A Study of Q-Contiguous Function Relations

Harsh, Harsh Vardhan; Kim, Yong Sup; Rakha, Medhat A.; Rathie, Arjun K.

doi:10.4134/ckms.2016.31.1.065

Cited by 7 publications

(9 citation statements)

References 14 publications

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“…Theorem 1, for k = 1, 2, 3, 4 and 5, produces the results given in [19] pp. 892-83, Equations (6.100) to (6.104) , respectively.…”

mentioning

confidence: 64%

“…For example, in the context of q-Kummer's theorems (6), ( 9) and (10), Kim et al [18] re-derived the q-analogue of Kummer's first theorem (6) by using Thomae's q-integral representation given by Equation (13) and also obtained two particular contiguous results [18] p. 154, Equations (3.1) and (3.2) . In the same context, recently, Harsh et al [19] (pp. 81-92) derived some more particular contiguous extensions of the q-Kummer's first theorem (6), one-by-one from certain q-contiguous relations.…”

Section: Introduction Motivation and Preliminariesmentioning

confidence: 92%

“…In another paper, Harsh et al [20] (pp. [29][30][31] derived some particular contiguous extensions of the q-Kummer second summation theorem (9) and the q-Kummer third summation theorem (10) by implementing the summation formulas contiguous to the q-Kummer first theorem, derived in the previous paper [19] (pp. 81-92), along the lines of the derivations by Andrews [15].…”

Section: Introduction Motivation and Preliminariesmentioning

confidence: 99%

“…81-92), along the lines of the derivations by Andrews [15]. The particular contiguous extensions for the three q-Kummer summation theorems, investigated in [19,20], are very limited in numbers and are in scattered forms due to the one-by-one use of q-contiguous relations. In conclusion, as per our observations, the general contiguous extensions for the three q-Kummer summation theorems, which are presented in this paper in the form of Theorems 1 to 6 have not appeared previously in the literature.…”

Section: Introduction Motivation and Preliminariesmentioning

confidence: 99%

“…892-83, Equations (6.100) to (6.104) , respectively. For example, the case k = 1 of Theorem 1 is as shown below [19] p. 82, Equation (6.100) :…”

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confidence: 99%

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

et al. 2021

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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.

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“…Theorem 1, for k = 1, 2, 3, 4 and 5, produces the results given in [19] pp. 892-83, Equations (6.100) to (6.104) , respectively.…”

mentioning

confidence: 64%