A new representation of the extended k‐gamma function with applications

Abstract

In this study, a new series representation of the extended k‐gamma function is investigated. A new representation expresses this function as an infinite sum of delta functions. Several researchers have examined this family of functions, but no study has been conducted dealing with the fractional kinetic equation. A new representation proves useful to solve the fractional kinetic equation involving an extended k‐gamma function. Particular cases involving the original gamma function are discussed as corollaries. Such an application of the gamma function is not possible by using known representations; nevertheless, using this representation, new fractional transform formulae are evaluated. A new representation is also worthwhile to compute new integrals of products of the extended k‐gamma functions, proving to be reliable with known identities. Several distributional properties of the extended k‐gamma function are also discussed using Fourier transform.

Unified Approach to Fractional Calculus Images Involving the Pathway Transform of Extendedk-Gamma Function and Applications

“…The interested reader is directed to [29][30][31][32][33] and associated references therein for a more extensive and exhaustive review of related work. In recent years, Tassaddiq [34] has investigated a distributional representation of the extended k-gamma function in terms of complex delta function as follows:…”

“…Hence, its action on a suitably chosen function over a specific domain is easily obtainable by means of the standard techniques applied to delta function, and it is found that [34] 2…”

“…For further information on such representations concerning other special functions, and for properties of delta functions, one may refer to [34][35][36][37][38][39][40][41][42][43][44][45].…”

“…More recently, it has been computed by Tassaddiq in [34] by using the distributional representation of the extended k-gamma function. Present motivation is to further explore and extend the results of [34] keeping in view their novelty and significance to crack the problems that remained unsolved for many years.…”

“…More recently, it has been computed by Tassaddiq in [34] by using the distributional representation of the extended k-gamma function. Present motivation is to further explore and extend the results of [34] keeping in view their novelty and significance to crack the problems that remained unsolved for many years. For instance, quite a few recent articles by Kiryakova ([46,47] and references therein) point out that the special functions are either generalized fractional calculus operators of the basic functions (or the generalized fractional calculus operators are defined as the action of a specific special function on a general class of functions and vice versa).…”