• 제목/요약/키워드: extended generalized beta functions

검색결과 13건 처리시간 0.028초

EXTENDED HYPERGEOMETRIC FUNCTIONS OF TWO AND THREE VARIABLES

  • AGARWAL, PRAVEEN;CHOI, JUNESANG;JAIN, SHILPI
    • 대한수학회논문집
    • /
    • 제30권4호
    • /
    • pp.403-414
    • /
    • 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.

ON GENERALIZED EXTENDED BETA AND HYPERGEOMETRIC FUNCTIONS

  • Shoukat Ali;Naresh Kumar Regar;Subrat Parida
    • 호남수학학술지
    • /
    • 제46권2호
    • /
    • pp.313-334
    • /
    • 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.

Certain Fractional Integral Operators and Extended Generalized Gauss Hypergeometric Functions

  • CHOI, JUNESANG;AGARWAL, PRAVEEN;JAIN, SILPI
    • Kyungpook Mathematical Journal
    • /
    • 제55권3호
    • /
    • pp.695-703
    • /
    • 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 NEW EXTENSION OF HURWITZ-LERCH ZETA FUNCTION

  • KHAN, WASEEM A.;GHAYASUDDIN, M.;AHMAD, MOIN
    • Journal of applied mathematics & informatics
    • /
    • 제37권1_2호
    • /
    • pp.13-21
    • /
    • 2019
  • In the present research paper, we introduce a further extension of Hurwitz-Lerch zeta function by using the generalized extended Beta function defined by Parmar et al.. We investigate its integral representations, Mellin transform, generating functions and differential formula. In view of diverse applications of the Hurwitz-Lerch Zeta functions, the results presented here may be potentially useful in some related research areas.

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

  • Kim, Yongsup
    • 대한수학회논문집
    • /
    • 제32권2호
    • /
    • pp.321-332
    • /
    • 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.

SOME INTEGRAL TRANSFORMS AND FRACTIONAL INTEGRAL FORMULAS FOR THE EXTENDED HYPERGEOMETRIC FUNCTIONS

  • Agarwal, Praveen;Choi, Junesang;Kachhia, Krunal B.;Prajapati, Jyotindra C.;Zhou, Hui
    • 대한수학회논문집
    • /
    • 제31권3호
    • /
    • pp.591-601
    • /
    • 2016
  • Integral transforms and fractional integral formulas involving well-known special functions are interesting in themselves and play important roles in their diverse applications. A large number of integral transforms and fractional integral formulas have been established by many authors. In this paper, we aim at establishing some (presumably) new integral transforms and fractional integral formulas for the generalized hypergeometric type function which has recently been introduced by Luo et al. [9]. Some interesting special cases of our main results are also considered.

AN EXTENSION OF THE BETA FUNCTION EXPRESSED AS A COMBINATION OF CONFLUENT HYPERGEOMETRIC FUNCTIONS

  • Marfaing, Olivier
    • 호남수학학술지
    • /
    • 제43권2호
    • /
    • pp.183-197
    • /
    • 2021
  • Recently several authors have extended the Beta function by using its integral representation. However, in many cases no expression of these extended functions in terms of classic special functions is known. In the present paper, we introduce a further extension by defining a family of functions Gr,s : ℝ*+ → ℂ, with r, s ∈ ℂ and ℜ(r) > 0. For given r, s, we prove that this function satisfies a second-order linear differential equation with rational coefficients. Solving this ODE, we express Gr,s as a combination of confluent hypergeometric functions. From this we deduce a new integral relation satisfied by Tricomi's function. We then investigate additional specific properties of Gr,1 which take the form of new non trivial integral relations involving exponential and error functions. We discuss the connection between Gr,1 and Stokes' first problem (or Rayleigh problem) in fluid mechanics which consists in determining the flow created by the movement of an infinitely long plate. For $r{\in}{\frac{1}{2}}{\mathbb{N}}^*$, we find additional relations between Gr,1 and Hermite polynomials. In view of these results, we believe the family of extended beta functions Gr,s will find further applications in two directions: (i) for improving our knowledge of confluent hypergeometric functions and Tricomi's function, (ii) and for engineering and physics problems.

GENERALIZATION OF EXTENDED APPELL'S AND LAURICELLA'S HYPERGEOMETRIC FUNCTIONS

  • Khan, N.U.;Ghayasuddin, M.
    • 호남수학학술지
    • /
    • 제37권1호
    • /
    • pp.113-126
    • /
    • 2015
  • Recently, Liu and Wang generalized Appell's and Lauricella's hypergeometric functions. Motivated by the work of Liu and Wang, the main object of this paper is to present new generalizations of Appell's and Lauricella's hypergeometric functions. Some integral representations, transformation formulae, differential formulae and recurrence relations are obtained for these new generalized Appell's and Lauricella's functions.

EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES

  • Arshad, Muhammad;Mubeen, Shahid;Nisar, Kottakkaran Sooppy;Rahman, Gauhar
    • 대한수학회논문집
    • /
    • 제33권1호
    • /
    • pp.143-155
    • /
    • 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.

A NEW EXTENSION OF THE MITTAG-LEFFLER FUNCTION

  • Arshad, Muhammad;Choi, Junesang;Mubeen, Shahid;Nisar, Kottakkaran Sooppy;Rahman, Gauhar
    • 대한수학회논문집
    • /
    • 제33권2호
    • /
    • pp.549-560
    • /
    • 2018
  • Since Mittag-Leffler introduced the so-called Mittag-Leffler function in 1903, due to its usefulness and diverse applications, a variety and large number of its extensions (and generalizations) and variants have been presented and investigated. In this sequel, we aim to introduce a new extension of the Mittag-Leffler function by using a known extended beta function. Then we investigate ceratin useful properties and formulas associated with the extended Mittag-Leffler function such as integral representation, Mellin transform, recurrence relation, and derivative formulas. We also introduce an extended Riemann-Liouville fractional derivative to present a fractional derivative formula for a known extended Mittag-Leffler function, the result of which is expressed in terms of the new extended Mittag-Leffler functions.