Uniﬁed Approach to Fractional Calculus Images of Special Functions—A Survey

: Evaluation of images of special functions under operators of fractional calculus has become a hot topic with hundreds of recently published papers. These are growing daily and we are able to comment here only on a few of them, including also some of the latest of 2019–2020, just for the purpose of illustrating our uniﬁed approach. Many authors are producing a ﬂood of results for various operators of fractional order integration and differentiation and their generalizations of different special (and elementary) functions. This effect is natural because there are great varieties of special functions, respectively, of operators of (classical and generalized) fractional calculus, and thus, their combinations amount to a large number. As examples, we mentioned only two such operators from thousands of results found by a Google search. Most of the mentioned works use the same formal and standard procedures. Furthermore, in such results, often the originals and the images are special functions of different kinds, or the images are not recognized as known special functions, and thus are not easy to use. In this survey we present a uniﬁed approach to fulﬁll the mentioned task at once in a general setting and in a well visible form: for the operators of generalized fractional calculus (including also the classical operators of fractional calculus); and for all generalized hypergeometric functions such as p Ψ q and p F q , Fox H - and Meijer G -functions, thus incorporating wide classes of special functions. In this way, a great part of the results in the mentioned publications are well predicted and appear as very special cases of ours. The proposed general scheme is based on a few basic classical results (from the Bateman Project and works by Askey, Lavoie–Osler–Tremblay, etc.) combined with ideas and developments from more than 30 years of author’s research, and reﬂected in the cited recent works. The main idea is as follows: From one side, the operators considered by other authors are cases of generalized fractional calculus and so, are shown to be ( m -times) compositions of weighted Riemann–Lioville, i.e., Erdélyi–Kober operators. On the other side, from each generalized hypergeometric function p Ψ q or p F q ( p ≤ q or p = q + 1) we can reach, from the ﬁnal number of applications of such operators, one of the simplest cases where the classical results are known, for example: to 0 F q − p (hyper-Bessel functions, in particular trigonometric functions of order ( q − p ) ), 0 F 0 (exponential function), or 1 F 0 (beta-distribution of form ( 1 − z ) α z β ) . The ﬁnal result, written explicitly, is that any GFC operator (of multiplicity m ≥ 1) transforms a generalized hypergeometric function into the same kind of special function with indices p and q increased by m .


Introduction
Special functions (SF) have always been unavoidable tools for mathematicians, physicists, astronomers, applied scientists and engineers while looking to express and study (theoretically, in tables or by numerical algorithms) the solutions of treated mathematical models. On the other side, recently there has been an increased interest in fractional calculus (FC) and its applications, as evidence for which we refer the readers to the data in the survey by . Fractional calculus is nowadays a favorite, and even a sort of fashionable research area, although the boom of publications and attempts to "fractalize" any kinds of integer order models can bring some threats to the prestige of this discipline, especially in cases of weak or wrong results and not adequate innovations. Let us mention also the phenomenon of hundreds of papers of the last few years (only a few of them can be cited here) dealing with "evaluation of FC images of SF", most of which use the same standard techniques with changing only the particular special function (SF) and the particular case of the FC operator. Furthermore, it often happens that in such results the originals and the images are special functions of different kinds, or the images are not recognized as known special functions, and thus are not easy to use. In recent papers, such as [2][3][4][5], we share our criticism on this practice and show that all such results can be derived at once by following a general approach, based on ideas from older author's works on generalized fractional calculus (GFC), since [6].
Here we try to collect the ideas, results and examples from our recent works on the subject. The survey starts with Preliminaries (Section 2) providing a short background on the considered SF and FC operators; followed by Section 3 with results for images of the generalized hypergeometric functions p F q and p Ψ q and their simpler cases under the operators the classical FC operators (Riemann-Liouville and Erdélyi-Kober integrals and derivatives of fractional order). Then, in Section 4 we present our unified approach for evaluation of GFC operators of arbitrary generalized hypergeometric functions ( p F q and p Ψ q ), resulting in the main Theorems 3 and 4. This allows to handle very wide classes of operators of generalized (m-tuple, m ≥ 1) fractional integration and differentiation and of considered special functions. In Sections 5-7 we consider specifications of these results for the Erdélyi-Kober, Saigo and Marichev-Saigo-Maeda (M-S-M) operators, that appear as cases of our GFC, resp. for m = 1, m = 2, m = 3, give their images for the Wright generalized hypergeometric functions, and many illustrative examples for particular results by other authors. Section 8 considers more general cases of GFC operators with arbitrary multiplicity m ≥ 1, as the multiple Gel'fond-Leontiev operators related to the multi-index Mittag-Leffler functions, and the hyper-Bessel operators related to the hyper-Bessel functions of Delerue. In Section 9 we comment on works of other authors on introducing some "new" special functions and show that these are again Wright generalized hypergeometric functions p Ψ q . Therefore, the various FC images they propose come as simple corollaries of our general results. To show the effectiveness of the proposed unified approach, in this survey we collected some 21 examples for FC images of SF, and referred to a long list of other authors' works on the subject. Section 10 summarizes some conclusions.

