SOME PARAMETRIC AND ARGUMENT VARIATIONS OF THE OPERATORS OF FRACTIONAL CALCULUS AND RELATED SPECIAL FUNCTIONS AND INTEGRAL TRANSFORMATIONS

One of the main objects of this paper is to show that, in recent years, there is an on-going trend toward extensions and generalizations of known and readily accessible definitions and results by introducing some obviously redundant and seemingly inconsequential parameters or by changing the variable of integration in an integral definition. In particular, we investigate and closely examine the so-called k-gamma function and the corresponding k-Pochhammer symbol and k-Laplace transform, the pathway integral version and the conformable or non-conformable version as well as the so-called (k, 0 -extension of the operators of the traditional Riemanu-Liouville fractional calculus and such other familiar operators of fractional calculus as the Liouville-Caputo fractional derivative operator, the Sumudu transform and the P-delta-version of the classical Laplace transform, the so-called post-quantum or, briefly, the (p, q)-version of the familiar basic or quantum (or q-) analysis, the parametric variation of the Bessel and related functions, and so on. We also look into the current literature which is full of repeated or translational usages of the classical Laplace transform operator L in order to successfully solve initial-value problems involving ordinary and partial differential equations. Finally, in the concluding section, we present some recent developments and potential directions for further researches which can be based upon a certain general non-trivial family of the Riemann-Liouville type fractional integrals and fractional derivatives.