scholarly journals Generating Functions for Some Hypergeometric Functions of Four Variables via Laplace Integral Representations

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Jihad Younis ◽  
Maged Bin-Saad ◽  
Ashish Verma
Keyword(s):  
Generating Functions ◽  
Hypergeometric Functions ◽  
Essential Role ◽  
Integral Representations ◽  
Laplace Integral ◽  
Special Cases ◽  
Generating Relations

Generating functions plays an essential role in the investigation of several useful properties of the sequences which they generate. In this paper, we establish certain generating relations, involving some quadruple hypergeometric functions introduced by Bin-Saad and Younis. Some interesting special cases of our main results are also considered.

scholarly journals A (p,v)-Extension of Hurwitz-Lerch Zeta Function and its Properties

2018 ◽  
Author(s):  
Gauhar Rahman ◽  
KS Nisar ◽  
Shahid Mubeen
Keyword(s):  
Zeta Function ◽  
Beta Function ◽  
Integral Representations ◽  
Basic Properties ◽  
Mellin Transformation ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Derivative Formulas ◽  
Generating Relations

In this paper, we define a (p,v)-extension of Hurwitz-Lerch Zeta function by considering an extension of beta function defined by Parmar et al. [J. Classical Anal. 11 (2017) 81–106]. We obtain its basic properties which include integral representations, Mellin transformation, derivative formulas and certain generating relations. Also, we establish the special cases of the main results.

scholarly journals On a Certain Extension of the Hurwitz-Lerch Zeta Function

2014 ◽  
Vol 52 (2) ◽  
Author(s):  
Rakesh K. Parmar ◽  
R. K. Raina
Keyword(s):  
Zeta Function ◽  
Probability Distributions ◽  
Integral Representations ◽  
Mellin Transforms ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Generating Relations

AbstractOur purpose in this paper is to consider a generalized form of the extended Hurwitz-Lerch Zeta function. For this extended Hurwitz-Lerch Zeta function, we obtain some classical properties which includes various integral representations, a differential formula, Mellin transforms and certain generating relations. We further consider an application to probability distributions and also point out some important special cases of the main results.

scholarly journals Some Incomplete Hypergeometric Functions and Incomplete Riemann-Liouville Fractional Integral Operators

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 483 ◽  
Author(s):  
Mehmet Ali Özarslan ◽  
Ceren Ustaoğlu
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Recurrence Relations ◽  
Beta Function ◽  
Integral Operators ◽  
Integral Representations ◽  
Fractional Integral Operators ◽  
Derivative Formulas ◽  
Generating Relations ◽  
Transformation Formulas

Very recently, the incomplete Pochhammer ratios were defined in terms of the incomplete beta function B y ( x , z ) . With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric, and Appell’s functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, and recurrence relations. Furthermore, incomplete Riemann-Liouville fractional integral operators are introduced. This definition helps us to obtain linear and bilinear generating relations for the new incomplete Gauss hypergeometric functions.

scholarly journals Generalization of Konhauser Polynomials

2020 ◽  
Vol 2 (11) ◽  
pp. 8-14
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Biorthogonal System ◽  
Integral Representations ◽  
New Class ◽  
Generating Relations ◽  
Biorthogonal Polynomials

In this paper, we are showing study of biorthogonal polynomials associated with generalization of Laguere polynomials of Srivastava and Singhal [14]. It happens to generalized Konhauser. here we are trying to obtain the generating functions, recurrence relations, biorthogonality relations, integral representations and also bilinear and bilateral generating relations for the new class of biorthogonal system.

scholarly journals On the integral representations for the confluent hypergeometric function

2016 ◽  
Vol 472 (2193) ◽  
pp. 20160421 ◽  
Author(s):  
Bujar Xh. Fejzullahu
Keyword(s):  
Integral Representation ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Contour Integral ◽  
Confluent Hypergeometric Functions ◽  
Special Cases

In this paper, we derive a new contour integral representation for the confluent hypergeometric function as well as for its various special cases. Consequently, we derive expansions of the confluent hypergeometric function in terms of functions of the same kind. Furthermore, we obtain a new identity involving integrals and sums of confluent hypergeometric functions. Our results generalized several well-known results in the literature.

q-Integral and Moment Representations for q-Orthogonal Polynomials

2002 ◽  
Vol 54 (4) ◽  
pp. 709-735 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
Dennis Stanton
Keyword(s):  
Orthogonal Polynomials ◽  
Exponential Function ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Integral Representations ◽  
Basic Hypergeometric Functions ◽  
Transformation Formulas

AbstractWe develop a method for deriving integral representations of certain orthogonal polynomials as moments. These moment representations are applied to find linear and multilinear generating functions for q-orthogonal polynomials. As a byproduct we establish new transformation formulas for combinations of basic hypergeometric functions, including a new representation of the q-exponential function εq.