Preliminaries
Here we provide a short and only necessary background on the considered classes of special functions (SF) and of operators of classical FC and of generalized fractional calculus (GFC), so as to explain the general ideas. All details on defining the single-valued branches of the considered functions, functional spaces, and necessary conditions on appearing parameters, can be found in our previous works, as cited, and for example in ( [6], Section 5.5.i). Basically, we consider functions in the complex plane of the form f (z) = z µ f (z), µ ≥ 0, f (z) analytic and single valued in Ω , where Ω is a starlike domain with respect to z = 0, usually a disk ∆ R : |z| < R. Most of the considered special functions are entire functions, or analytic ones in disks in C.
The results we consider are for the classes of so-called generalized hypergeometric functions (g.h.f) with Mellin-Barnes type integral representations, namely the Fox H-function, Meijer G-function and their most widely used cases of Wright g.h.f. p Ψ q and g.h.f. p F q . Even if our aim is to incorporate as large as possible classes of special functions, let us mention that other transcendental functions as the elliptic integrals, Lambert W-, Mathiew-, Zeta-, etc. functions are outside of our studies. Also, we emphasize on results for LHS integrals, although for the RHS ones similar techniques and results are applied; and consider Riemann-Liouville type fractional derivatives. For the Caputo-type differentiation operators, similar but different results will be exposed in a separate work.

Special Functions of Fractional Calculus
Under "classical" Special Functions (SF) we mean these "mathematical functions" and orthogonal polynomials of which the origin goes back to 18th and 19th centuries and are named after great mathematicians like Euler, Gauss, Riemann, Bessel, Kummer, Legendre, Laguerre. These "Special Functions of Mathematical Physics" appeared with the needs of applied sciences and serve as solutions of integer order (most commonly 2nd order) differential equations from models in mathematical physics. In the last two centuries it was observed that modeling of many phenomena of the physical and social world can be reflected much more adequately by means of differential equations of arbitrary fractional or higher integer orders, and the so-called special functions of fractional calculus (SF of FC) as providing tools for their explicit solutions became unavoidable tools in the hands of theoretical and applied scientists recognizing the power of fractional calculus (FC).
Recently, many handbooks and surveys appeared as dedicated not only to classical SF but also to the SF of FC, to mention some of them: Prudnikov-Brychkov-Marichev [7], Marichev [8], Srivastava-Gupta-Goyal [9], Kilbas-Srivastava-Trujillo [10], Podlubny [11], Kiryakova [6], Yakubovich-Luchko [12], Mathai-Haubold [13], Gorenflo-Kilbas-Mainardi-Rogosin [14]. Such a list cannot be full here, and for more sources see also the survey paper . In the papers on the topic and in this survey, we limit ourselves to the Fox H-functions of one complex variable, as enough of a general level to expose the proposed approach.
The details on the properties of the Fox H-function can be found in many contemporary handbooks on SF such as [7,9,10], where its behavior is described in term of the denotations: Note that the H-function is an analytic function of z in circle domains |z| < ρ or outside them (or in sectors of them, or in the whole C), depending on the above parameters and the contours.
If all A 1 = · · · = A p = 1, B 1 = · · · = B q = 1, the Wright g.h.f. reduces to the generalized hypergeometric p F q -function, which is a case of the G-function (3), see details in ( [15], Vol.1): The Mittag-Leffler (M-L) function, introduced by G. Mittag-Leffler (1902-1905, with extended 2-parameters' definition by R.P. Agarwal (1953), was presented yet in Bateman Project's [15], Vol. 3 (1954), in a chapter for "Miscellaneous Functions". However, it was ignored for a long time in the books on special functions because the applied scientists suffered from a lack of tables for its Laplace transforms. Although appearing from studies not related to fractional calculus, nowadays the M-L function has become the most popular and most exploited SF of FC, honored to be the "Queen"-function of FC. See details, for example, in [14], also in [6,17,18]. Definition 3. The Mittag-Leffler (M-L) functions E α and E α,β , are entire functions of order ρ = 1/α and type 1, defined by the power series , α > 0, β > 0.
As "fractional index" (α > 0) analogs of the exponential and trigonometric functions that satisfy ODEs of 1st and 2nd order (α = 1, 2), the M-L functions serve as solutions of fractional order differential equations. A M-L type function with three indices, known as the Prabhakar function (T.R. Prabhakar, 1971) is also often studied and used, for details see [14,[17][18][19], and other contemporary books and surveys on M-L type functions: where (γ) 0 = 1, (γ) k = Γ(γ + k)/Γ(γ) denotes the Pochhammer symbol. For γ = 1 we get the M-L function E α,β , and if additionally β = 1, it is E α . These M-L type functions are simple cases of the Wright g.h.f. and of the H-function, namely: .
In another survey paper, Kiryakova [27], we are exposing many other details on the theory of the SF of FC, in the sense of Wright generalized hypergeometric functions p Ψ q and multi-index Mittag-Leffler functions, and provide an extensive list of their particular cases, studied in theoretical and applicable aspects by various authors. is one of the main tools to evaluate integrals and various integral transforms of special functions, including their images under operators of FC. After some classical publications of previous centuries, the main contribution to this approach is due to Marichev [8]. He proposed a natural but wide ranged scheme, based on the contour integral representations of Mellin-Barnes type for the H-and G-functions, like (1) and (3). Note that the integrands H m,n p,q and G m,n p,q are their Mellin transforms (of variable s → −s) are fractions of products of 2 × 2 groups of Gamma-functions, and each special function being a special case of the generalized hypergeometric functions, has a particular representation of that kind. For example ([10], (1.11.24)): For variations of results, one can use in addition the relations (see, e.g., in ( [28], (2.6)-(2.8))): Examples for the use of the Mellin transform in this respect are given (among many others works) in: Luchko and Kiryakova ([28], Section 4) (general scheme and examples with the M-L and Wright functions), Agarwal, Rogosin, and Trujillo [29] and Paneva-Konovska and Kiryakova [30] (images for multi-index M-L functions and their particular cases).

