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 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 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.

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 Certain transformations for hypergeometric series in the p-adic setting

2015 ◽  
Vol 11 (02) ◽  
pp. 645-660 ◽  
Author(s):  
Rupam Barman ◽  
Neelam Saikia
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Gamma Function ◽  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Transformation Formulas ◽  
Trace Of Frobenius

In [The trace of Frobenius of elliptic curves and the p-adic gamma function, Pacific J. Math. 261(1) (2013) 219–236], McCarthy defined a function nGn[⋯] using the Teichmüller character of finite fields and quotients of the p-adic gamma function. This function extends hypergeometric functions over finite fields to the p-adic setting. In this paper, we give certain transformation formulas for the function nGn[⋯] which are not implied from the analogous hypergeometric functions over finite fields.

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.

Special values of the hypergeometric series II

2001 ◽  
Vol 131 (2) ◽  
pp. 309-319 ◽  
Author(s):  
I. J. ZUCKER ◽  
G. S. JOYCE
Keyword(s):  
Gamma Function ◽  
Hypergeometric Series ◽  
Elliptic Integral ◽  
Singular Values ◽  
Complete Elliptic Integral ◽  
Special Values ◽  
Gaussian Hypergeometric Series

Several authors [1, 5, 9] have investigated the algebraic and transcendental values of the Gaussian hypergeometric series(formula here)for rational parameters a, b, c and algebraic and rational values of z ∈ (0, 1). This led to several new identities such as(formula here)and(formula here)where Γ(x) denotes the gamma function. It was pointed out by the present authors [6] that these results, and others like it, could be derived simply by combining certain classical F transformation formulae with the singular values of the complete elliptic integral of the first kind K(k), where k denotes the modulus.Here, we pursue the methods used in [6] to produce further examples of the type (1·2) and (1·3). Thus, we find the following results:(formula here)The result (1·6) is of particular interest because the argument and value of the F function are both rational.

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