scholarly journals Some applications of Srivastava’s theorem involving a certain family of generalized and extended hypergeometric polynomials

Filomat ◽  
2015 ◽  
Vol 29 (8) ◽  
pp. 1811-1819 ◽  
Author(s):  
Shy-Der Lin ◽  
H.M. Srivastava ◽  
Mu-Ming Wong
Keyword(s):  
Generating Functions ◽  
Hypergeometric Functions ◽  
Integral Transforms ◽  
Pochhammer Symbol ◽  
Generalized Hypergeometric Functions ◽  
Fundamental Properties ◽  
Hypergeometric Polynomials ◽  
Special Cases

Recently, Srivastava et al. [H. M. Srivastava, M. A. Chaudhry and R. P. Agarwal, The incomplete Pochhammer symbols and their applications to hypergeometric and related functions, Integral Transforms Spec. Funct. 23 (2012), 659-683] introduced and initiated the study of many interesting fundamental properties and characteristics of a certain pair of potentially useful families of the so-called generalized incomplete hypergeometric functions. Ever since then there have appeared many closely-related works dealing essentially with notable developments involving various classes of generalized hypergeometric functions and generalized hypergeometric polynomials, which are defined by means of the corresponding incomplete and other novel extensions of the familiar Pochhammer symbol. Here, in this sequel to some of these earlier works, we derive several general families of hypergeometric generating functions by applying Srivastava?s Theorem. We also indicate various (known or new) special cases and consequences of the results presented in this paper.

scholarly journals The incomplete Srivastava’s triple hypergeometric functions γHB and ΓHB

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1779-1787 ◽  
Author(s):  
Junesang Choi ◽  
Rakesh Parmar ◽  
Purnima Chopra
Keyword(s):  
Recurrence Relation ◽  
Hypergeometric Functions ◽  
Integral Representations ◽  
Reduction Formula ◽  
Gamma Functions ◽  
Fundamental Properties ◽  
Special Cases ◽  
Incomplete Gamma Functions ◽  
Derivative Formula

Recently Srivastava et al. [26] introduced the incomplete Pochhammer symbols by means of the incomplete gamma functions ?(s,x) and ?(s,x), and defined incomplete hypergeometric functions whose a number of interesting and fundamental properties and characteristics have been investigated. Further, ?etinkaya [6] introduced the incomplete second Appell hypergeometric functions and studied many interesting and fundamental properties and characteristics. In this paper, motivated by the abovementioned works, we introduce two incomplete Srivastava?s triple hypergeometric functions ?HB and ?HB by using the incomplete Pochhammer symbols and investigate certain properties, for example, their various integral representations, derivative formula, reduction formula and recurrence relation. Various (known or new) special cases and consequences of the results presented here are also considered.

scholarly journals Further generalization of the extended Hurwitz-Lerch Zeta functions

2017 ◽  
Vol 37 (1) ◽  
pp. 177-190
Author(s):  
Rakesh K. Parmar ◽  
Junesang Choi ◽  
Sunil Dutt Purohit
Keyword(s):  
Generating Functions ◽  
Fractional Derivatives ◽  
Mellin Transform ◽  
Probability Distributions ◽  
Zeta Functions ◽  
Integral Representations ◽  
Special Cases

Recently various extensions of Hurwitz-Lerch Zeta functions have been investigated. Here, we first introduce a further generalization of the extended Hurwitz-Lerch Zeta functions. Then we investigate certain interesting and (potentially) useful properties, systematically, of the generalization of the extended Hurwitz-Lerch Zeta functions, for example, various integral representations, Mellin transform, generating functions and extended fractional derivatives formulas associated with these extended generalized Hurwitz-Lerch Zeta functions. An application to probability distributions is further considered. Some interesting special cases of our main results are also pointed out.

scholarly journals A new extension of Srivastava’s triple hypergeometric functions and their associated properties

Analysis ◽  
2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Abdus Saboor ◽  
Gauhar Rahman ◽  
Zunaira Anjum ◽  
Kottakkaran Sooppy Nisar ◽  
Serkan Araci
Keyword(s):  
Hypergeometric Functions ◽  
Recurrence Relations ◽  
Integral Representations ◽  
Pochhammer Symbol ◽  
Basic Properties ◽  
Special Cases ◽  
Derivative Formulas

AbstractIn this paper, we define a new extension of Srivastava’s triple hypergeometric functions by using a new extension of Pochhammer’s symbol that was recently proposed by Srivastava, Rahman and Nisar [H. M. Srivastava, G. Rahman and K. S. Nisar, Some extensions of the Pochhammer symbol and the associated hypergeometric functions, Iran. J. Sci. Technol. Trans. A Sci. 43 2019, 5, 2601–2606]. We present their certain basic properties such as integral representations, derivative formulas, and recurrence relations. Also, certain new special cases have been identified and some known results are recovered from main results.

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