Operators of Generalized Fractional Calculus
In fractional calculus (FC), meant as a theory of the integration and differentiation of arbitrary (including fractional, not obligatorily integer) order, there are several almost equivalent definitions for "fractional" integrals and derivatives, applied in various functional spaces. Here we are interested in evaluating FC operator images of special functions, defined by power series, most of which are entire functions, or at least analytic ones inside/outside disks in a complex plane. Therefore, we restrict our statements to such functions, although they hold also for spaces of weighted continuous or Lebesgue integrable functions on the real half-line.
We state results for the Riemann-Liouville (R-L) operator for integration R δ of order δ > 0, the corresponding R-L fractional derivative D δ , and its counterpart * D δ in the Caputo sense, that is only the left-hand sided operators of FC (and skip similar details for the Weyl-type, right-hand sided operators).
The main operator of fractional integration we consider is the Erdélyi-Kober operator (E-K) of integration of order δ > 0, depending on two additional parameters γ ∈ R and β > 0, note it is the identity for δ = 0. Especially for functional spaces of weighted analytic functions of the form f (z) = z µ f (z), µ ≥ 0 (see beginning of Section 2), to be preserved by this operator, we require γ > −1 − µ β , in addition to δ ≥ 0, β > 0. This operator, more general than the R-L integral, and having many more applications, was introduced in Sneddon's works, such as [33], and is considered in books ( [6,10,31], Ch.2), and recently in many other works on fractional order models. The Erdélyi-Kober-type fractional integrals, or briefly Erdélyi-Kober integrals, of the form are basic in our studies, and are called classical fractional integrals, and we consider their commutable compositions that are presented as our generalized fractional integrals, Kiryakova [6,34]. The Erdélyi-Kober operator (13) reduces to the R-L operator of integration for γ = 0, β = 1, δ 0 = δ, Note that some authors often refer to the Erdélyi-Kober integral (12) as Euler integral transformation, when they are to handle various integral transforms of special functions.
The fractional order derivative of R-L type corresponding to the E-K integral (12), called E-K fractional derivative D Instead of (d/dz) n , a suitably chosen auxiliary differential operator D n of integer order is used, a polynomial of the Euler differential operator (z d/dz). It has been introduced and studied in the works of Kiryakova and Luchko et al., ([6], Ch.2) and ( [12], Ch.3) and in the next ones, as [35], The more formal representation ( [6], Ch.1, Equation (1.6.7)) serves to provide a better understanding on the structure and nature of (15). The Caputo-type analogs of the R-L and E-K fractional derivatives are defined in the same way but with exchanged order of the nonnegative order integration and the integer order differentiation, see, for example, [36], also [35], namely, The notion of generalized fractional integration operators was introduced by S. Kalla (1969Kalla ( -1979, who suggested the common form of such operators (see details and references in [37]), where Φ(σ) is an arbitrary continuous (analytical) function for which the integral makes sense, most commonly a special function as the Bessel, Gauss, Gor H-function. The operators of such generalized fractional calculus (GFC) are expected to include, in particular, these of the classical FC and should satisfy the main axioms for the FC theory. Note that for a rather general or rather narrow choice of the special function Φ, only some formal operational rules for the generalized fractional integrals (17) can be provided. Therefore, in our generalized fractional calculus (GFC), Kiryakova [6], the suitable choice of the kernel-functions Φ as G m,0 m,m -and H m,0 m,m -functions was crucial. In that case, the generalized fractional integrals can be decomposed into commutative products of operators of classical FC (Erdélyi-Kober operators). Thus, the tools of the special functions and the wide usage of the classical FC are combined into a GFC with developed detailed theory and many established applications.
The following decomposition property is proved in [6], etc. (see, e.g., decomposition Th.5.2.1 in [6]). It is important because the GFC integrals (18) and (19) can be represented not only by using the kernel Fox Hand G-functions, but also by means of the repeated integral representations for the commutative product of classical E-K operators (12): In the book Kiryakova [6] and subsequent papers, we have provided the operational properties of the operators (18) and (19) as semigroup property, formal inversion formula, reduction to identity or to the conventional integration operators for special parameters' choice. This is to justify their names as operators of GFC.
Following the idea of how the R-L and E-K fractional derivatives are defined, we have proposed the definition of the corresponding generalized fractional derivatives. To this end, the auxiliary Definition 6 (Kiryakova ([6], Ch.1,Ch.5), [34,35]). The multiple (m-tuple) Erdélyi-Kober fractional derivative of R-L type of multi-order δ = (δ 1 ≥ 0, . . . , δ m ≥ 0) is defined by means of the differ-integral operator: Analogously, the Caputo-type generalized fractional derivative has been introduced in Kiryakova and Luchko [35], as * D For all equal β's: β 1 = ... = β m = β > 0, the R-L and Caputo-type "derivatives" corresponding to the generalized fractional integral (19) has a simpler form with Meijer G-function in the kernel: Under generalized (multiple, multi-order) fractional derivatives of the R-L type, resp. of the Caputo type, we have in mind all the differ-integral/integro-differential operators of the form A basic formula for the image of a power function in the general case of (18) and (19) (say from Kiryakova [6]) reads as and a similar one holds for the generalized fractional derivatives, both analogous to the same formulas for the classical Erdélyi-Kober operators. These results are in the base of the standard techniques applied by other authors for the evaluation of FC operators of special functions, in various particular cases. Using (27), in particular for p = k = 0, 1, 2, ..., n, ..., then interchanging the integration and summation of the power series for a particular special function, the authors of mentioned papers obtain a new power series to be recognized as another special function (in the successful cases, or the result is useless, left just as such series or as some p Ψ q -function). Our general result states as follows.
Theorem 1 (Kiryakova, since 1988, see, e.g., ( [6,38], Ch.5)). Let the conditions .., m, be satisfied for the parameters of the multiple E-K integral (18). Then, it preserves the class of weighted analytic Namely, the images of such functions have the same form: with the same radius of convergence R > 0 and the same signs of the coefficients in their series expansions.

