• Title/Summary/Keyword: Fox-Wright hypergeometric function

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NEW SEVEN-PARAMETER MITTAG-LEFFLER FUNCTION WITH CERTAIN ANALYTIC PROPERTIES

  • Maryam K. Rasheed;Abdulrahman H. Majeed
    • Nonlinear Functional Analysis and Applications
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    • v.29 no.1
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    • pp.99-111
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    • 2024
  • In this paper, a new seven-parameter Mittag-Leffler function of a single complex variable is proposed as a generalization of the standard Mittag-Leffler function, certain generalizations of Mittag-Leffler function, hypergeometric function and confluent hypergeometric function. Certain essential analytic properties are mainly discussed, such as radius of convergence, order, type, differentiation, Mellin-Barnes integral representation and Euler transform in the complex plane. Its relation to Fox-Wright function and H-function is also developed.

FRACTIONAL INTEGRATION AND DIFFERENTIATION OF THE (p, q)-EXTENDED BESSEL FUNCTION

  • Choi, Junesang;Parmar, Rakesh K.
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.2
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    • pp.599-610
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    • 2018
  • We aim to present some formulas for Saigo hypergeometric fractional integral and differential operators involving (p, q)-extended Bessel function $J_{{\nu},p,q}(z)$, which are expressed in terms of Hadamard product of the (p, q)-extended Gauss hypergeometric function and the Fox-Wright function $_p{\Psi}_q(z)$. A number of interesting special cases of our main results are also considered. Further, it is emphasized that the results presented here, which are seemingly complicated series, can reveal their involved properties via those of the two known functions in their respective Hadamard product.

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.

EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES

  • Arshad, Muhammad;Mubeen, Shahid;Nisar, Kottakkaran Sooppy;Rahman, Gauhar
    • Communications of the Korean Mathematical Society
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    • v.33 no.1
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    • pp.143-155
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    • 2018
  • In this present paper, our aim is to introduce an extended Wright-Bessel function $J^{{\lambda},{\gamma},c}_{{\alpha},q}(z)$ which is established with the help of the extended beta function. Also, we investigate certain integral transforms and generalized integration formulas for the newly defined extended Wright-Bessel function $J^{{\lambda},{\gamma},c}_{{\alpha},q}(z)$ and the obtained results are expressed in terms of Fox-Wright function. Some interesting special cases involving an extended Mittag-Leffler functions are deduced.

CERTAIN IMAGE FORMULAS OF (p, 𝜈)-EXTENDED GAUSS' HYPERGEOMETRIC FUNCTION AND RELATED JACOBI TRANSFORMS

  • Chopra, Purnima;Gupta, Mamta;Modi, Kanak
    • Communications of the Korean Mathematical Society
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    • v.37 no.4
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    • pp.1055-1072
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    • 2022
  • Our aim is to establish certain image formulas of the (p, 𝜈)-extended Gauss' hypergeometric function Fp,𝜈(a, b; c; z) by using Saigo's hypergeometric fractional calculus (integral and differential) operators. Corresponding assertions for the classical Riemann-Liouville(R-L) and Erdélyi-Kober(E-K) fractional integral and differential operators are deduced. All the results are represented in terms of the Hadamard product of the (p, 𝜈)-extended Gauss's hypergeometric function Fp,𝜈(a, b; c; z) and Fox-Wright function rΨs(z). We also established Jacobi and its particular assertions for the Gegenbauer and Legendre transforms of the (p, 𝜈)-extended Gauss' hypergeometric function Fp,𝜈(a, b; c; z).

Generalizations of Ramanujan's Integral Associated with Infinite Fourier Cosine Transforms in Terms of Hypergeometric Functions and its Applications

  • Qureshi, Mohammad Idris;Dar, Showkat Ahmad
    • Kyungpook Mathematical Journal
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    • v.60 no.4
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    • pp.781-795
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    • 2020
  • In this paper, we obtain an analytical solution for an unsolved definite integral RC (m, n) from a 1915 paper of Srinivasa Ramanujan. We obtain our solution using the hypergeometric approach and an infinite series decomposition identity. Also, we give some generalizations of Ramanujan's integral RC (m, n) defined in terms of the ordinary hypergeometric function 2F3 with suitable convergence conditions. Moreover as applications of our result we obtain nine new infinite summation formulas associated with the hypergeometric functions 0F1, 1F2 and 2F3.

SOME PROPERTIES OF GENERALIZED BESSEL FUNCTION ASSOCIATED WITH GENERALIZED FRACTIONAL CALCULUS OPERATORS

  • Jana, Ranjan Kumar;Pal, Ankit;Shukla, Ajay Kumar
    • Communications of the Korean Mathematical Society
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    • v.36 no.1
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    • pp.41-50
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    • 2021
  • This paper devoted to obtain some fractional integral properties of generalized Bessel function using pathway fractional integral operator. We also find the pathway transform of the generalized Bessel function in terms of Fox H-function.

On Certain Integral Transforms Involving Hypergeometric Functions and Struve Function

  • Singhal, Vijay Kumar;Mukherjee, Rohit
    • Kyungpook Mathematical Journal
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    • v.56 no.4
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    • pp.1169-1177
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    • 2016
  • This paper is devoted to the study of Mellin, Laplace, Euler and Whittaker transforms involving Struve function, generalized Wright function and Fox's H-function. The main results are presented in the form of four theorems. On account of the general nature of the functions involved here in, the main results obtained here yield a large number of known and new results in terms of simpler functions as their special cases. For the sake of illustration some corollaries have been recorded here as special cases of our main findings.

AN EXTENSION OF THE EXTENDED HURWITZ-LERCH ZETA FUNCTIONS OF TWO VARIABLES

  • Choi, Junesang;Parmar, Rakesh K.;Saxena, Ram K.
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.6
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    • pp.1951-1967
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    • 2017
  • We aim to introduce a further extension of a family of the extended Hurwitz-Lerch Zeta functions of two variables. We then systematically investigate several interesting properties of the extended function such as its integral representations which provide extensions of various earlier corresponding results of two and one variables, its summation formula, its Mellin-Barnes type contour integral representations, its computational representation and fractional derivative formulas. A multi-parameter extension of the extended Hurwitz-Lerch Zeta function of two variables is also introduced. Relevant connections of certain special cases of the main results presented here with some known identities are pointed out.