scholarly journals Integrals involving Hermite polynomials, generalized hypergeometric series and Fox's H-function, and Fourier-Hermite series for products of generalized hypergeometric functions

1991 ◽  
Vol 56 (1) ◽  
pp. 19-28 ◽  
Author(s):  
Sadhana Mishra
Keyword(s):  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Hermite Polynomials ◽  
Generalized Hypergeometric Functions ◽  
Generalized Hypergeometric Series ◽  
Fox’S H Function ◽  
Hermite Series

Zeros of generalized hypergeometric functions

1973 ◽  
Vol 74 (2) ◽  
pp. 269-276
Author(s):  
A. D'Adda ◽  
R. D'Auria
Keyword(s):  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Generalized Hypergeometric Functions ◽  
Generalized Hypergeometric Series ◽  
Image Position

In this paper we derive the conditions which have to be satisfied in order to obtain some classes of zeros of the generalized hypergeometric series of the typeThese conditions read:

scholarly journals An extension of Pochhammer’s symbol and its application to hypergeometric functions, II

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6505-6517
Author(s):  
Mohammad Masjed-Jamei ◽  
Gradimir Milovanovic
Keyword(s):  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Arbitrary Order ◽  
Beta Function ◽  
Pochhammer Symbol ◽  
Generalized Hypergeometric Functions ◽  
Generalized Hypergeometric Series ◽  
General Properties ◽  
Different Types ◽  
Additive Form

Recently we have introduced a productive form of gamma and beta functions and applied them for generalized hypergeometric series [Filomat, 31 (2017), 207-215]. In this paper, we define an additive form of gamma and beta functions and study some of their general properties in order to obtain a new extension of the Pochhammer symbol. We then apply the new symbol for introducing two different types of generalized hypergeometric functions. In other words, based on the defined additive beta function, we first introduce an extension of Gauss and confluent hypergeometric series and then, based on two additive types of the Pochhammer symbol, we introduce two extensions of generalized hypergeometric functions of any arbitrary order. The convergence of each series is studied separately and some illustrative examples are given in the sequel.

scholarly journals Operational Methods in the Study of Sobolev-Jacobi Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 124 ◽  
Author(s):  
Nicolas Behr ◽  
Giuseppe Dattoli ◽  
Gérard Duchamp ◽  
Silvia Penson
Keyword(s):  
Generating Functions ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Hermite Polynomials ◽  
Umbral Calculus ◽  
Generalized Hypergeometric Functions ◽  
Normal Ordering ◽  
Gamma Functions ◽  
Image Technique ◽  
Operational Methods

Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multi-variate version of umbral calculus and of the so-called umbral image technique. Besides providing a class of new formulae for generalized hypergeometric functions and an implementation of series manipulations for computing lacunary generating functions, our main application of these techniques is the study of Sobolev-Jacobi polynomials. Motivated by applications to theoretical chemistry, we moreover present a deep link between generalized normal-ordering techniques introduced by Gurappa and Panigrahi, two-variable Hermite polynomials and our integral-based series transforms. Notably, we thus calculate all K-tuple L-shifted lacunary exponential generating functions for a certain family of Sobolev-Jacobi (SJ) polynomials explicitly.

scholarly journals Generalized hypergeometric series for Racah matrices in rectangular representations

2018 ◽  
Vol 33 (04) ◽  
pp. 1850020 ◽  
Author(s):  
A. Morozov
Keyword(s):  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Mathematical Physics ◽  
Natural Question ◽  
Generalized Hypergeometric Series ◽  
Wilson Polynomials ◽  
The One ◽  
Quantum Dimensions ◽  
Skew Characters ◽  
Trivial Fact

One of the spectacular results in mathematical physics is the expression of Racah matrices for symmetric representations of the quantum group [Formula: see text] through the Askey–Wilson polynomials, associated with the [Formula: see text]-hypergeometric functions [Formula: see text]. Recently it was shown that this is in fact the general property of symmetric representations, valid for arbitrary [Formula: see text] — at least for exclusive Racah matrices [Formula: see text]. The natural question then is what substitutes the conventional [Formula: see text]-hypergeometric polynomials when representations are more general? New advances in the theory of matrices [Formula: see text], provided by the study of differential expansions of knot polynomials, suggest that these are multiple sums over Young sub-diagrams of the one which describes the original representation of [Formula: see text]. A less trivial fact is that the entries of the sum are not just the factorized combinations of quantum dimensions, as in the ordinary hypergeometric series, but involve non-factorized quantities, like the skew characters and their further generalizations — as well as associated additional summations with the Littlewood–Richardson weights.