Some Special Cases of GFC Operators
We emphasize here only some operators of FC that are recently exploited very often in publications on FC operators of SF. In [6] and the author's other papers as well as in works by other authors, there many other particular cases of linear integral and differential operators provided and used with applications in geometric (univalent) function theory, in differential and integral equations of integer and fractional order, operational calculus, transmutation theory, special functions theory, mathematical models of phenomena of fractional order, etc.
For m = 1 the kernel-functions of the generalized fractional integrals and derivatives (18) and (23) can be represented as therefore we have the E-K and R-L (γ = 0, β = 1) operators of classical FC. Many other integration and differentiation operators introduced and used by different authors appear as special cases of I γ,δ β , R δ and D δ . When m = 2, the kernels H 2,0 2,2 and G 2,0 2,2 reduce to a Gauss hypergeometric lfunction: In this case, the generalized fractional integrals are known as hypergeometric fractional integrals, and some of them are introduced and studied by Love, Saxena, Kalla, Saigo (see in next Section 6), Hohlov, etc.
Let m ≥ 1 be an arbitrary integer, but all δ's are integers, say δ 1 = ... = δ m = 1. Then we have the Bessel type integral and differential operators of arbitrary (higher) integer order, introduced by Dimovski [41] (see also [42]) and named as hyper-Bessel operators by Kiryakova ([6], Ch.3), as shown related to the hyper-Bessel functions of Delerue [43] as their eigenfunctions (see Example 16, in next Section 8). The studies on these operators gave rise to our GFC, since they appeared as "fractional" integrals and derivatives of integer multi-orders (1, 1, ..., 1) and for λ > 0 their fractional powers have multi-orders (λ, λ, ..., λ). In Section 8, we will discuss also the Gelfond-Leontiev operators generated by the multi-index M-L functions, as more general operators of arbitrary multiplicity m > 1 and arbitrary fractional multi-order.
As mentioned, here we stress on only a few particular examples of GFC operators I ,m , that are involved in results on to the topic of this survey. This is because many other authors' works handle the evaluation of images of various elementary or special functions under the classical or some "generalized operators of FC"-such as the operators of R-L, E-K, Saigo, Marichev-Saigo-Maeda. Say, one takes first the cosine or Bessel function, later the generalized Bessel (Bessel-Maitland) function, then an M-L or generalized M-L function, etc., so as to produce new publications by same standard techniques. Very rarely observed, or mostly is ignored, the fact from relation (21) that these are 2-tuple (m = 2), respectively 3-tuple (m = 3), or m-tuple (arbitrary m > 1) compositions of Erdélyi-Kober operators. Therefore, the task can be done at once, if one knows how an E-K operator acts on such special functions, all being cases of Wright g.h.f. (4), and then applying the procedure a suitable number of times (2-, 3-, or m). Thus, the result can be predicted in advance, having in mind the general statements in the next sections.

