• Title/Summary/Keyword: generalized Pochhammer symbol

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EXTENSIONS OF MULTIPLE LAURICELLA AND HUMBERT'S CONFLUENT HYPERGEOMETRIC FUNCTIONS THROUGH A HIGHLY GENERALIZED POCHHAMMER SYMBOL AND THEIR RELATED PROPERTIES

  • Ritu Agarwal;Junesang Choi;Naveen Kumar;Rakesh K. Parmar
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.3
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    • pp.575-591
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    • 2023
  • Motivated by several generalizations of the Pochhammer symbol and their associated families of hypergeometric functions and hypergeometric polynomials, by choosing to use a very generalized Pochhammer symbol, we aim to introduce certain extensions of the generalized Lauricella function F(n)A and the Humbert's confluent hypergeometric function Ψ(n) of n variables with, as their respective particular cases, the second Appell hypergeometric function F2 and the generalized Humbert's confluent hypergeometric functions Ψ2 and investigate their several properties including, for example, various integral representations, finite summation formulas with an s-fold sum and integral representations involving the Laguerre polynomials, the incomplete gamma functions, and the Bessel and modified Bessel functions. Also, pertinent links between the major identities discussed in this article and different (existing or novel) findings are revealed.

Extension of Generalized Hurwitz-Lerch Zeta Function and Associated Properties

  • Choi, Junesang;Parmar, Rakesh Kumar;Raina, Ravinder Krishna
    • Kyungpook Mathematical Journal
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    • v.57 no.3
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    • pp.393-400
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    • 2017
  • Very recently, Srivastava et al. [8] introduced an extension of the Pochhammer symbol and used it to define a generalization of the generalized hypergeometric functions. In this paper, by using the generalized Pochhammer symbol, we extend the generalized Hurwitz-Lerch Zeta function by Goyal and Laddha [6] and investigate some interesting properties which include various integral representations, Mellin transforms, differential formula and generating function. Some interesting special cases of our main results are also considered.

THE INCOMPLETE LAURICELLA AND FIRST APPELL FUNCTIONS AND ASSOCIATED PROPERTIES

  • Choi, Junesang;Parmar, Rakesh K.;Chopra, Purnima
    • Honam Mathematical Journal
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    • v.36 no.3
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    • pp.531-542
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    • 2014
  • Recently, Srivastava et al. [18] introduced the incomplete Pochhammer symbol and studied some fundamental properties and characteristics of a family of potentially useful incomplete hypergeometric functions. Here we introduce the incomplete Lauricella function ${\gamma}_D^{(n)}$ and ${\Gamma}_D^{(n)}$ of n variables, and investigate certain properties of the incomplete Lauricella functions, for example, their various integral representations, differential formula and recurrence relation, in a rather systematic manner. Some interesting special cases of our main results are also considered.

INCOMPLETE EXTENDED HURWITZ-LERCH ZETA FUNCTIONS AND ASSOCIATED PROPERTIES

  • Parmar, Rakesh K.;Saxena, Ram K.
    • Communications of the Korean Mathematical Society
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    • v.32 no.2
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    • pp.287-304
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    • 2017
  • Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [Integral Transforms Spec. Funct. 23 (2012), 659-683] by means of the incomplete Pochhammer symbols $({\lambda};{\kappa})_{\nu}$ and $[{\lambda};{\kappa}]_{\nu}$, we first introduce incomplete Fox-Wright function. We then define the families of incomplete extended Hurwitz-Lerch Zeta function. We then systematically investigate several interesting properties of these incomplete extended Hurwitz-Lerch Zeta function which include various integral representations, summation formula, fractional derivative formula. We also consider an application to probability distributions and some special cases of our main results.

THE INCOMPLETE GENERALIZED τ-HYPERGEOMETRIC AND SECOND τ-APPELL FUNCTIONS

  • Parmar, Rakesh Kumar;Saxena, Ram Kishore
    • Journal of the Korean Mathematical Society
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    • v.53 no.2
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    • pp.363-379
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    • 2016
  • Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [Integral Transforms Spec. Funct. 23 (2012), 659-683] and the second Appell function [Appl. Math. Comput. 219 (2013), 8332-8337] by means of the incomplete Pochhammer symbols $({\lambda};{\kappa})_{\nu}$ and $[{\lambda};{\kappa}]_{\nu}$, we introduce here the family of the incomplete generalized ${\tau}$-hypergeometric functions $2{\gamma}_1^{\tau}(z)$ and $2{\Gamma}_1^{\tau}(z)$. The main object of this paper is to study these extensions and investigate their several properties including, for example, their integral representations, derivative formulas, Euler-Beta transform and associated with certain fractional calculus operators. Further, we introduce and investigate the family of incomplete second ${\tau}$-Appell hypergeometric functions ${\Gamma}_2^{{\tau}_1,{\tau}_2}$ and ${\gamma}_2^{{\tau}_1,{\tau}_2}$ of two variables. Relevant connections of certain special cases of the main results presented here with some known identities are also pointed out.

The p-deformed Generalized Humbert Polynomials and Their Properties

  • Savalia, Rajesh V.;Dave, B.I.
    • Kyungpook Mathematical Journal
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    • v.60 no.4
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    • pp.731-752
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    • 2020
  • We introduce the p-deformation of generalized Humbert polynomials. For these polynomials, we derive the differential equation, generating function relations, Fibonacci-type representations, and recurrence relations and state the companion matrix. These properties are illustrated for certain polynomials belonging to p-deformed generalized Humbert polynomials.

The Incomplete Lauricella Functions of Several Variables and Associated Properties and Formulas

  • Choi, Junesang;Parmar, Rakesh K.;Srivastava, H.M.
    • Kyungpook Mathematical Journal
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    • v.58 no.1
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    • pp.19-35
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    • 2018
  • Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [30] and the second Appell function [6], we introduce here the incomplete Lauricella functions ${\gamma}^{(n)}_A$ and ${\Gamma}^{(n)}_A$ of n variables. We then systematically investigate several properties of each of these incomplete Lauricella functions including, for example, their various integral representations, finite summation formulas, transformation and derivative formulas, and so on. We provide relevant connections of some of the special cases of the main results presented here with known identities. Several potential areas of application of the incomplete hypergeometric functions in one and more variables are also pointed out.

CERTAIN FORMULAS INVOLVING EULERIAN NUMBERS

  • Choi, Junesang
    • Honam Mathematical Journal
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    • v.35 no.3
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    • pp.373-379
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    • 2013
  • In contrast with numerous identities involving the binomial coefficients and the Stirling numbers of the first and second kinds, a few identities involving the Eulerian numbers have been known. The objective of this note is to present certain interesting and (presumably) new identities involving the Eulerian numbers by mainly making use of Worpitzky's identity.

NOTE ON STIRLING POLYNOMIALS

  • Choi, Junesang
    • Journal of the Chungcheong Mathematical Society
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    • v.26 no.3
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    • pp.591-599
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    • 2013
  • A large number of sequences of polynomials and numbers have arisen in mathematics. Some of them, for example, Bernoulli polynomials and numbers, have been investigated deeply and widely. Here we aim at presenting certain interesting and (potentially) useful identities involving mainly in the second-order Eulerian numbers and Stirling polynomials, which seem to have not been given so much attention.

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.