1. Historical Introduction
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, astronomy, statistics or other applications (Wikipedia: Special Functions [
1]). It might be Euler, who started to talk, since 1720, about lots of the standard special functions. He defined the Gamma-function as a continuation of the factorial, also the Bessel functions and looked after the elliptic functions. Several (theoretical and applied) scientists started to use such functions, introduced their notations and named them after famous contributors. Thus, the notions as the Bessel and cylindrical functions; the Gauss, Kummer, Tricomi, confluent and generalized hypergeometric functions; the classical orthogonal polynomials (as Laguerre, Jacobi, Gegenbauer, Legendre, Tchebisheff, Hermite, etc.); the incomplete Gamma- and Beta-functions; and the Error functions, the Airy, Whittaker, etc. functions appeared and a long list of handbooks on the so-called “
Special Functions of Mathematical Physics” or “
Named Functions” (we call them also “
Classical Special Functions”) were published. We mention only some of them in this survey.
As Richard Askey (
to whose memory we dedicate this survey) confessed in his lectures [
2] on orthogonal polynomials and special functions: “Now, there are relatively large number of people who know a fair amount about this topic. Nevertheless, …most of the mathematicians are totally unaware of the power of the special functions. They react to a paper which contains a Bessel function or Legendre polynomial by turning immediately to the next paper”, and continued: “Hopefully these lectures will show … how useful hypergeometric functions can be. Very few facts about them are known, but these few facts can be very useful in many different contexts. So, my advice is to learn something about hypergeometric functions: or, if this seems too hard or dull a task, get to know someone who knows something about them. If you already know something about these functions, share your knowledge with a colleague or two, or a group of students. Every large university and research laboratory should have a person who not only finds things in the Bateman Project (i.e., [
3]), but can fill in a few holes in this set of books … In any case, I hope my point has been made; special functions are useful and those who need them and those who know them should start to talk to each other … The mathematical community at large needs the education on the usefulness of special functions more than most other people who could use them …”.
As a participant in the NATO International Conference on Special Functions and Applications 2000 (Arizona State University), the author had the chance to witness the late night long discussions (mainly between Richard Askey and Oleg Marichev) for the merits and competition of the two great projects on Special Functions, on which the Computer Algebra Systems packages
Mathematica and
Maple are based, the
Bateman Project [
3] and the
NIST Project [
4] based on the Abramowitz–Stegun handbook [
5] and on a more recent one, edited by Olver–Lozier–Boisvert–Clark [
6].
The author of this survey was tempted to start paying attention to Special Functions by having the handbook [
3] on her desk, while working on a M.Sc. thesis. We cite some texts from the Preface of this encyclopedia book, known as the
Bateman Project: “…During his last years he (Professor Harry Bateman) had embarked upon a project whose successful completion, he believed, would prove of great value to scientists in all fields. He planned an extensive compilation of “special functions”—solutions of a wide class of mathematically and physically relevant functional equations. He intended to investigate and to tabulate properties of such functions, inter-relations between such functions, their representations in various forms, their macro- and micro-scopic behavior, and to construct tables of important definite integrals involving such functions …anyone who has been faced with the task of handling and discussing and understanding in detail the solution to an applied problem which is described by a differential equation is painfully familiar with the disproportionately large amount of scattered research on special functions one must wade through in the hope of extracting the desired information …”. In the time of Bateman’s death (1946) his notes amounted to a veritable mountain of paper. His card-catalogue alone filled several dozen cardboard boxes (the famous
“shoe-boxes”). …“Bateman planned his Project as a
‘Guide to the Functions’ on a gigantic scale ….the great importance of such a work hardly needs emphasis …(this) would have made this book as a kind of ‘Greater Oxford Dictionary of Special Functions’ (from the Introduction to [
3])”. This project resulted in publication of five important reference volumes ([
3,
7]), under the editorship of Arthur Erdélyi, in association with W. Magnus, F. Oberhettinger and F.G. Tricomi.
In 2007, the
Askey–Bateman Project was announced by Mourad Ismail as a five- or six-volume encyclopedic book series on special functions, based on the works of both Harry Bateman and Richard Askey. Starting in 2020, Cambridge University Press began publishing Volumes 1 and 2 of this Encyclopedia of Special Functions with series editors Mourad Ismail and Walter Van Assche [
8].
4. Mittag–Leffler Functions and Their Extensions
The
Mittag–Leffler (M-L) function was introduced by G. Mittag–Leffler ([
67], 1902–1905), extended to 2-parameters as
by A. Wiman [
68] and studied later by P. Humbert and R.P. Agarwal [
69]. It was presented in the Bateman Project [
3], Vol. 3 (1954), in a chapter for “Miscellaneous Functions”. However, for long time, it was ignored in the other handbooks on special functions because the applied scientists suffered from the lack of tables for its Laplace transforms. Although arising from the studies of Mittag–Leffler on a problem not related to fractional calculus, but on analytical continuation of series to maximal starlike domain (Mittag–Leffler star), nowadays, the M-L function is the most popular and most exploited SF of FC. It was titled as the
“Queen”-function of FC by Gorenflo and Mainardi in 1997 (see also the very recent survey by Mainardi [
34]). The basic theory and more details, can be found, for example, in [
22,
43] (see also, e.g., [
9,
16,
70,
71]).
