Significant Behavior Of Generalized Fractional Operators With Bessel K-Function And Their Consequences

Abstract

In this paper, we discuss the generalized behavior of fractional operators with series type of special functions. The Saigo’s generalized k-fractional integral operators involving khypergeometric function as its kernel and its applications are also under discussion. Moreover, we resolve some results of generalized fractional integral operators with power function in terms of k are solved. Such type of results are represented in terms of generalized k-Wright hypergeometric functions. We also discuss the behavior of fractional operators with k-Bessel function and further in terms of generalized k-hypergeometric function. All the results are obtained have immense applications in the field of analysis of operator theory as well as special functions in mathematics. Keywords: generalized k-fractional integral operators; k-hypergeometric function; generalized k-Wright hypergeometric function; k-Bessel function of the first kind; generalized k-hypergeometric function 26A33; 33C05; 33C10; 33C20; 33C60; 26A09 2 Introduction Fractional calculus, which extends the classical notions of differentiation and integration to non-integer orders, has become an indispensable tool in modern mathematical analysis due to its capability of modeling memory and hereditary effects. Such properties are essential in the mathematical formulation of physical, engineering, and biological processes governed by nonlocal dynamics [8–10]. As a result, fractional integral and differential operators have been widely studied from both theoretical and applied perspectives. Among the classical fractional operators, the Riemann–Liouville fractional integral of order α > 0 is defined by (I α a+ f)(x) = 1