• 제목/요약/키워드: confluent hypergeometric functions

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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
    • 대한수학회보
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    • 제60권3호
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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 EXTENDED BETA, HYPERGEOMETRIC AND CONFLUENT HYPERGEOMETRIC FUNCTIONS

  • Choi, Junesang;Rathie, Arjun K.;Parmar, Rakesh K.
    • 호남수학학술지
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    • 제36권2호
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    • pp.357-385
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    • 2014
  • Recently several authors have extended the Gamma function, Beta function, the hypergeometric function, and the confluent hypergeometric function by using their integral representations and provided many interesting properties of their extended functions. Here we aim at giving further extensions of the abovementioned extended functions and investigating various formulas for the further extended functions in a systematic manner. Moreover, our extension of the Beta function is shown to be applied to Statistics and also our extensions find some connections with other special functions and polynomials such as Laguerre polynomials, Macdonald and Whittaker functions.

Confluent Hypergeometric Distribution and Its Applications on Certain Classes of Univalent Functions of Conic Regions

  • Porwal, Saurabh
    • Kyungpook Mathematical Journal
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    • 제58권3호
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    • pp.495-505
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    • 2018
  • The purpose of the present paper is to investigate Confluent hypergeometric distribution. We obtain some basic properties of this distribution. It is worthy to note that the Poisson distribution is a particular case of this distribution. Finally, we give a nice application of this distribution on certain classes of univalent functions of the conic regions.

SOME FRACTIONAL INTEGRAL FORMULAS INVOLVING THE PRODUCT OF CONFLUENT HYPERGEOMETRIC FUNCTIONS

  • Kim, Yongsup
    • 호남수학학술지
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    • 제39권3호
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    • pp.443-451
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    • 2017
  • Very recently, Agarwal gave remakably a scads of fractional integral formulas involving various special functions. Using the same technique, we obtain certain(presumably) new fractional integral formulas involving the product of confluent hypergeometric functions. Some interesting special cases of our two main results are considered.

TURÁN-TYPE INEQUALITIES FOR GAUSS AND CONFLUENT HYPERGEOMETRIC FUNCTIONS VIA CAUCHY-BUNYAKOVSKY-SCHWARZ INEQUALITY

  • Bhandari, Piyush Kumar;Bissu, Sushil Kumar
    • 대한수학회논문집
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    • 제33권4호
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    • pp.1285-1301
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    • 2018
  • This paper is devoted to the study of $Tur{\acute{a}}n$-type inequalities for some well-known special functions such as Gauss hypergeometric functions, generalized complete elliptic integrals and confluent hypergeometric functions which are derived by using a new form of the Cauchy-Bunyakovsky-Schwarz inequality. We also apply these inequalities for some sample of interest such as incomplete beta function, incomplete gamma function, elliptic integrals and modified Bessel functions to obtain their corresponding $Tur{\acute{a}}n$-type inequalities.

GENERALIZATION OF EXTENDED BETA FUNCTION, HYPERGEOMETRIC AND CONFLUENT HYPERGEOMETRIC FUNCTIONS

  • Lee, Dong-Myung;Rathie, Arjun K.;Parmar, Rakesh K.;Kim, Yong-Sup
    • 호남수학학술지
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    • 제33권2호
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    • pp.187-206
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    • 2011
  • The main object of this paper is to present generalization of extended beta function, extended hypergeometric and confluent hypergeometric function introduced by Chaudhry et al. and obtained various integral representations, properties of beta function, Mellin transform, beta distribution, differentiation formulas transform formulas, recurrence relations, summation formula for these new generalization.

A TYPE OF FRACTIONAL KINETIC EQUATIONS ASSOCIATED WITH THE (p, q)-EXTENDED 𝜏-HYPERGEOMETRIC AND CONFLUENT HYPERGEOMETRIC FUNCTIONS

  • Khan, Owais;Khan, Nabiullah;Choi, Junesang;Nisar, Kottakkaran Sooppy
    • Nonlinear Functional Analysis and Applications
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    • 제26권2호
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    • pp.381-392
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    • 2021
  • During the last several decades, a great variety of fractional kinetic equations involving diverse special functions have been broadly and usefully employed in describing and solving several important problems of physics and astrophysics. In this paper, we aim to find solutions of a type of fractional kinetic equations associated with the (p, q)-extended 𝜏 -hypergeometric function and the (p, q)-extended 𝜏 -confluent hypergeometric function, by mainly using the Laplace transform. It is noted that the main employed techniques for this chosen type of fractional kinetic equations are Laplace transform, Sumudu transform, Laplace and Sumudu transforms, Laplace and Fourier transforms, P𝛘-transform, and an alternative method.

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

  • Marfaing, Olivier
    • 호남수학학술지
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    • 제43권2호
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    • pp.183-197
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    • 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.

INEQUALITIES OF EXTENDED (p, q)-BETA AND CONFLUENT HYPERGEOMETRIC FUNCTIONS

  • Mubeen, Shahid;Nisar, Kottakkaran Sooppy;Rahman, Gauhar;Arshad, Muhammad
    • 호남수학학술지
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    • 제41권4호
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    • pp.745-756
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    • 2019
  • In this paper, we establish the log convexity and Turán type inequalities of extended (p, q)-beta functions. Likewise, we present the log-convexity, the monotonicity and Turán type inequalities for extended (p, q)-confluent hypergeometric function by utilizing the inequalities of extended (p, q)-beta functions.

INTEGRAL REPRESENTATIONS FOR SRIVASTAVA'S HYPERGEOMETRIC FUNCTION HA

  • Choi, June-Sang;Hasanov, Anvar;Turaev, Mamasali
    • 호남수학학술지
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    • 제34권1호
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    • pp.113-124
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    • 2012
  • While investigating the Lauricella's list of 14 complete second-order hypergeometric series in three variables, Srivastava noticed the existence of three additional complete triple hypergeometric series of the second order, which were denoted by $H_A$, $H_B$ and $H_C$. Each of these three triple hypergeometric functions $H_A$, $H_B$ and $H_C$ has been investigated extensively in many different ways including, for example, in the problem of finding their integral representations of one kind or the other. Here, in this paper, we aim at presenting further integral representations for the Srivatava's triple hypergeometric function $H_A$.