Definition 7. The Mittag–Leffler (M-L) functions and , are entire functions of order and type 1, defined by the power series As
“fractional index” analogs of the exponential and trigonometric functions that satisfy ODEs of first and second order (
), the M-L functions serve as solutions of
fractional order differential and integral equations. An example is the Rabotnov function, called also “fractional exponent”,
that solves the simplest fractional order differential equation
. Let us refer also to the pioneering work by Hille–Tamarkin [
72], where the solution of the Abel integral equation of the second kind was provided in terms of a M-L function. As far as the Laplace transform images are mentioned, one can find these for the M-L type functions and their
kth derivatives in the work of Podlubny ([
16] (S.1.2.2)):
A Mittag–Leffler type function with three indices, known as the
Prabhakar function [
73], is also often studied and used (for details, see [
22,
70,
71,
74,
75] and other contemporary books and surveys on M-L type functions):
where
denotes the Pochhammer symbol. Its Laplace transform has the form
For , we get the M-L function , and, if additionally , then it is .
These M-L type functions are simple cases of the Wright g.h.f. and of the
H-function, namely:
Another generalization of M-L function (
37) with additional parameters, for example
, was considered by Gorenflo–Kilbas–Rogosin [
76], and its relations to FC operators:
A
vector index extension of (
37) appeared in the works by Luchko et al. (e.g., [
15,
77,
78]) on operational calculus’ methods for some fractional order PDE and multi-term FO differential equations. Under the name
multi-index (multiple) M-L function, it was introduced by Kiryakova [
63,
79] using a different approach, as to be the generating function of Gelfond–Leontiev generalized integration and differentiation operators (
47) (see Definition 9) and inspired from the paper by Dzrbashjan [
44] on M-L type function with
indices. Further, this class of functions were studied in details by Kiryakova [
59,
80], Kilbas–Koroleva–Rogosin [
81], Paneva–Konovska [
74] and many other followers. Luchko et al. also considered multivariate analogs of the so-called vector index M-L functions [
78].
Definition 8. (Kiryakova [
59,
80])
Let be an integer, and be arbitrary real parameters. By means of these sets of “multi-indices”, the multi-index Mittag–Leffler function (abbrev. as multi-M-L f.) is defined as: Under weakened restrictions on αs (or their real parts) not to be obligatory all non-negative, the study was extended by Kilbas et al. [81]. As a further extension of both Prabhakar function (
38) and of the (
) multi-index M-L functions (
39), Paneva–Konovska [
74,
82] introduced and studied the so-called
-parametric (multi-index) Mittag–Leffler functions, similar to (
39) but with additional set of parameters
:
For
, one has the Prabhakar function, and, for
, these are (
39).
The Mellin transforms of (
39), (
40) and their particular cases can be found in [
83].
The so-called
Le Roy type function has been an object of several recent studies, e.g., by Gerhold [
84], Garra–Polito [
85], Garrappa–Rogosin–Mainardi [
86], Garrappa–Orsingher–Polito [
87], as a new special function
which is an entire function of
for parameters
,
and
. This resembles to the M-L function (for
) and to the multi-index M-L function (
39) (for integer
, all
). The function (
41) appeared as extension of the function
, introduced by E. Le Roy [
88] (1899), similarly to the purposes of G. Mittag–Leffler [
67] (1903) to study analytical continuations of the sums of power series, and it seems they were working in competition on such ideas. Similar to the M-L type functions, (
41) is involved in solutions of various problems, including a Convey–Maxwell–Poison distribution for different degrees of over- and under-dispersion.
Some Basic Properties of the Multi-Index Mittag-Leffler Functions
The basic properties and results for the functions (
39) and long lists of their examples, all of them having wide applications in solutions of integer- and fractional-order models, are provided in our previous papers (e.g., [
59,
60,
79,
80]). Some of them are reminded here.
Theorem 1. The multi-index M-L functions (39) are entire functions with the following order ρ and type σ:respectively with αis replaced by s. Note that the type for and only for (classical case (37)): . The following asymptotic estimate holds: The
-parameters M-L type functions (
40) are also entire functions with the same order and type as in (
42), see [
74,
82].
Lemma 1. The multi-index M-L functions (39) are important examples of the Wright generalized hypergeometric functions and of the Fox H-functions: Thus, the following Mellin–Barnes type integral representation holds (cf. with (1)):based on the Mellin transform (see [59,83]; also [18] (p. 48)): Additionally, as shown by Paneva–Konovska [
74,
82], the
-parametric functions (
40) can be represented as
which is in agreement with (
43) for
.
As an analog of the Laplace transform (
), relationship between the classical M-L function (
37) and the classical Wright function:
(see in the books [
16,
18]), we derive the following
new relation.
Note that we can consider the
-functions on the left-hand side as
“fractional indices” analogs of the -functions, that is
of the hyper-Bessel functions of Delerue [
89], related to the hyper-Bessel operators (
9) as their eigenfunctions, and discussed further as special cases of (
39). For details on these special functions, see Kiryakova [
9] (Ch.3).