Erdélyi-Kober and Riemann-Liouville Images of p Ψ q , p F q and Simpler Special Functions
Some basic classical results on the topic exist from the previous century that should not be forgotten and on which our approach was built. Namely, the image of a generalized hypergeometric function p F q , with p ≤ q + 1, under the R-L fractional integral/ derivative is shown to be the same special function with indices p and q increased by 1: with Re δ > 0, Re ν > 0, p ≤ q + 1; λ = 0, z ∈ C and if p = q + 1: |λz| < 1 is additionally required. For this, we can refer to Erdélyi Note that (33) is an interpretation of the Poisson integral formula for the Bessel function, that has been generalized in [42] and ( [6], Ch.4) to represent the hyper-Bessel functions J (m) ν 1 ,...,ν m with multi-indices (ν 1 , ..., ν m ) (the Bessel function J ν is the case for m = q− p = 1 with one index ν), that is, to represent the 0 F q−p -functions by means of "generalized cosine" cos m .
R-L integrals/ derivatives of the most general Gand H-functions are also well known in the literature (for example, from [44]), and these are the same type of functions but with increased orders and additional parameters.
Along with the mentioned old classical results, recently, new articles are published on the evaluation of classical (R-L, E-K) or generalized FC operators of classical SF or of SF of FC almost every day (e.g., in 2020: [47,48]), and also of their multivariate or matrix variants. Just as one example on fractional operators for the matrix Wright hypergeometric functions (5), is a 2020 paper [49].
The classical results (32) and (33)- (35) have been extended in our works (as in ( [6], Ch.4), [22,24,50]) in terms of the Erdélyi-Kober operators (12) and (15) and for their counterparts of the GFC: I (β k ),m , not only for p F q but for p Ψ q as well. To reduce the Wright g.h.f. p Ψ q in the general case to three basic simplest functions with lowest indices p and q, we also apply modifications as the Wright-Erdélyi-Kober multiple operators with a Bessel-Maitland kernel-function and in general, GFC operators with H-functions like in (20) but with different parameters 1/β k > 0 and 1/λ k > 0 in the upper and low row. Details are in Kiryakova [24]. Now we provide some basic statements necessary for the topic of this survey, repeating in a few lines the ideas of the proofs, so as to clarify the approach used. Lemma 1. The image of a Wright g.h.f. p Ψ q under the Erdélyi-Kober fractional integral (12) is the same type of function in which the indices p-and q are increased by one, and so, has two additional parameters: It is supposed that Re δ > 0, Re γ > −1, µ > 0, λ = 0, c is arbitrary (real), and if p = q + 1, then |λz µ | < 1 is additionally required.
Proof. In a simpler case with c = 0, this is Lemma 1 from Kiryakova [2]. There, a proof is based on the Formula (44) (Section 4) for the integral (the Mellin transform) of a product of two H-functions, since both the p Ψ q -function and the kernel of the E-K operator are cases of H-functions; compare (5) and Section 2.3. This approach will be discussed later for the more general case of GFC operators.
Proof. For c = 0 this is Lemma 3 in Kiryakova [2], and for the case of γ = 0, β = 1 we have the formula for the R-L fractional derivative from Kilbas ([16], Th.4) (where ν−1 := c, Re c > −1, and the same other conditions): (40) The same standard term-by-term integration/differentiation technique can be used for the proof of (39).
Here we demonstrate a new proof of Lemma 2 for the more general case of Wright function p Ψ q . For simplicity, β = 1 and µ = 1. Interpreting the E-K derivative (15) as in (16), we have subsequently: The L.H.S. of (41) λz .
Now, we use the representation (5) of the Wright g.h.f. as an H-function, and may apply a formula for differentiation of integer order n of the H-function, say Equation (1.69) from Mathai-Saxena-Haubold [51], to continue as follows: and because of the coincidence of the terms (−γ − n − c, 1) in upper and low parameters' rows, according to the reduction order formula for the H-function: (1.56) in [51], see also (E.8) in [6], and [7,9]), we have , which, by using again (5) to go back to a Wright g.h.f., gives the result (39). In case β = 1, substitution z → z 1/β is necessary, and same for µ = 1.
Yet another approach to check the validity of (39) is to use the identity D B 1 ), ..., (b q , B q ) λz µ , the result from Lemma 1 and reduction of the intermediate result p+2 Ψ q+2 to a p+1 Ψ q+1 -function, since the last two equal parameters in the upper and low rows of its series eliminate each other.

Remark 2.
The corresponding result for the Caputo-type E-K derivative * D γ,δ β for images of the Wright p Ψ q -functions, and in particular also for the p F q -functions and for the simpler case of operators with β 1 = ... = β m = β > 0, will be presented in another separate work.

