• Title/Summary/Keyword: Hypergeometric function

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A NEW EXTENSION OF BESSEL FUNCTION

  • Chudasama, Meera H.
    • Communications of the Korean Mathematical Society
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    • v.36 no.2
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    • pp.277-298
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    • 2021
  • In this paper, we propose an extension of the classical Bessel function by means of our ℓ-hypergeometric function [2]. As the main results, the infinite order differential equation, the generating function relation, and contour integral representations including Schläfli's integral analogue are derived. With the aid of these, other results including some inequalities are also obtained. At the end, the graphs of these functions are plotted using the Maple software.

HYPERGEOMETRIC FUNCTIONS AND EICHLER INTEGRALS

  • Lim, Su-Bong
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.12 no.4
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    • pp.223-226
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    • 2008
  • Duke and Imamo$\bar{g}$lu express the Eichler integrals associated to modular forms of weight 3 in terms of generalized hypergeometric functions. We extend this result to most general modular forms of weight 3.

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On p-adic analogue of hypergeometric series

  • Kim, Yong-Sup;Song, Hyeong-Kee
    • Communications of the Korean Mathematical Society
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    • v.12 no.1
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    • pp.11-16
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    • 1997
  • In this paper we will study a p-adic analogue of Kummer's theorem[6],[7], which gives the value at x = -1 of a well-piosed $_2F_1$ hypergeometric series.

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THE HARMONIC ANALYSIS ASSOCIATED TO THE HECKMAN-OPDAM'S THEORY AND ITS APPLICATION TO A ROOT SYSTEM OF TYPE BCd

  • Trimeche, Khalifa
    • Korean Journal of Mathematics
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    • v.27 no.1
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    • pp.221-267
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    • 2019
  • In the five first sections of this paper we define and study the hypergeometric transmutation operators $V^W_k$ and $^tV^W_k$ called also the trigonometric Dunkl intertwining operator and its dual corresponding to the Heckman-Opdam's theory on ${\mathbb{R}}^d$. By using these operators we define the hypergeometric translation operator ${\mathcal{T}}^W_x$, $x{\in}{\mathbb{R}}^d$, and its dual $^t{\mathcal{T}}^W_x$, $x{\in}{\mathbb{R}}^d$, we express them in terms of the hypergeometric Fourier transform ${\mathcal{H}}^W$, we give their properties and we deduce simple proofs of the Plancherel formula and the Plancherel theorem for the transform ${\mathcal{H}}^W$. We study also the hypergeometric convolution product on W-invariant $L^p_{\mathcal{A}k}$-spaces, and we obtain some interesting results. In the sixth section we consider a some root system of type $BC_d$ (see [17]) of whom the corresponding hypergeometric translation operator is a positive integral operator. By using this positivity we improve the results of the previous sections and we prove others more general results.

A Study of Marichev-Saigo-Maeda Fractional Integral Operators Associated with the S-Generalized Gauss Hypergeometric Function

  • Bansal, Manish Kumar;Kumar, Devendra;Jain, Rashmi
    • Kyungpook Mathematical Journal
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    • v.59 no.3
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    • pp.433-443
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    • 2019
  • In this work, we evaluate the Mellin transform of the Marichev-Saigo-Maeda fractional integral operator with Appell's function $F_3$ type kernel. We then discuss six special cases of the result involving the Saigo fractional integral operator, the $Erd{\acute{e}}lyi$-Kober fractional integral operator, the Riemann-Liouville fractional integral operator and the Weyl fractional integral operator. We obtain new and known results as special cases of our main results. Finally, we obtain the images of S-generalized Gauss hypergeometric function under the operators of our study.

FURTHER HYPERGEOMETRIC IDENTITIES DEDUCIBLE BY FRACTIONAL CALCULUS

  • Gaboury, Sebastien;Rathie, Arjun K.
    • Communications of the Korean Mathematical Society
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    • v.29 no.3
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    • pp.429-437
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    • 2014
  • Motivated by the recent investigations of several authors, in this paper we present a generalization of a result obtained recently by Choi et al. ([3]) involving hypergeometric identities. The result is obtained by suitably applying fractional calculus method to a generalization of the hypergeometric transformation formula due to Kummer.

SOME INTEGRAL REPRESENTATIONS AND TRANSFORMS FOR EXTENDED GENERALIZED APPELL'S AND LAURICELLA'S HYPERGEOMETRIC FUNCTIONS

  • Kim, Yongsup
    • Communications of the Korean Mathematical Society
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    • v.32 no.2
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    • pp.321-332
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    • 2017
  • In this paper, we generalize the extended Appell's and Lauricella's hypergeometric functions which have recently been introduced by Liu [9] and Khan [7]. Also, we aim at establishing some (presumbly) new integral representations and transforms for the extended generalized Appell's and Lauricella's hypergeometric functions.

A TRANSFORMATION FORMULA ASSOCIATED WITH THE GENERALIZED HYPERGEOMETRIC SERIES

  • Lee, Keumsik;Cho, Young-Joon;Seo, Tae-Young
    • Communications of the Korean Mathematical Society
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    • v.15 no.4
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    • pp.707-714
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    • 2000
  • The authors aim at presenting a presumably new transformation formula involving generalized hypergeometric series by making use of series rearrangement technique which is one of the most effective methods for obtaining generating functions or other identities associated with (especially) the hypergeometric series. They also consider a couple of interesting special cases of their main result.

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A PROOF OF THE MOST IMPORTANT IDENTITY INVOLVED IN THE BETA FUNCTION

  • Choi, June-Sang
    • The Pure and Applied Mathematics
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    • v.4 no.1
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    • pp.71-76
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    • 1997
  • A new proof of the well-known identity involved in the Beta function B(p, q) is given by using the theory of hypergeometric series and a brief history of Gamma function is also provided. The method here is shown to be able to apply to evaluate some definite integrals.

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