Various relations for the multi-M-L functions in terms of the operators of classical FC and GFC have been derived in our previous works (e.g., [
59,
80]). First, let us consider the so-called
Gelfond–Leontiev (G-L) operators of generalized integration and differentiation, generated by the coefficients of an entire function
. For the theory of the G-L operators in general, see Gelfond and Leontiev’s paper [
62]) of 1951, and for details in the case when the mentioned entire function is taken to be the M-L function or multi-index Mittag–Leffler function, we refer to Kiryakova [
9] (Ch.1), [
59,
63,
79]. Here, we only remind the definition of the G-L operators related to
whose coefficients
are taken as multipliers’ sequences below.
Definition 9. (Kiryakova [
63,
64])
For functions analytic in a disk , we consider the operatorsand call them multiple (multi-index) Dzrbashjan–Gelfond–Leointiev (D-G-L) differentiations and integrations, respectively. These are generated by the multi-index M-L functions and the name of Dzrbashjan is used in addition to Gelfond–Leontiev to honor his contribution to one of the first deep studies on M-L type functions, the book [43]. Evidently,
, and it is proven that the radii of convergence (and analyticity) of resulting analytical functions in (
47) are the same
R as for
. According to Theorem 3 in [
79], operators (
47) can be analytically extended outside the disks to starlike domains and represented as operators of GFC, as follows:
To start with the classical FC operators for the multi-index M-L functions, we state the following
Lemma 3. (Kiryakova [
80] (Lemma 3.2))
For any fixed and integration order , we have for the E-K fractional integral the relation that is, a fractional integration can transform a multi-M-L function into another one with same αis and corresponding parameter γl increased by the order of integration to . Applying E-K fractional integrals of the form
successively
m-times (
) to (
39) and using the composition (decomposition) property (
26), we obtain for the generalized fractional integrals (
23) the image:
Then, for
, and applying the operational rules for the operators
and
of GFC, the following generalized fractional integration and differentiation relations follow:
as analogs of the classical relation
for the R-L derivative
.
It remains to combine the results (
48) and (
50) to verify the fact that
the multi-index M-L functions that generate the G-L operators (47) appear as their eigenfunctions:
Theorem 2. The multi-index Mittag–Leffer function (39) satisfies the differential equation of fractional multi-order : The classical
Poisson integral formula, representing the Bessel function via the cosine-function ([
3] (Vol. 2)), can be written in terms of an E-K fractional integral, as
This representation has been extended in our works [
9] (Ch.4), [
90] for the
hyper-Bessel functions (
58),
, that is for the
-functions, via generalized fractional integrals (
24) of the function
. The details follow in
Section 8. For the multi-index M-L functions, a Poisson type integral representation of the kind of (
52) has to explore the more general fractional calculus operators from Definition 6. This is a part of the general results discussed in
Section 8, but we expose it here as to close (at least partly) the topic with some properties of the multi-index Mittag–Leffler functions.
Theorem 3. (Kiryakova [
59])
Let , , . Then, we have the following Poisson-type integral representation of the multi-index M-L functions my means of multiple W-E-K fractional integrals (33) of the cosine function (54) of order m (from the next section): Remark 1. The above result is parallel with (52) for the Bessel functions. If we take , , the above GFC operator, the multiple W-E-K fractional integral, has a multi-order and since also , it turns into identity. Then, the -function reduces to the -function. It is similar in the simplest case to the Bessel function with index : . More generally, it is also known that the Bessel functions of “semi-integer” indices (called also“spherical functions”
for their use in theory of spherical waves) are reducible to trigonometric functions or to integer order operators of them: . In the case of multi-index M-L functions (39), we can call multi-indices of the form , ; for , as“semi-integer multi-indices”
. A corollary of Theorem 3 tells that for such multi-indices the functions reduce either directly to generalized trigonometric functions, or to integer order integral or differential operators of them. The results for the
images of the multi-index Mittag–Leffler functions (
39) and (
40)
under GFC integrals and derivatives, or under their particular cases a R-L, E-K, Saigo, Marichev–Saigo–Maeda operators, etc. can be written from the general results in
Section 7 according to definition via the Wright g.h.f.
.
Series in systems of special functions, in the general cases of
- and
-parameters M-L functions and their particular case (mentioned in next section) as the M-L function, Parbhakar function, multi-index and fractional analogs of the Bessel- and hyper-Bessel functions, were studied recently in details by Paneva–Konovska in a series of papers and in the book [
74], especially with respect to their convergence in complex domain, including Cauchy–Hadamard, Abel, Tauber type, Hardy–Littlewood and Ostrovski type theorems.
5. Examples of M-L Type and Multi-Index M-L Functions
5.1. For , this is the classical M-L function with all its special cases:
•: ; ; , ; = = = (the error functions, or incomplete gamma functions);
•: ; ; ; the Miller-Ross function ; etc.;
•: the α-exponential (Rabotnov) function .
• The
trigonometric functions of order m, and, respectively the
hyperbolic functions of order m:
can also be expressed in terms of the M-L function (see in [
3] (Vol. 3) and [
16] (Ch.1)); and the same for their
fractionalized versions, as by Plotnikov [
91] and Tseytlin [
92]:
and by Luchko–Srivastava [
77]:
(see details again in Podlubny [
16] (Ch.1)).