Results for the Generalized Fractional Calculus Operators of Special Functions
Here we present our results on evaluating operators of generalized fractional calculus (in the sense of [6] and of Riemann-Liouville type) of wide classes of special functions as the Wright generalized hypergeometric functions p Ψ q and even of the Fox H-functions (thus incorporating the SF of FC) and in particular, of the p F q -and Meijer G-functions (thus having general results also for the "classical" SF).
. (43) Note that three of the orders of the H-function are increased by the multiplicity m, and additional m+m parameters appear depending on those of the operator.
Corollary 1 (Lemma 1.2.2, [6])). The I (γ k ),(δ k ) β,m -image of a G-function is also a G-function in which the three orders are increased by the multiplicity m and has additional m + m parameters depending on those of the GFC operator: Proof. In this case one can use a formula for the integral of product of two arbitrary G-functions, simpler than (44) (see for example, ([7], Section 2.24, (1)]) and with a proof in ( [6], App., (A.29))), and be reminded again that the G m,0 m,m -function vanishes outside the unit disc. Thus the integral (45) gives the required image G-function. Because the G s,t u,v -function is from the space of an analytic function in a disk centered at the origin, and has the following asymptotic behavior the conditions for the used formula to hold on are satisfied.
Formulas (43) and (45) can be used to evaluate practically all (classical and generalized) operators of FC of arbitrary SF (which are representable either as a Gor as a more general H-function).
Now we present the main result on the topic of this survey paper, which comes from Kiryakova [2], Th.1.

Theorem 3.
Assume that the conditions δ k ≥ 0, γ k > −1, β k > 0, k = 1, ..., m and µ > 0, λ = 0 hold. The image of a Wright g.h.f. p Ψ q (z) by a generalized fractional integral (20) (multiple, m-tuple Erdélyi-Kober integral) is another Wright g.h.f. with indices p and q increased by the multiplicity m and with additional parameters coming from those of the GFC integral: Proof. Here we briefly repeat the proof from Kiryakova [2], Th.1, in order to exhibit the main ideas on which this survey paper is based.
As one approach to prove (46), the general integral formula (44) can be used. This theorem can be seen also as a consequence of Theorem 2. It is because the kernel-function of the operator is a H m,0 m,m -function and the p Ψ q -function is a H 1,p p,q+1 -function, see (5). Then, according to (45) the result will be a H 1,p+m p+m,q+1+m -function that should be recognized as a p+1 Ψ q+1 -function, because it is reduced to a H 1,p+1 p+1,q+2 -function in view of the coincidence of (m−1) parameters in the upper and low row (use "reduction order" property of the H-function, [7], Section 8.3, 6.; [6], App. (E.8), etc.).
However, to clarify our main idea it is more instructive to refer to the decomposition property (21) presenting the generalized fractional integral (18) as a product of commuting (classical) Erdélyi-Kober operators. In the simplest case, we use subsequently m-times (36) from Lemma 1, to get: To derive the general relation (46) we apply to the above result the property for "generalized commutation" from Kiryakova ([6], Ch.5, (5.1.28)), namely: (1,...,1),m , (19) with (for simplicity) all β k = β = 1, k = 1, ..., m, is another g.h.f. of the same kind with indices increased by the multiplicity m: The above results (46) and (49) can be interpreted alternatively as the assertions stated in our earlier works ( [6,50], Ch.4) (in the simpler case of Corollary 2), and later in [24] (in more general case of Theorem 3). That is, a p+m Ψ q+m -function (resp. a p+m F q+m -function) of the form below can be represented by means of a multiple (m-tuple) operator of GFC of a p Ψ q -function (resp. a p F q -function), with orders reduced by m, namely: In the case of Wright function with arbitrary parameters A p+i , B q+i , i = 1, ..., m: z , such kind of result is presented in [24] by means of more general operators I, the so-called Wright-Erdélyi-Kober operators. This means that using a suitable number of times of a procedure similar to that in proof of Theorem 3, from any p Ψ q -function (resp. p F q -function) we can go down to one of the three basic generalized hypergeometric functions, depending on if p < q, p = q or p = q+1: 0 Ψ q−p , 1 Ψ 1 , 2 Ψ 1 ; resp. to: 0 F q−p (hyper-Bessel f. and cos m -f.), 1 F 1 (confluent h.f. and exp-f.), 2 F 1 (Gauss f. and beta-distribution of form z α (1 − z) β ). This is the reason that we classified the g.h.f. to be of three basic types, as: "g.h.f. of Bessel/cosine type", "g.h.f. of confluent/exp type" and "g.h.f. of Gauss/beta-distribution type". Details on this approach and such a classification of the SF can be found in Kiryakova ([6,22,24,50], Ch.4). Analogously to Theorem 3, we have also a relation (image) for the generalized fractional derivatives of g.h.f., presented as Theorem 2 in Kiryakova [2]. The more general formula (as below) is available in Kiryakova, ( [3], Theorem 4.2.).

Theorem 4.
Proof. One possible approach to derive this, is to use a decomposition formula for the generalized (multiple) fractional derivatives (23), as sequential derivatives. Then, we apply m-times the result (39) of Lemma 2.
We can verify (51) also directly, in the same way as in the end of the proof of Lemma 2, using the basic relation D  (24) will be discussed in a separate work, see also Remark 2.

