General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications

Yashoverdhan Vyas,Shivani Pathak,Kalpana Fatawat,Hari M Srivastava

doi:10.3390/sym13061102

Yashoverdhan Vyas, Shivani Pathak + Show 2 more

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https://doi.org/10.3390/sym13061102

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Abstract

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.

Highlights

The celebrated Gauss hypergeometric function 2 F1, the Kummer hypergeometric function 1 F1, the Clausen hypergeometric function 3 F2, and various other mathematical functions of hypergeometric type, are all contained in the generalized hypergeometric function r Fs, involving r numerator parameters a1, · · ·, ar and s denominator parameters b1, · · ·, bs, as special cases
This paper establishes a fact that the known techniques, which were used in [15,16,18] to derive the three q-Kummer summation theorems, can be applied in a straightforward manner to develop the general contiguous extensions of the q-Kummer summation theorems (6), (9) and (10)
In a forthcoming sequel to this article, we aim at investigating several general contiguous extensions of the q-analogues of

Summary

Φ1 aq q

The proof of the result stated in Equation (6) was based upon the specialization of parameters in Jackson’s summation of the well-poised 6 Φ5 [7] p. 356, Equation (II.). Andrews [15] derived the q-analogue of Kummer’s second summation theorem [6]. 361, Equation (III.21) with bases q and q2 , respectively This solitary transformation filled in a century-old gap in the theory of basic (or q-) series. Theorems 1 and 2 (see Section 2) of this paper provide two general contiguous extensions of the q-Kummer first summation theorem in the form of general summations for the following 2 Φ1 basic (or q-) hypergeometric series: q, −.

General Contiguous Extensions of the q-Kummer First Theorem

Special Cases and Additional q-Summations

Concluding Remarks and Observations