• Here, we mention also the so-called
Lorenzo–Hartley functions [
93], the
F-function and its generalization the
R-function, shown to be solutions of some linear fractional differential equations. We can represent them in terms of M-L function, namely, for
,
,
,
:
5.2. For
: We start with the not enough popular
M-L type function of Dzrbashjan [
44], with
indices, which he denoted alternatively by (we need to set
):
Dzrbashjan found the order and type of this entire function, claimed on few simple particular cases, and considered some integral relations between (
55) and Mellin transforms on a set of axes. Then, he developed a theory of integral transforms in the class
L2, involving kernel close to functions (
55) and, further, proposed approximations of entire functions in
L2 for an arbitrary finite system of axes in complex plane starting from the origin.
The
-indices M-L type functions (
55) were also studied in detail by Luchko in the recent paper [
94]. He allowed the parameters
to be also negative or zero and called them “
4-parameters Wright functions of second kind”, separating the cases when
,
or
.
Some of the simple cases of (
55), as mentioned and denoted in Dzrbashjan [
44], are:
• the M-L function itself: ; ; the Bessel function: ; etc.
To these examples, we added (see, e.g., Kiraykova [
59]) the following cases:
• The
Struve and Lommel functions (see [
3] (Vol. 2); and details in [
9] (App.,(C.8)), [
79,
80]):
• The
“classical” Wright function that arose in the studies of Fox ([
95], 1928), Wright ([
31], 1933) and Humbert and Agarwal ([
69], 1953) and was also referred to in Erdélyi et al. [
3] (Vol. 3). Initially, Wright [
31] defined this function only for
, then extended its definition for
[
32]. Now, we see this is a case of multi-index M-L function with
:
which is entire function of order
. The survey papers by Gorenflo–Luchko–Mainardi [
96] and Mainardi–Consiglio [
97] reflect in detail its analytical properties and applications, see also the book [
22] as well as the related literature. In the case
, the Wright function is said to be of first kind, and for
of second kind. The latter survey [
97] concentrates on the Wright function of second kind. It is noted that the first kind Wright function is of exponential order, while the second kind is not of exponential order, and naturally they have different asymptotic behaviors, Laplace transforms, etc. (see also Luchko [
94]). The function (
56) plays an important role in the solutions of linear partial fractional differential equations as the
fractional diffusion-wave equation studied by Nigmatullin (1984–1986, to describe the diffusion process in media with fractal geometry,
) and by Mainardi et al. (since 1994, for propagation of mechanical diffusive waves in viscoelastic media,
). In the form
, this function is recently called as the
Mainardi function (see [
16] (Ch.1)). In our denotations, it is a multi-index M-L function with
and a Dzrbashjan function (
55):
and has its own particular cases, such as
and the
Airy function,
. Note also that, for
, the Wright function (
56) reduces to the
exponent, since
.
In alternative form and denotation, the Wright function (
56) is known as the
Wright–Bessel function or is misnamed as the
Bessel–Maitland function:
again as an example of the Dzrbashjan function. It is an obvious (and was introduced as such by Sir E. Maitland Wright [
32])
“fractional index” analog of the classical Bessel function , more exactly, of the Bessel–Clifford function
.
Several further
“fractional-indices” generalizations of
and
are found in the studies of other authors (details are in [
59]), and we can represent all of them as multi-index M-L functions. One of them is the so-called
generalized Wright–Bessel(–Lommel) functions, introduced by Pathak ([
98], 1966),
For
, it includes the mentioned Lommel and Struve functions, e.g.,
. A next example is the
generalized Lommel–Wright function with four indices, introduced by de Oteiza, Kalla and Conde ([
99], 1986), with
:
5.3. The above is an interesting example of a multi-M-L function with .
Other particular cases of multi-index (-parameters) M-L functions with greater multiplicity are:
• For arbitrary
: let
and
,
. Then, from definition (
39), we get again the
geometric series• Consider the case
,
,
. Then, the function
reduces to
-function and also to a
Meijer’s -function. Denote
, and let additionally one of the
be 1, e.g.,
, i.e.,
. Then, the multi-index M-L function becomes a
-function, that is, a
hyper-Bessel function in the sense of Delerue [
89] (see also [
9] (Ch.3)):
where
is called as
normalized hyper-Bessel function.
This representation suggests that the multi-index M-L functions (
39) with arbitrary
≠
can be interpreted as
fractional-indices analogs of the hyper-Bessel functions (
58) and (
59), which themselves are
multi-index (but integer)
analogs of the Bessel function. Functions (
58) and (
59) are closely related to the
hyper-Bessel differential operators (
9) (see
Section 3.1), and form a fundamental system of solutions of the differential equations of the form
; the details are found in Kiryakova [
9] (Ch.3, Th.3.4.3). For example, if the hyper-Bessel operator (
9) is with
,
, the solution of the Cauchy problem
is given by the
normalized hyper-Bessel function (
59):
. Closely related functions are also the
Bessel–Clifford functions of order m:
Let us mention the special functions appearing in a very recent paper by Ricci [
100]. He considered the so-called
Laguerre derivative and its iterates
, same as the particular hyper-Bessel differential operators (
19) considered in operational calculus by Ditkin and Prudnikov [
50], as mentioned in
Section 3.1. Then, the
L-exponentials , which are eigenfunctions of
, that is,
, are shown in [
100] to have the form
Then, these are examples of the hyper-Bessel functions (
58) and of the multi-index Mittag–Leffler functions
as well. In [
100], applications to population dynamics and in solutions of linear dynamical systems of these SF and of the related Laguerre-type Bell polynomials and Laguerre-type generalized hypergeometric functions are discussed.