Examples of Erdélyi-Kober and Riemann-Liuoville Operators of Some Special Functions
In the beginning of Section 3 we already acknowledged the contributions by some classical authors, such as Erdélyi et al., Askey, Lavoie-Osler-Tremblay, to provide the images of some special and elementary functions under the Riemann-Liuoville fractional integral/derivative, see Formulas (32)- (35). We may refer also to works where detailed tables of images under Riemann-Liuoville operators are provided, for example the book Erdélyi et al. [44], some recent surveys, including in this Journal, such as by Rogosin [18] (as for M-L type functions), Garrappa-Kaslik-Popolizio [52] (images of elementary functions expressed by M-L functions).
As mentioned in Section 3, the proof of Lemma 1, a most general result for Riemann-Liuoville operators of special functions (in sense of g.h.f.) is formula (37) from Kilbas ([16], Th.2): and for the R-L derivative, the corresponding result is as in Equation (40).
It may be instructive to repeat (as from [2]) some very special cases of the images (37) and (40) under the R-L integral R δ f (z) = z δ I 0,δ 1 f (z), that have been derived by the cited authors by the standard term-by-term integration/differentiation. Naturally, these come also as specifications of our results from Lemmas 1 and 2 for the Erdélyi-Kober operators (case m = 1).
The formulas in Theorems 3 and 4 can easily be reduced to corresponding results for generalized fractional integrals and derivatives (20) and (23)

Saigo Hypergeometric Operators of Various Special Functions
In a series of papers since 1978, such as [57] (for more references see in [6,58]), Saigo introduced a linear integral operator with Gauss function in the kernel, and applied it first for studying BVP for PDE as the Euler-Darboux equation. Later on, this operator was used by him and collaborators in geometric function theory (classes of univalent functions). It happens that, as a case of the hypergeometric integral operators, the Saigo operator has also a role as an FC operator and this has recently become a reason for great interest for researchers in FC, and mainly to authors whose job is to evaluate images of Saigo operator(s) of various special functions. A search in Google for the phrase "Saigo operator" + "function" returns now more than 1060 results (of course some of them may also concern the more general Marichev-Saigo-Maeda, discussed in next Section 7).
First, let us remind the definition and two basic properties of the Saigo operators. For complex α, β, η and Re α > 0, the Saigo fractional integration operator (the LHS version) is As mentioned in Section 2.3, the Saigo operators are cases of the hypergeometric operators of FC, and of the GFC operators for m = 2, simply because according to (30) the Gauss kernel function is representable as the kernel of (19) and (18) Thus, the Saigo operator is a generalized (2-tuple fractional integral) of the form (20) and therefore in view of (21), it is also a commutable composition of two classical E-K fractional integrals, see for example ([6], Ch.1): The relation between the first and second lines follows by application of the "generalized commutation" between (multiple) Erdélyi-Kober operators and power functions (( [6], Ch.1, (1.3.3)), ( [34], Th.4), etc.). For particular parameters α, β, η, the Saigo operator can reduce to one E-K operator or an R-L operator, say for β = −α, η = 0 it is an R-L integral; and for η = −α, an E-K integral: Therefore, the Saigo image of some special function, which can be represented as a Wright function p Ψ q , can be written as a particular case of the general formulas (46), resp. (51), or also, as a subsequent two-times application of classical E-K operators. Therefore, the Saigo image of a p Ψ q -function can always be predicted to result into a p+2 Ψ q+2 -function (unless some parameters in upper and lower rows eliminate each other, and so the indices can be reduced). Our result, as a corollary of Theorem 3 and Corollary 2 states as follows.
The following examples for Saigo operators of particular functions from our previous papers [2,3] are repeated here as an illustration for the general result in Lemma 3.
The Saigo operator is also derived by Lavault in [55]: for the M-series-in Th. 4.2, and for the K-series-in Cor. 4.3. Namely, Equation (4.10), [55] reads as: Let us note that the K-series is a p+2 Ψ q+2 -function (66), and from our Lemma 3 the expected result should be a p+4 Ψ q+4 -function, with indices increased by two. However, pairs of upper and lower rows' parameters appear the same and eliminate each other, therefore the result reduces to a p+3 Ψ q+3 , as above.
For complex parameters a, a , b, b , c, Re c > 0, the Marichev-Saigo-Maeda (M-S-M) integral operator, of which the kernel is the Appel function, or Horn's function F 3 (see ( [15], Vol.1), also [7]) is defined as the linear integral operator Observing the representation (31) of the kernel F 3 -function as a kernel of the generalized fractional integrals (20) (see Section 2.3), it is evident that the M-S-M operator is nothing but their special case for m = 3. Then, in view of (21), it is also a composition of three commutable classical E-K integrals (see can be found in Purohit-Suthar-Kalla ( [60], Th.2.1, (10)). It is supposed that Re c > 0, Re ν > −1, The same result, however, can be obtained in the way as discussed in Example 5, using the M-S-M operator's representation (69) Evidently, it is a variant (up to variable substitution) of the Bessel-Maitland-Wright function J κ ν (z) and of the Wright function φ(z), representable as 0 Ψ 1 -function of (z 2 /4). Then as well expected, the result comes as a 3 Ψ 4 -function, since the indices are increased by 3.
is handled in Agarwal-Rogosin-Trujillo [29]. When m = 1 it was studied also by Srivastava-Tomovski [65]. Note that for γ = κ = 1 the above function reduces to the (2m) multi-index Mittag-Leffler function (9). This appeared also in Saxena-Nishimoto [66] and was studied in Saxena-Pogany-Ram-Daiya [67]. The result from ( [29], Th.3.1, (3.2)) is the following: Using the representations of the M-S-M operator as three-tuple generalized fractional integral (69) and of this special function as a 1 Ψ m -function, the same formula can be evaluated by the general result in Theorem 3, that is the image is again a Wright g.h.f. but its indices are increased by three.
The authors did not observe the fact that the considered M-L type function is a case of the generalized Wright function (4), the definition of which is also given in the mentioned paper, namely: (1, 1); (θ + 1, 1), ..., (θ + 1, 1), (ω + θ + 1, ϕ); −z 2 /4 , which is a Wright g.h.f. (see Equation (1.1) there). We can note that it is also example of the multi-index M-L function (9), namely J (1,...,1,ϕ),(θ+1,...,θ+1,ω+θ+1) −( z 2 ) 2 . Then the result, as calculated by the authors, follows directly from Theorem 3 and especially from Lemma 4 (below, A := to be again a Wright g.h.f. but with indices increased by three, that is, a 4 Ψ m+4 -function. In [62] also many special cases are derived, such as Beta-transform (that is E-K integral), Saigo operator, path integral, of the function (73) and of its particular cases. As in the Commentary [5] we discussed the possibilities to use our unified approach, the authors tried to argue with the facts in their Response [61]. The curious readers are recommended to read Kiryakova's footnote comments at the bottom to this Response [61].
Next, let us consider the special cases of GFC operators for which the multi-index M-L functions (9) appear as eigenfunctions, that is, these special functions are transformed into the same kind of functions with the same multi-indices.
Therefore, Example 16 appears a special case of Example 15, and shows that the multi-index Mittag-Leffler functions (9) can be seen also as "fractional indices" analogs, extensions of the hyper-Bessel functions (82), which themselves are multi-index variants of the classical Bessel function.

