• Title/Summary/Keyword: Generalized Hypergeometric Function

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ON GENERALIZED EXTENDED BETA AND HYPERGEOMETRIC FUNCTIONS

  • Shoukat Ali;Naresh Kumar Regar;Subrat Parida
    • Honam Mathematical Journal
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    • v.46 no.2
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    • pp.313-334
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    • 2024
  • In the current study, our aim is to define new generalized extended beta and hypergeometric types of functions. Next, we methodically determine several integral representations, Mellin transforms, summation formulas, and recurrence relations. Moreover, we provide log-convexity, Turán type inequality for the generalized extended beta function and differentiation formulas, transformation formulas, differential and difference relations for the generalized extended hypergeometric type functions. Also, we additionally suggest a generating function. Further, we provide the generalized extended beta distribution by making use of the generalized extended beta function as an application to statistics and obtaining variance, coefficient of variation, moment generating function, characteristic function, cumulative distribution function, and cumulative distribution function's complement.

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.

EXTENDED HYPERGEOMETRIC FUNCTIONS OF TWO AND THREE VARIABLES

  • AGARWAL, PRAVEEN;CHOI, JUNESANG;JAIN, SHILPI
    • Communications of the Korean Mathematical Society
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    • v.30 no.4
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    • pp.403-414
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    • 2015
  • Extensions of some classical special functions, for example, Beta function B(x, y) and generalized hypergeometric functions $_pF_q$ have been actively investigated and found diverse applications. In recent years, several extensions for B(x, y) and $_pF_q$ have been established by many authors in various ways. Here, we aim to generalize Appell's hypergeometric functions of two variables and Lauricella's hypergeometric function of three variables by using the extended generalized beta type function $B_p^{({\alpha},{\beta};m)}$ (x, y). Then some properties of the extended generalized Appell's hypergeometric functions and Lauricella's hypergeometric functions are investigated.

SOME PROPERTIES OF BIVARIATE GENERALIZED HYPERGEOMETRIC PROBABILITY DISTRIBUTIONS

  • Kumar, C. Satheesh
    • Journal of the Korean Statistical Society
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    • v.36 no.3
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    • pp.349-355
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    • 2007
  • In this paper we study some important properties of the bivariate generalized hypergeometric probability (BGHP) distribution by establishing the existence of all the moments of the distribution and by deriving recurrence relations for raw moments. It is shown that certain mixtures of BGHP distributions are again BGHP distributions and a limiting case of the distribution is considered.

CERTAIN UNIFIED INTEGRAL FORMULAS INVOLVING THE GENERALIZED MODIFIED k-BESSEL FUNCTION OF FIRST KIND

  • Mondal, Saiful Rahman;Nisar, Kottakkaran Sooppy
    • Communications of the Korean Mathematical Society
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    • v.32 no.1
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    • pp.47-53
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    • 2017
  • Generalized integral formulas involving the generalized modified k-Bessel function $J^{b,c,{\gamma},{\lambda}}_{k,{\upsilon}}(z)$ of first kind are expressed in terms generalized Wright functions. Some interesting special cases of the main results are also discussed.

ON GENERALIZED WRIGHT'S HYPERGEOMETRIC FUNCTIONS AND FRACTIONAL CALCULUS OPERATORS

  • Raina, R.K.
    • East Asian mathematical journal
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    • v.21 no.2
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    • pp.191-203
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    • 2005
  • In the present paper we first establish some basic results for a substantially more general class of functions defined below. The results include simple differentiation and fractional calculus operators(integration and differentiation of arbitrary orders) for this class of functions. These results are then invoked in determining similar properties for the generalized Wright's hypergeometric functions. Further, norm estimate of a certain class of integral operators whose kernel involves the generalized Wright's hypergeometric function, and its composition(and other related properties) with the fractional calculus operators are also investigated.

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SOME PROPERTIES OF GENERALIZED HYPERGEOMETRIC FUNCTION

  • Rao, Snehal B.;Patel, Amit D.;Prajapati, Jyotindra C.;Shukla, Ajay K.
    • Communications of the Korean Mathematical Society
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    • v.28 no.2
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    • pp.303-317
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    • 2013
  • In present paper, we obtain functions $R_t(c,{\nu},a,b)$ and $R_t(c,-{\mu},a,b)$ by using generalized hypergeometric function. A recurrence relation, integral representation of the generalized hypergeometric function $_2R_1(a,b;c;{\tau};z)$ and some special cases have also been discussed.

Certain Fractional Integral Operators and Extended Generalized Gauss Hypergeometric Functions

  • CHOI, JUNESANG;AGARWAL, PRAVEEN;JAIN, SILPI
    • Kyungpook Mathematical Journal
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    • v.55 no.3
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    • pp.695-703
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    • 2015
  • Several interesting and useful extensions of some familiar special functions such as Beta and Gauss hypergeometric functions and their properties have, recently, been investigated by many authors. Motivated mainly by those earlier works, we establish some fractional integral formulas involving the extended generalized Gauss hypergeometric function by using certain general pair of fractional integral operators involving Gauss hypergeometric function $_2F_1$, Some interesting special cases of our main results are also considered.

CERTAIN INTEGRATION FORMULAE FOR THE GENERALIZED k-BESSEL FUNCTIONS AND DELEURE HYPER-BESSEL FUNCTION

  • Kim, Yongsup
    • Communications of the Korean Mathematical Society
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    • v.34 no.2
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    • pp.523-532
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    • 2019
  • Integrals involving a finite product of the generalized Bessel functions have recently been studied by Choi et al. [2, 3]. Motivated by these results, we establish certain unified integral formulas involving a finite product of the generalized k-Bessel functions. Also, we consider some integral formulas of the (p, q)-extended Bessel functions $J_{{\nu},p,q}(z)$ and the Delerue hyper-Bessel function which are proved in terms of (p, q)-extended generalized hypergeometric functions, and the generalized Wright hypergeometric functions, respectively.