• The
Rabotnov function (the
-exponential function), presented in
5.1., appeared in Rabotnov’s works on application of fractional order operators in mechanics of solids. It is interesting to consider its
multi-index analog, that is the case with all
. This is the function
Observe that, for
, we get the Ricci function (
60), namely:
, and also a case of the
original Le Roy function with
.
• In general, for rational values of
, the functions (
39) are reducible to
generalized hypergeometric functions and to
Meijer’s G-functions , that is, to classical special functions.
Remark 2. Note that all the results we derived for the multi-index M-L functions can be applied for their particular cases mentioned above.
6. Other Special Cases of the Wright Generalized Hypergeometric Functions
6.1.Virchenko and Ricci generalized hypergeometric functions. In [
101] and some other papers, Virchenko studied some generalized hypergeometric functions denoted by
and
, as well as their integral representations, relations and applications to the generalized Legendre functions
, gamma functions, Laguerre’s functions, etc.
For
, and
- complex,
;
,
;
, it is rewritten as
which is nothing but the Wright g.h.f.
. Virchenko also proposed some examples of elementary functions for these special functions, e.g.,
and
; some generalized incomplete
B-function; the Gauss function
; etc.
•
and, in Virchenko [
101], generalizations of the gamma function, incomplete gamma function, probability integrals and Laguerre’s functions are introduced by means of
, which is a Wright g.h.f. of the form
, and, according to our classifications in
Section 8, a confluent type g.h.f.
• In
5.3., the recent paper by Ricci [
100] is mentioned for the Laguerre-type derivatives and related special functions. Along with the functions (
60), there he also considered the Laguerre-type (
L-) Bessel functions,
L-type Gauss hypergeometric functions and the
Laguerre-type generalized hypergeometric functions . They can be shown to be representable by
, thus also as
, namely:
6.2.Mainardi-Masina and Paris generalized exponential integrals. In [
102], Mainardi and Masina introduced a generalized exponential integral
by replacing the exponential function in the complementary exponential integral
by the Mittag–Leffler function
and mentioned the physical applications for
in the studies of the creep features of linear viscoelastic models. In the recent paper [
103], Paris made the next step to involve the two-parameter M-L function, namely to consider the generalized exponential integral
As observed, this function can be seen as a case of the Wright g.h.f. with
, namely
Paris studied in details the asymptotic expansion of (
63) for
. In [
102,
103], generalized Sine and Cosine integrals are also considered (of the kind mentioned in
5.1.), for example
, with their asymptotics and plots for different values of parameters.
6.3.The so-called k-analogs of special functions. Claims on inventing and studying “new” classes of special functions in several recent papers have been based on the extended notion of the
k-Gamma function,
. However, in all such works, its representation in terms of the classical Gamma-function is explicitly written there, and then is ignored:
where
is the classical Gamma-function.
In addition, the
k-Pochhammer symbol is used in the next denotations:
In [
104], using the above two definitions, we showed that
most of these “new” functions are in fact some known special functions, namely Wright g.h.f. and its cases. For the details of establishing the mentioned relations, see Kiryakova [
104]. In addition, in the references lists of [
104,
105], one can find the particular authors/sources mentioned below.
•
A generalized k-Bessel function was introduced by Gehlot ([
106], 2014), and studied by Mondal ([
107], 2016) and Shaktawat et al. ([
108], 2017). It is defined by
However, after simple exercise, the function (
66) can be represented as a Wright g.h.f.
, and even as the simpler g.h.f.
of the same type as the classical Bessel function:
Indeed, if we take
and
, this function reduces to the classical Bessel function:
. For
and
Gehlot [
106] used (
66) as a solution of a
k-Bessel differential equation. Mondal [
107] studied some properties of (
66) for arbitrary
. Shaktawat et al. [
108] evaluated the
Marichev–Saigo–Maeda (M-S-M) operators of FC
of this function. Since its kernel Appel
-function is a
H-function (
36) with
, in view of author’s result from Corollary 3 in
Section 7, it is well expected that the result appears in terms of a
-function (because the indices of
are increased by 3 under the 3-tuple FC integral).
•
Generalized k-Mittag–Leffler function. It was studied by many authors, for example in its simplest case by Gupta and Parihar ([
109], 2014) in the form
This function has various further extensions, such as the generalized
k-Mittag–Leffler function by Nisar–Eata–Dhaifalla–Choi ([
110], 2016):
Again, by using (
64) and (
65), it can be transformed into a Wright g.h.f. (see [
104], Case 5.2), namely:
Nisar–Eata–Dhaifalla-Choi [
110] put efforts to evaluate FC operators’ images of (
69) by the standard techniques, and as expected in view of the general results in next
Section 7 Theorem 5, Corollarys 1–3) these appear in terms of
-functions (for the M-S-M operators (
68)), in particular, as
-functions (for the Saigo operators (
78)) and
-functions (for the R-L and E-K operators). In addition, the pathway integrals (that are related to E-K integrals) are calculated there.
