalized) hypergeometric series. In 2001, Rathie and Kim have obtained two results closely related to the above quadratic transformation for $_2F_1$. Our main objective of this paper is to deduce some interesting known or new results for the series $_2F_1(x)$ by using the above Gauss's quadratic transformation and its contiguous relations and then apply our results to provide a list of a large number of integrals involving confluent hypergeometric functions, some of which are (presumably) new. The results established here are (potentially) useful in mathematics, physics, statistics, engineering, and so on.