k-Hypergeometric Series Solutions to One Type of Non-Homogeneous k-Hypergeometric Equations

Li, Shengfeng; Dong, Yi

doi:10.3390/sym11020262

Cited by 13 publications

(20 citation statements)

References 31 publications

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“…We have spotted that by setting p = 1, the various outcomes presented in this article will reduce to some the corresponding outcomes derived earlier in [16,18,19,24,25]. Further, if we let k = 1, then we obtain several interesting new outcomes for the p-extended Gauss and Kummer hypergeometric functions.…”

Section: Discussionmentioning

confidence: 62%

“…The k-analogue of theta operator kΘ as given in [18,19,25], takes the form kΘ = k ξ dξ . This operator has the particularly pleasant property that kΘξ m = kmξ m , which makes it handy to be used on power series.…”

Section: Contiguous Function Relations and Differential Equationsmentioning

confidence: 99%

“…In particular, Diaz and Pariguan [16] introduced the k-analogue of gamma, beta and hypergeometric functions and proved a number of their properties. Since that period, many different results concerning the k-hypergeometric function and related functions have been considered by many researchers, for instance, Agarwal et al [17], Mubeen et al [18][19][20], Rahman et al [21], Chinra et al [22], Korkmaz-Duzgun and Erkus-Duman [23], Nisar et al [24], Li and Dong [25], Yilmaz et al [26] and Yilmazer and Ali [27].…”

Section: Introductionmentioning

confidence: 99%

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Some results on (p, k)-extension of the hypergeometric functions

Abdalla

Hidan

2021

Preprint

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In this study, we investigate a new natural extension of hypergeometric functions with the two parameters p and k which is so called (p, k)-extended hypergeometric functions”. In particular, we introduce the (p, k)-extended Gauss and Kummer (or confluent) hypergeometric functions. The basic properties of the (p, k)-extended Gauss and Kummer hypergeometric functions, including convergence properties, integral and derivative formulas, contiguous function relations and differential equations. Since the latter functions contain many of the familiar special functions as sub-cases, this extension is enriches theory of k-special functions.

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Section: Discussionmentioning

confidence: 62%

Section: Contiguous Function Relations and Differential Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Some results on (p, k)-extension of the hypergeometric functions

Abdalla

Hidan

2021

Preprint

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“…In [18], some families of multilinear and multilateral generating functions for the k-analogue of the hypergeometric functions were obtained. Studies on this subject are not limited to these papers; for details, see [19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative

2020

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Our present investigation is mainly based on the k-hypergeometric functions which are constructed by making use of the Pochhammer k-symbol Diaz et al. 2017, which are one of the vital generalizations of hypergeometric functions. We introduce k-analogues of F2and F3 Appell functions denoted by the symbols F2,kand F3,k,respectively, just like Mubeen et al. did for F1 in 2015. Meanwhile, we prove integral representations of the k-generalizations of F2and F3 which provide us with an opportunity to generalize widely used identities for Appell hypergeometric functions. In addition, we present some important transformation formulas and some reduction formulas which show close relation not only with k-Appell functions but also with k-hypergeometric functions. Finally, employing the theory of Riemann–Liouville k-fractional derivative from Rahman et al. 2020, and using the relations which we consider in this paper, we acquire linear and bilinear generating relations for k-analogue of hypergeometric functions and Appell functions.

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“…The solution in the form of the so-called k-hypergeometric series of k-hypergeometric differential equation by utilizing the Frobenius method can be found in the work of Mubeen et al [23,22]. Recently, Li and Dong [14] investigated the hypergeometric series solutions for the second-order non-homogeneous khypergeometric differential equation with the polynomial term. Rahman et al [27,21] proposed the generalization of Wright hypergeometric k-functions and derived its various basic properties.…”

Section: Introductionmentioning

confidence: 99%