•
The generalized multi-index Bessel function. In a series of papers, Nisar et al. ([
111], 2017, 2019) introduced and studied the function
with the Pochhammer symbol denotation (
65) for
; and for
,
;
;
,
,
. As shown in Kiryakova [
104], this is
only a very special case of the Wright generalized hypergeometric function , namely:
, that is, it is also a Fox
H-function.
Then, the R-L fractional integral (
21) can be evaluated as part of Kiryakova’s general results in next
Section 7 (Theorem 5, in particular Corollary 1 for
,
), or directly from Kilbas’ Theorem 2 in [
33], which is a variant of Lemma 1 in Kiryakova [
112]. Taking there
,
,
and
, one obtains the following R-L image for the multi-index Bessel function (
70):
This was to be the result in Theorem 1, Equation (2.4) in
arXiv:1706.08039 [
111], its v1: 2017, but was
written wrongly there—similarly looking but involving a
-function. The evident true result involves the Wright function
(see Kiryakova [
104] (5.3.)), as later corrected in v2: 2019 of [
111].
• A special case of (
70) appears as a kind of
generalized multi-index Mittag–Leffler function. It was introduced by Saxena and Nishimoto ([
113], 2010). As mentioned by Agarwal–Rogosin–Trujillo ([
114], 2015), it is representable also as a Wright g.h.f.
, namely:
Therefore, all the GFC operators (say the R-L, E-K, Saigo, M-S-M operators) of this special function can be evaluated by means of the general results in
Section 7, Corollaries 1–3 there. For
, this is the SF considered by Srivastava and Tomovski ([
115], 2009):
.
• Similar, but simpler, is the case of the
generalized Lommel-Wright function from the paper by Agarwal–Jain–Agarwal–Baleanu ([
116], 2018), which is commented in Kiryakova [
117]. It has a representation as a Wright g.h.f. as follows:
Note, additionally, that (
73) is an example of the multi-index Mittag–Leffler function (
39), namely:
. Then, all the FC images of (
73) evaluated in the commented paper follow at once by our general results (see details in [
117]).
6.4.The S-function. It was introduced by Saxena-Daiya ([
118], 2015) as a “new” special function extending the M-L function (
), the Prabhakar function (
38), the
M-series (
76) of Sharma and Jain ([
119], 2009) with
,
, etc., as follows:
For
, it reduces to the generalized
k-Mittag–Leffler function
, a variant of (
69). However, it can be easily seen to be special case of the generalized hypergeometric function of Wright of the form
. Unfortunately, this fact has not been observed, neither by the authors introducing (
74) nor by their numerous followers. Namely, one can write (
74) as follows (see details in [
104]):
That is, the “new” special function
is nothing
but a case of the Wright function . Then, all results for images of FC operators, as R-L, E-K, Saigo, M-S-M and the Euler-transform, follow from the statements in
Section 7.
• Special cases of the
S-function in
6.4. are the
generalized K-series and the
M-series. Recently, (K.) Sharma ([
120], 2012) introduced an extension of both g.h.f.
and Prabhakar function
:
with integers
(and additional requirement
if
). For
, this gives the
M-series (
76) of (M.) Sharma and Jain ([
119], 2009):
We can mention its particular cases, for example: (1) for
, the (simpler)
M-series, introduced by M. Sharma (2008); (2) for
(no upper and lower parameters), M-L function
; (3) for
, the Wright function
, or the generalized Bessel-Maitland function (
57); (4) for
,
in (
75), the Prabhakar type function (
38); and (5) for
, the g.h.f.
.
In the recent
arXiv preprint [
121], Lavault represented (
75) as a Wright g.h.f.:
although this can also be reduced to:
, since the two pairs
of parameters in the upper and low rows eliminate each other. In [
121] some FC operators of this
K-series are calculated, as the R-L, Saigo and M-S-M operators. Naturally, a R-L integral is transforming a
-function into a
-function (Theorem 4.1, there), similarly to our Example 11 in [
112] for the
M-series. Next, in Theorem 4.2 of [
121] for the
M-series and Corollary 4.3 for the
K-series, the
Saigo operator (
78) (with Gauss hypergeometric function (
35), GFC with
) is derived,
Since the
K-series (
75) is a
-function, from our results (and Corollary 3 [
112]; see also Corollary 2 in the next section), it is expected that the result should be given as a
-function (the indices are to be increased by 2), which is the result (4.10) in [
121]:
Similarly, the M-S-M-images (
68) follow as
-functions, according to Corollary 3 in next section.
6.5.k-Wright generalized hypergeometric function . Purohit and Badguzer ([
122], 2018) introduced the
generalized k-Wright function, as a
k-extension (
) of the Wright g.h.f. (
4), by
Replacing the
k-Gamma function by the classical Gamma function according to (
64), it is seen that the “new” function is
again a Wright generalized hypergeometric function, of the form
7. Results for the FC and GFC Images of SF of FC
Recently, there have appeared too many papers that deal with evaluation of FC and GFC operators of various special functions. They use the same standard techniques—replace the particular function by its power series, then interchange the orders of integration (fractional order integrals) and summation, etc. Usually only the special functions are changed and also the FC operators—with more and more general ones (but all these happen to be cases of our GFC operators). The great number of combinations “special function + particular operator” explains the dramatically increasing production of such works.
