Going Next after “A Guide to Special Functions in Fractional Calculus”: A Discussion Survey

Abstract

In the survey Kiryakova: “A Guide to Special Functions in Fractional Calculus” (published in this same journal in 2021) we proposed an overview of this huge class of special functions, including the Fox H-functions, the Fox–Wright generalized hypergeometric functions pΨq and a large number of their representatives. Among these, the Mittag-Leffler-type functions are the most popular and frequently used in fractional calculus. Naturally, these also include all “Classical Special Functions” of the class of the Meijer’s G- and pFq-functions, orthogonal polynomials and many elementary functions. However, it so happened that almost simultaneously with the appearance of the Mittag-Leffler function, another “fractionalized” variant of the exponential function was introduced by Le Roy, and in recent years, several authors have extended this special function and mentioned its applications. Then, we introduced a general class of so-called (multi-index) Le Roy-type functions, and observed that they fall in an “Extended Class of SF of FC”. This includes the I-functions of Rathie and, in particular, the H¯-functions of Inayat-Hussain, studied also by Buschman and Srivastava and by other authors. These functions initially arose in the theory of the Feynman integrals in statistical physics, but also include some important special functions that are well known in math, like the polylogarithms, Riemann Zeta functions, some famous polynomials and number sequences, etc. The I- and H¯-functions are introduced by Mellin–Barnes-type integral representations involving multi-valued fractional order powers of Γ-functions with a lot of singularities that are branch points. Here, we present briefly some preliminaries on the theory of these functions, and then our ideas and results as to how the considered Le Roy-type functions can be presented in their terms. Next, we also introduce Gelfond–Leontiev generalized operators of differentiation and integration for which the Le Roy-type functions are eigenfunctions. As shown, these “generalized integrations” can be extended as kinds of generalized operators of fractional integration, and are also compositions of “Le Roy type” Erdélyi–Kober integrals. A close analogy appears with the Generalized Fractional Calculus with H- and G-kernel functions, thus leading the way to its further development. Since the theory of the I- and H¯-functions still needs clarification of some details, we consider this work as a “Discussion Survey” and also provide a list of open problems.

The Generalized Fox–Wright Function: The Laplace Transform, the Erdélyi–Kober Fractional Integral and Its Role in Fractional Calculus

“…Section 6 discusses in brief the role of the p Ψ qfunctions (in particular, of the multi-index Le Roy-type functions) as eigenfunctions of operators of new Fractional Calculus with Rathie I-functions as singular kernels, which we have introduced to extend our generalized fractional integrals involving Fox H-functions. Finally, in Section 7, we quickly comment on how our results work for the particular cases, quickly mention that some open problems are already stated in a previous survey [12], and discuss matters concerning numerical algorithms for classes of such special functions.…”

“…Namely, series (1) is absolutely convergent in C if µ > 0; it absolutely converges in the open disk |z| < R when µ = 0, but if µ < 0, the series converges only at zero. In our recent paper [12], we introduced a generalization of p Ψ q with arbitrary positive parameters A j and B i as in the "classical" case (1), but with additional "fractional" power parameters α j > 0 and β i > 0 for the Γ-function members in the numerator and denominator, respectively. Then, here, as new contributions, we study its analytical properties, its images under the Laplace transform and FC operators, and its role as an eigenfunction of some new operators of FC.…”

“…Definition 2 (Kiryakova, Paneva-Konovska [12]). The generalized Fox-Wright function is defined by the power series p Ψ q (a j , A j ; α j ) p j=1…”

“…Below, we skip the subtle details on the definitions, Mellin-Barnes-type contours and conditions for the existence and behavior of these rather general and not popular enough special functions, referring to the pioneering works by Rathie [13] and Inayat-Hussain [14], and to the comments in our previous survey [12]. We only recall the formal definitions.…”

“…According to [12] (Def. 6), the generalized Fox-Wright function ( 3) is representable in terms of the following Iand H-functions:…”