Double Meijer transformations of certain hypergeometric functions

1968 ◽  
Vol 64 (2) ◽  
pp. 425-430 ◽  
Author(s):  
H. M. Srivastava ◽  
J. P. Singhal
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Image Position ◽  
Usual Notation

Following the usual notation for generalized hypergeometric functions we let(a) denotes the sequence of A parametersthat is, there are A of the a parameters and B of the b parameters. Thus ((a))m has the interpretationwith a similar interpretation for ((b))m; Δ(k; α) stands for the set of k parametersand for the sake of brevity, the pair of parameters like α + β, α − β will be written as α ± β, the gamma product Γ(α + β) Γ(α − β) as Γ(α ± β), and so on.

The integration of generalized hypergeometric functions

1966 ◽  
Vol 62 (4) ◽  
pp. 761-764 ◽  
Author(s):  
H. M. Srivastava
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Image Position ◽  
Usual Notation

In the usual notation for generalized hypergeometric functions we letwhereand (a) denotes the sequence of parametersThroughout the present paper we shall suppose that there are A of the a parameters, Bof the b parameters, and so on. Thus ((a))m is to be interpreted asand similar interpretations hold for ((b))m, etc.

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:

The integration of generalized hypergeometric functions

1969 ◽  
Vol 65 (3) ◽  
pp. 591-595 ◽  
Author(s):  
G. E. Barr
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Generalized Hypergeometric Function ◽  
Pochhammer Symbol ◽  
Generalized Hypergeometric Functions ◽  
Image Position

Let the generalized hypergeometric function of one variable be denoted bywhere (a)m is the Pochhammer symbol ((1, 3)).

Expansions of Generalized Hypergeometric Functions in Series of Products of Generalized Whittaker Functions

1962 ◽  
Vol 58 (2) ◽  
pp. 239-243 ◽  
Author(s):  
F. M. Ragab
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Whittaker Function ◽  
Generalized Hypergeometric Function ◽  
Whittaker Functions ◽  
Generalized Hypergeometric Functions ◽  
In Series ◽  
Image Position

In a previous paper (l) in this journal L. J. Slater gave expansions of the generalized Whittaker functions . She gave this name to the generalized hypergeometric function in the sense that it is a generalization of the well-known Whittaker function . In this paper series of products of generalized Whittaker functions will be evaluated in terms of such functions or in terms of generalized hypergeometric functions pFp(x). These expansions are These formulae will be proved in § 2 and particular cases will be given in § 3.

Some expansions in products of hypergeometric functions

1966 ◽  
Vol 62 (2) ◽  
pp. 245-247 ◽  
Author(s):  
H. M. Srivastava
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Generalized Hypergeometric Function ◽  
Image Position ◽  
Usual Notation

In the usual notation letwhere .Also, for the generalized hypergeometric function pFq(x) let us employ a contracted notation and writeThroughout the present paper i will run from 1 to p, I from 1 to P, and so on. Thus ((a) )m is to be interpreted as,and similar interpretations hold for ((A))m, etc.

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.

Clebsh–Gordan coefficients for the algebra 𝔤𝔩₃ and hypergeometric functions

2021 ◽  
Vol 33 (1) ◽  
pp. 1-22
Author(s):  
D. Artamonov
Keyword(s):  
Explicit Formula ◽  
Lie Algebra ◽  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Content Type

The Clebsh–Gordan coefficients for the Lie algebra g l 3 \mathfrak {gl}_3 in the Gelfand–Tsetlin base are calculated. In contrast to previous papers, the result is given as an explicit formula. To obtain the result, a realization of a representation in the space of functions on the group G L 3 GL_3 is used. The keystone fact that allows one to carry the calculation of Clebsh–Gordan coefficients is the theorem that says that functions corresponding to the Gelfand–Tsetlin base vectors can be expressed in terms of generalized hypergeometric functions.

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