Based on our older results on GFC for SF, since the work in [
9], in the papers [
64,
104,
112,
117,
123], and in a recent survey paper [
105] in this same journal, we propose an unified approach how this job can be done at once, for all SF of FC (we mean the
H- and
G-functions and in particular the Wright g.h.f., multi-index M-L functions and all their particular cases) and for all operators of GFC (we mean the generalized fractional integrals and derivatives of the form (
25) and (
28), thus including the R-L, E-K, Saigo, Marichev–Saigo–Maeda operators, etc.). For the initiating idea, we need to pay tribute to the initial classical results of 20th century in the Bateman Project on Integral Transforms [
7] and in works by Askey [
2], Lavoie–Osler–Tremblay [
124], etc. for the R-L images of many elementary functions and of the simplest
-functions, as:
,
and
. We combined these with the composition/decomposition rule (
26) presenting the GFC operators as compositions of weighted R-L/E-K operators. As a recent survey on FC images of elementary functions, we mention also the work of Garrappa–Kaslik–Popolizio [
125].
Below, we remind only the statements of the main results from the mentioned author’s papers, as surveyed in [
105], in this same journal.
Theorem 4. The -image (23) of a H-function is also a H-function whose last three components of the order are increased by m (the multiplicity in GFC operators), and with additional parameters depending on those of the generalized fractional integration. Namely, Then, GFC images of almost all SF of FC can be evaluated from (
81). This result is based on a formula for the integral of product of two arbitrary
H-functions, namely for the Mellin transform of such a product ([
9] (App., (E.21
), [
12] (§5.1, (5.1.1)), [
14] (§2.25, (1))). A similar formula presents the GFC operators (with
-kernel) of arbitrary
G-function, in terms of another
G-function with increased orders and additional parameters ((Lemma 1.2.2 in [
9] and Corollary 1 in [
105]).
Since most of the considered SF of FC are Wright g.h.f., the main and most useful result is as follows.
Theorem 5. The image of a Wright g.h.f. by a generalized fractional integral (23) (multiple, m-tuple Erdélyi-Kober integral), provided , , , is another Wright g.h.f. with indices p and q increased by the multiplicity m and additional parameters related to these of the GFC integral: Specially, for , , this result is simplified to , as above.
Similarly (Theorem 4.2 in [
104]; Theorem 4 in [
105]),
The
simpler results for the -functions read by analogy (Corollarys 4.1 and 4.2 in [
104]), for example with
, as:
We also describe the corollaries of the results (
82) and (
83)
for the particular cases of most often FC operators on which the other authors have exercised their evaluations, say for:
(R-L and E-K),
(Saigo operators) and
(M-S-M operators). These results for arbitrary Wright g.h.f. are mentioned below.
Corollary 1. For the Riemann–Liouville (R-L) integrals and derivatives, the simplest results are parts of Lemmas 1 and 2 in Kiryakova [105]: The results for the E-K operators have same expressions as in (82) and (83) with . Corollary 2. The images of the Wright g.h.f. and, in particular, of the g.h.f. under the Saigo operators (78) are given by the formulas:(for , , this is Corollary 3 in [112]) and Corollary 3. The Marichev–Saigo–Maeda (M-S-M) operators (68) transform a Wright g.h.f. function into same kind of special function but with indices increased by 3: We state here also the more general result for images of arbitrary Wright generalized hypergeometric function
in the case of multiple Wright–Erdélyi–Kober operators (
33).
Theorem 6. (Kiryakova, [
60], Theorem 9)
The image of a Wright generalized function by a multiple W-E-K operator (33) has the formConversely, the alternatively stated result reads as: each -function can be represented by means of a multiple (m-tuple) operator of GFC, of a -function, the orders of which are reduced by m:with A long list of examples how these general results work at once for any of the SF of FC mentioned in previous sections is provided in author’s works [
104,
105,
112,
117,
123], including some of the particular cases of W.g.h.f. and of multi-index M-L f., mentioned in
Section 5 and
Section 6. There we also provided the details on the references items for the authors cited here only with years.
8. Theory of SF of FC in View of GFC Operators
Usually, the special functions of mathematical physics are defined by means of power series representations. However, some alternative representations can be used as their definitions. Let us mention the well-known
Poisson integral (
52) for the Bessel function and the analytical continuation of the Gauss hypergeometric function via the
Euler integral formula. The
Rodrigues differential formulas, involving repeated or fractional differentiation are also used as definitions of the classical orthogonal polynomials and their generalizations. As to the other special functions (most of them being
- and
-functions), such representations have been less popular and even unknown in the general case. There exist various integral and differential formulas, but, unfortunately, quite peculiar for each corresponding special function and scattered in the literature without any common idea to relate them.
In our works since 1985 (e.g., [
9] (Ch.4), [
58,
60]), we showed that all the classical SF and the SF of FC (in the sense of generalized hypergeometric functions
and
) can be presented by means of generalized fractional integrals or derivatives of three basic elementary functions. On this basis, these special functions have been classified into three specific classes, and several new integral and differential representations have been proposed under a unified idea. Besides, for these three classes of SF,
we provide analogs of the mentioned Poisson and Euler integral formulas and of the Rodrigues differential formulas, which can also be used for
alternative definitions of these special functions, their analytical extensions or for numerical algorithms.
