scholarly journals Euler Type Integrals and Integrals in Terms of Extended Beta Function

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Subuhi Khan ◽  
Mustafa Walid Al-Saad
Keyword(s):  
Hypergeometric Series ◽  
Beta Function ◽  
Generalized Hypergeometric Series ◽  
Special Polynomials ◽  
Extended Beta Function

We derive the evaluations of certain integrals of Euler type involving generalized hypergeometric series. Further, we establish a theorem on extended beta function, which provides evaluation of certain integrals in terms of extended beta function and certain special polynomials. The possibility of extending some of the derived results to multivariable case is also investigated.

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.

scholarly journals On the Fock Kernel for the Generalized Fock Space and Generalized Hypergeometric Series

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Jong-Do Park
Keyword(s):  
Hypergeometric Series ◽  
Reproducing Kernel ◽  
Fock Space ◽  
Error Function ◽  
Close Connection ◽  
Generalized Hypergeometric Series

In this paper, we compute the reproducing kernel B m , α z , w for the generalized Fock space F m , α 2 ℂ . The usual Fock space is the case when m = 2 . We express the reproducing kernel in terms of a suitable hypergeometric series   1 F q . In particular, we show that there is a close connection between B 4 , α z , w and the error function. We also obtain the closed forms of B m , α z , w when m = 1 , 2 / 3 , 1 / 2 . Finally, we also prove that B m , α z , z ~ e α z m z m − 2 as ∣ z ∣ ⟶ ∞ .

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