scholarly journals Certain Applications of Generalized Kummer’s Summation Formulas for 2F1

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1538
Author(s):  
Junesang Choi
Keyword(s):  
Hypergeometric Series ◽  
Gamma Functions ◽  
Generalized Hypergeometric Series ◽  
Finite Series ◽  
Summation Formulas

We present generalizations of three classical summation formulas 2F1 due to Kummer, which are able to be derived from six known summation formulas of those types. As certain simple particular cases of the summation formulas provided here, we give a number of interesting formulas for double-finite series involving quotients of Gamma functions. We also consider several other applications of these formulas. Certain symmetries occur often in mathematical formulae and identities, both explicitly and implicitly. As an example, as mentioned in Remark 1, evident symmetries are naturally implicated in the treatment of generalized hypergeometric series.

scholarly journals Reduction Formulas for Generalized Hypergeometric Series Associated with New Sequences and Applications

2021 ◽  
Vol 5 (4) ◽  
pp. 150
Author(s):  
Junesang Choi ◽  
Mohd Idris Qureshi ◽  
Aarif Hussain Bhat ◽  
Javid Majid
Keyword(s):  
Positive Integer ◽  
Hypergeometric Series ◽  
Stirling Numbers ◽  
Generalized Hypergeometric Series ◽  
Summation Formulas ◽  
Specific Parameter ◽  
Finite Sums ◽  
Potential Use

In this paper, by introducing two sequences of new numbers and their derivatives, which are closely related to the Stirling numbers of the first kind, and choosing to employ six known generalized Kummer’s summation formulas for 2F1(−1) and 2F1(1/2), we establish six classes of generalized summation formulas for p+2Fp+1 with arguments −1 and 1/2 for any positive integer p. Next, by differentiating both sides of six chosen formulas presented here with respect to a specific parameter, among numerous ones, we demonstrate six identities in connection with finite sums of 4F3(−1) and 4F3(1/2). Further, we choose to give simple particular identities of some formulas presented here. We conclude this paper by highlighting a potential use of the newly presented numbers and posing some problems.

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 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.

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