The idea is briefly explained as follows: (i) most of the classical SF (SF of mathematical physics) and SF of FC are nothing but modifications of the g.h.f. or ; (ii) each -function or -function can be represented as an E-K fractional differintegral (i.e., integral or derivative) of a -function or , respectively; (iii) a finite number of steps (ii) leads to one of the basic g.h.f. ( (for : Bessel function); (confluent h.f.) and (exponent); and (Gauss h.f.) and (beta-distribution) to the simplest functions , , , respectively); (iv) the above three basic g.h.f. can be considered themselves as fractional differintegrals of the three elementary functions, depending on whether , or ; and (v) the compositions of E-K operators arising in Step (iii) give generalized (q-tuple) fractional integrals or derivatives.
Thus, for the simpler case of -functions, we have the following general proposition.
Theorem 7. (Kiryakova [
58])
All the generalized hypergeometric functions can be considered as generalized (q-tuple) fractional differintegrals (24), (30) (with -kernels) of one of the elementary functions depending on whether . It is based on the known auxiliary result coming yet from the Bateman Project on integral transforms [
7], Askey [
2], Lavoie–Osler–Tremblay [
124] for the R-L derivatives that we have
paraphrased in terms of E-K operators (e.g., Equation (4.2.2
) in [
9] and Lemma 3.2 in [
58]) as follows:
for all complex
z, and if
we require additionally
. Then, this basic fact is to be used repeatedly, and combined with the composition/decomposition property (
26) for the operators of GFC. In each of the three separate cases, we reach to one of the basic functions (
92) with smallest possible first index
p, namely:
;
and then
; and
and then
.
For the
Wright generalized hypergeometric functions (
4), this proposition reads almost the same, only the third basic function (for
) is more general, namely
, and the GFC operators have as kernel the
-function with different parameters
βs and
λs in the upper and low rows.
Theorem 8. (Kiryakova [
60] (Theorem 14))
All the Wright generalized hypergeometric functions can be represented as multiple (q-tuple) W-E-K fractional integrals (33), or their corresponding fractional derivatives, of one of the following three basic functions: In this case, the basic used result is Theorem 6, following similar Steps (i)–(v) as described above.
The three cases, for both Theorems 7 and 8, are considered in detail, in separate statements.
(1)
. The Poisson integral representation (
52) is extended in [
9] (Ch.4) and [
90] for the
hyper-Bessel functions (
58),
, that is for the
-functions, via generalized fractional integrals (
24) of the function
, (
54) as follows:
By analogy with the hyper-Bessel functions (
58), we consider what we call
the Wright hyper-Bessel functions:
The latter denotation is to remind of the analogy with the hyper-Bessel functions (
58), when
. It is easy to observe that (
96) appears as special case of the multi-index Mittag–Leffler functions (
39), namely:
.
We have then a result, analogous to (
95), and more general than (
53) for the multi-M-L functions, that:
each Wright hyper-Bessel function , , can be represented by means of a Poisson type integral of the -function, written in the form
Let us now apply to the function
above,
p-times the results (
90), (
91) (Theorem 6) with
, combined with the composition rule for the W-E-K integrals (
33). Then, we obtain the following:
Theorem 9. (Kiryakova [
60] (Theorem 15))
Each -function with is a generalized q-tuple W-E-K fractional (differ-)integral operator of , with the following parameters:If the following conditions are satisfied:then relation (98) gives a Poisson type integral representation; otherwise, the operator in the R.H.S. should be interpreted as a multiple W-E-K derivative (see, e.g., Definition 7 in [60]), and then (98) turns into a new Rodrigues type differential formula, or a mixed differ-integral representation. The particular case of Poisson type representation (
53) for the multi-index M-L function has been already stated as Theorem 3 in
Section 4.
In the other two cases,
and
, the starting results for
were formulated as Lemmas 11 and 12 in Kiryakova [
60]:
After additional
steps, from
passing via
to
, respectively, to
, the following explicit results for the statement in Theorem 8 are provided in [
60].
(2) .
Theorem 10. If , each g.h.f. is an p-tuple W-E-K fractional integral of the exponential function, namely, if , : If for some indices k, the above inequalities for parameters are not satisfied, representation (101) turns into differ-integral one, or in special cases to purely differential one. Theorem 10 suggests us to separate the g.h.f-s with in a class of so-called Wright g.h.f. of confluent type, involving the confluent hypergeometric function and as the simplest cases.
(3) . Analogously, we call the -functions with as Wright g.h.f. of Gauss type, since the simplest case of such special function is the Gauss function. We have following specific result.
Theorem 11. Each Wright g.h.f. of Gauss type , that is with , is a q-tuple Wright–Erdélyi–Kober fractional integral (or differ-integral) of the -function. Namely, for and : For other arrangements between and , the operator in (102) is a generalized fractional derivative. For particular choices of parameters and not satisfying the conditions , some integer order differentiations appear in place of the fractional integrals or derivatives and lead to Rodrigues type differential formulas, analogous to these for the classical orthogonal polynomials.
Note that the integral representation (
102) generalizes the
Euler integral formula for the Gauss hypergeometric functions that serves for its analytical extension outside
to the domain
:
This gave us the reason to name with as a Gauss type g.h.f.
In particular, for
, the basic function in the case
reduces to the geometric series: