Fractional Calculus of Some “New” but Not New Special Functions: K-, Multi-index-, and S-Analogues.

This paper is a continuation, as Part 2, of the recent author’s papers [18], [19], [20], where we have emphasized on our general and unified approach to evaluate operators of fractional calculus of a very general class of special functions. Namely, we have results for images of the Wright generalized hypergeometric functions (W. g.h.f.-s) under the operators of classical and generalized fractional calculus. Thus, great part of results published by other authors in numerous papers (for part of them - see references in the above mentioned 3 papers and also, herein) come as immediate particular cases. It is because the special functions considered there are all of them Wright g.h.f.-s, and the FC operators like the Riemann-Liouville (RL), Erdélyi-Kober (EK), Saigo, Marichev-Saigo-Maeda (MSM), etc., are all of them particular cases of the operators of Generalized Fractional Calculus (GFC), [11]. We gave previously long list of illustrative examples for the efficiency of the general approach, and now continue it. Recently, some authors repeated the job to evaluate FC operators of some special functions which they introduced and considered as “new” ones. Among them are some examples of the so-called k-analogues of the Bessel and Mittag-Leffler functions, generalized multi-index Bessel and Mittag-Leffler functions, K- and M-series and S -functions. In Section 5 we show that all these are just cases, again, of the Wright generalized hypergeometric function. Then, the results provided by the mentioned authors can come easily from our general ones. [ABSTRACT FROM AUTHOR]