scholarly journals Transformations of the Hypergeometric 4F3 with One Unit Shift: A Group Theoretic Study

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1966
Author(s):  
Dmitrii Karp ◽  
Elena Prilepkina
Keyword(s):  
Symmetric Group ◽  
Direct Product ◽  
Hypergeometric Functions ◽  
Integer Lattice ◽  
Transformation Groups ◽  
Group Property ◽  
Summation Formulas ◽  
Wolfram Mathematica ◽  
Contiguous Relations

We study the group of transformations of 4F3 hypergeometric functions evaluated at unity with one unit shift in parameters. We reveal the general form of this family of transformations and its group property. Next, we use explicitly known transformations to generate a subgroup whose structure is then thoroughly studied. Using some known results for 3F2 transformation groups, we show that this subgroup is isomorphic to the direct product of the symmetric group of degree 5 and 5-dimensional integer lattice. We investigate the relation between two-term 4F3 transformations from our group and three-term 3F2 transformations and present a method for computing the coefficients of the contiguous relations for 3F2 functions evaluated at unity. We further furnish a class of summation formulas associated with the elements of our group. In the appendix to this paper, we give a collection of Wolfram Mathematica® routines facilitating the group calculations.

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.

scholarly journals A New Identity for Generalized Hypergeometric Functions and Applications

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 12 ◽  
Author(s):  
Mohammad Masjed-Jamei ◽  
Wolfram Koepf
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Summation Formulas ◽  
Published Paper

We establish a new identity for generalized hypergeometric functions and apply it for first- and second-kind Gauss summation formulas to obtain some new summation formulas. The presented identity indeed extends some results of the recent published paper (Some summation theorems for generalized hypergeometric functions, Axioms, 7 (2018), Article 38).

On an extension of the generalized Pochhammer symbol and its applications to hypergeometric functions

2016 ◽  
Vol 09 (03) ◽  
pp. 1650064 ◽  
Author(s):  
Vivek Sahai ◽  
Ashish Verma
Keyword(s):  
Hypergeometric Function ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Generalized Hypergeometric Function ◽  
Pochhammer Symbol ◽  
Contiguous Relations

The main object of this paper is to present a generalization of the Pochhammer symbol. We present some contiguous relations of this generalized Pochhammer symbol and use it to give an extension of the generalized hypergeometric function [Formula: see text]. Finally, we present some properties and generating functions of this extended generalized hypergeometric function.

Fractional integral operators characterized by some new hypergeometric summation formulas

2017 ◽  
Vol 20 (2) ◽  
Author(s):  
Min-Jie Luo ◽  
Ravinder Krishna Raina
Keyword(s):  
Special Class ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Generalized Hypergeometric Functions ◽  
Fractional Integral Operators ◽  
Summation Formulas ◽  
Generalized Fractional Integral ◽  
Generalized Fractional Integral Operators

AbstractThe purpose of this paper is to study generalized fractional integral operators whose kernels involve a very special class of generalized hypergeometric functions

scholarly journals On Iteration of Bijective Functions with Discontinuities

2020 ◽  
Vol 34 (1) ◽  
pp. 51-72
Author(s):  
Harald Fripertinger
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Direct Product ◽  
Compact Interval ◽  
Iterative Root ◽  
Type Iii ◽  
First Case ◽  
Different Types ◽  
Increasing Mapping

AbstractWe present three different types of bijective functions f : I → I on a compact interval I with finitely many discontinuities where certain iterates of these functions will be continuous. All these examples are strongly related to permutations, in particular to derangements in the first case, and permutations with a certain number of successions (or small ascents) in the second case. All functions of type III form a direct product of a symmetric group with a wreath product. It will be shown that any iterative root F : J → J of the identity of order k on a compact interval J with finitely many discontinuities is conjugate to a function f of type III, i.e., F = φ−1 ∘ f ∘ φ where φ is a continuous, bijective, and increasing mapping between J and [0, n] for some integer n.

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