scholarly journals An extension of Pochhammer’s symbol and its application to hypergeometric functions

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 207-215 ◽  
Author(s):  
Mohammad Masjed-Jamei ◽  
Gradimir Milovanovic
Keyword(s):  
Gamma Function ◽  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Special Property ◽  
Convergence Problem ◽  
Pochhammer Symbol ◽  
Generalized Hypergeometric Series ◽  
General Properties

By using a special property of the gamma function, we first define a productive form of gamma and beta functions and study some of their general properties in order to define a new extension of the Pochhammer symbol. We then apply this extended symbol for generalized hypergeometric series and study the convergence problem with some illustrative examples in this sense. Finally, we introduce two new extensions of Gauss and confluent hypergeometric series and obtain some of their general properties.

scholarly journals Certain Unified Integrals Associated with Product of M-Series and Incomplete H-functions

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1191 ◽  
Author(s):  
Manish Kumar Bansal ◽  
Devendra Kumar ◽  
Ilyas Khan ◽  
Jagdev Singh ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Special Cases ◽  
Fox’S H Function

In this paper, we established some interesting integrals associated with the product of M-series and incomplete H-functions, which are expressed in terms of incomplete H-functions. Next, we give some special cases by specializing the parameters of M-series and incomplete H-functions (for example, Fox’s H-Function, Incomplete Fox Wright functions, Fox Wright functions and Incomplete generalized hypergeometric functions) and also listed few known results. The results obtained in this work are general in nature and very useful in science, engineering and finance.

An integral involving the G-function and Kampé de Fériet function

1968 ◽  
Vol 64 (4) ◽  
pp. 1041-1044 ◽  
Author(s):  
O Shanker
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Special Functions ◽  
General Function ◽  
Generalized Hypergeometric Functions ◽  
General Integral ◽  
Infinite Integral ◽  
Function Of Two Variables ◽  
General Hypergeometric Function

The object of this paper is to evaluate an infinite integral involving the product of Meijer's G-function (5) and Kampé de Fériet function (1) in terms of Kampé de Fériet function. A number of papers of Bailey (3,4), Ragab (7,8), Slater (9), and Srivastava (10) have appeared, evaluating an integral in terms of a hypergeometric function of two variables or in terms of an E-function. Their results are obviously the particular cases of my result. Since Meijer's G-function is the most general function of one variable which can be expressed in terms of special functions (5) and Kampé de Fériet's function being the most general hypergeometric function of two variables, the integral given by me is the most general integral ever obtained and generalizes most of the results obtained so far for the integral of Mellin type in terms of generalized hypergeometric series. This is because the Kampé de Fériet function reduces to the product of two generalized hypergeometric functions by choosing parameters suitably.

scholarly journals General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1102
Author(s):  
Yashoverdhan Vyas ◽  
Hari M. Srivastava ◽  
Shivani Pathak ◽  
Kalpana Fatawat
Keyword(s):  
Hypergeometric Functions ◽  
Theory Theory ◽  
Binomial Theorem ◽  
Generalized Hypergeometric Functions ◽  
Quantum Calculus ◽  
The Third ◽  
Summation Formulas ◽  
Symmetric Quantum ◽  
Summation Theorem

This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of well-known proofs of the q-Kummer summation theorem with a key role of the q-binomial theorem. In addition to the q-binomial theorem, the first proof makes use of Thomae’s q-integral representation and the second proof needs Heine’s transformation. Whereas the third proof utilizes only the q-binomial theorem. Subsequently, the applications of these summation formulas in obtaining the general contiguous extensions of the second and the third q-Kummer summation theorems are also presented. Furthermore, the investigated results are specialized to give many of the known as well as presumably new q-summation theorems, which are contiguous to the three q-Kummer summation theorems. This work is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including Number Theory, Theory of Partitions and Combinatorial Analysis as well as in the study of Combinatorial Generating Functions. Just as it is known in the theory of the Gauss, Kummer (or confluent), Clausen and the generalized hypergeometric functions, the parameters in the corresponding basic or quantum (or q-) hypergeometric functions are symmetric in the sense that they remain invariant when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. A case has therefore been made for the symmetry possessed not only by hypergeometric functions and basic or quantum (or q-) hypergeometric functions, which are studied in this paper, but also by the symmetric quantum calculus itself.

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