Some "New" Special Functions and Their FC Images
Recently, some authors claimed to introduce and consider "new" SF. Among these are examples of the so-called k-analogs of the Bessel and Mittag-Leffler functions, some generalized multi-index Bessel and Mittag-Leffer functions, and some S-functions. The mentioned k-analogs are based on the use of the k-Γ-function, which, however, can be rewritten in terms of the "classical" Γ-function: Γ k (s) = Usually, the denotations include also the k-analogs of the Pochhamer symbol: and in view of (83) are representable again by means of classical Gamma-functions. Then, one can easily observe that such "new SF" are just cases of the Wright generalized hypergeometric function p Ψ q . Therefore, all the results provided by the mentioned authors to evaluate FC operators of these special functions follow from our general ones, say from Theorems 3 and 4, or the special cases as Lemmas 1 and 2 (for E-K operators, incl. R-L ones), Lemma 3 (for Saigo operators), Lemma 4 (for M-S-M operators), and so on. As an illustration, we repeat some examples from Kiryakova [4].

Example 17.
A generalization of the Bessel function, called generalized k-Bessel function was introduced by Gehlot [70] and studied by Mondal [71], Shaktawat et al. [72], defined as Naturally, for k = 1, c = 1, (85) becomes the classical Bessel function: In the case c = 1, Gehlot [70] considered (85) as a solution of a k-Bessel differential equation. Mondal [71] studied its properties for complex c ∈ C. Shaktawat et al. [72] evaluated the M-S-M operators of FC of this function. In view of Lemma 4, the result there (Th.1, (18)) is well expected to appear in terms of the 3 Ψ 4 -function (because the 3-tuple FC integral increases by three the indices of the initial 0 Ψ 1 -function).
(93) 10.6. Many authors are publishing results on the images of particular special functions under some integral transforms like the Laplace transform, Mellin transform, Euler (Beta) transform, Whittaker transform. Observe that the Euler transform (called so after the Euler integral formula for the Gauss function) is just a case of the Erdélyi-Kober fractional integral (12), as an extension of the Riemann-Liuoville fractional integral (14). Therefore, there is no need to separately evaluate these two transforms (Euler transform and Riemann-Liouville operator), and what is more, to repeat such calculations for each particular special function. One can just apply the general result, as in Lemma 1. Note that the so-called pathway-transform is also closely related to the E-K integral. To evaluate a Laplace transform, say for any special function which is an H-function, one can use the general integral formula (44) and the representation of the kernel exponential function as a Wright g.h.f. (see (42), Example 1), then also as a H-function: exp(−z) = H 1,0 0,1 z −− (0, 1) . As already mentioned in Remark 1, a basic approach to evaluate integral transforms (also FC operators) of special functions relies on their images under the Mellin transform in terms of Gamma-functions, to which the fundamental book by Marichev [8] is devoted.
Author Contributions: The ideas and results in this paper survey and reflect the author's (V. K.) sole contributions, resulting from more than 30 years of research on the topic. The author haves read and agreed to the published version of the manuscript.
Funding: This research received no financial funding.