Workshop: Fractional CalculusInternational Conference on Fractional Calculus

(Ghent University, 9 – 10 June 2020)

“Ghent Analysis & PDEs” Res. Group

“THE MULTI-INDEX MITTAG-LEFFLERFUNCTIONS AS SPECIAL FUNCTIONS

OF FRACTIONAL CALCULUS:EXAMPLES AND APPLICATIONS”

Virginia Kiryakova

Institute of Mathematics and InformaticsBulgarian Academy of Sciences

1 / 40

The developments in theoretical and applied science requireknowledge of the properties of Mathematical Functions (followingthe notion from Wolfram Res. site,https://functions.wolfram.com/), from elementary trigonometricfunctions to the multivariety of Special Functions (SF), appearingwhenever natural and social phenomena are studied, engineeringproblems are formulated, and numerical simulations are performed.This survey aims to attract attention to classes of SF that werenot so popular (or some of them not introduced) until FractionalCalculus (FC) gained the important role, related to the boom ofapplications of fractional order models.

The so-called Special Functions of Fractional Calculus (SF of FC)are basically Fox H-functions, among them are the generalizedWright hypergeometric functions pΨq and in particular, theclassical Mittag-Leffler (M-L) functions and their variousextensions. The SF of FC are now unavoidable tool in solutions offractional order integral and differential equations and systems.

2 / 40

A very general class of the so-called multi-index (vector index) M-Lfunctions have been recently introduced (Luchko, Kiryakova) andstudied in details. Some of their basic properties, as results of ourworks, are discussed.

An unexpectedly long list of SF of FC with various applications infractional order models are shown to fall as particular cases of themulti-index M-L functions:

– The 1- and 2- parameter M-L functions and related special cases(as the error-, incomplete gamma, Rabotnov functions);– Prabhakar (3-parameter) function;– Wright (Bessel-Maitland) function;– Dzhrbasjan’s 2× 2-indices functions;– Mainardi function;– Bessel, Struve, Lommel and Airy functions;– Delerue / Kljuchantsev hyper-Bessel functions;– several other M-L type functions with 3, 4, etc. indices- by Pathak, Oteiza-Kalla-Conde, Kilbas-Srivastava-Trujillo,Paneva-Konovska, etc.

This talk is also a contribution to the research on COST Actionprogram CA 15225.

3 / 40

In this lecture we briefly survey some of our basic results on SF ofFC, so to attract attention of potential users:– all of them can be represented as GFC operators of basicelementary functions (among them the pdf functions of someprobability distributions like gamma-, beta-, etc.);– classification and basic properties;– evaluation of FC and GFC operators of these SF in the generalcase;– numerous examples. For their extended list and details, see Refs.

Here, under the term operators of Generalized Fractional Calculus(GFC) we mean integral, differ-integral and integro-differentialoperators which involve SF with singularities in the kernels, haveconvolutional structure and satisfy the basic “axioms” of theclassical FC. The notion “generalized operators of fractionalintegration” was introduced by Kalla, 1969-1979. He suggestedtheir general form, studied some formal properties and exampleswhen the kernels were SF as the Gauss and generalizedhypergeometric functions, including arbitrary G - and H-functions.

4 / 40

The ideas of S.L. Kalla provoked the author to choose a morepeculiar case of G - and H-kernels (Gm,n

m+n,m+n or Hm,nm+n,m+n) and

this choice allowed developing a full theory of the correspondingGFC with many particular cases and applications:

• V. Kiryakova, Generalized Fractional Calculus and Applications.Longman & J. Wiley, 1994.

As particular cases of GFC operators, they appear:

the operators of classical FC (m = 1) of Riemann-Liouville andCaputo, Erdelyi-Kober (E-K) operators and their variouscompositions, named as:Saigo (m = 2), Marichev-Saigo-Maeda (m = 3), hyper-Besseloperators (Dimovski) (m ≥ 1, arbitrary integer), etc.,

as well as many other operators of generalized integration anddifferentiation studied and used in Calculus, Differential andIntegral Equations and their applications in mathematical models.

5 / 40

SF of FC, in Brief – Generalized Hypergeometric Functions

The Wright generalized hypergeometric function pΨq(z), calledalso Fox-Wright function (F-W g.h.f.) is defined as:

pΨq

[(a1,A1), . . . , (ap,Ap)(b1,B1), . . . , (bq,Bq)

∣∣∣∣ z]=∞∑k=0

Γ(a1+kA1) . . . Γ(ap+kAp)

Γ(b1+kB1) . . . Γ(bq+kBq)

zk

k!

(1)

= H1,pp,q+1

[−z∣∣∣∣ (1− a1,A1), . . . , (1− ap,Ap)

(0, 1), (1− b1,B1), . . . , (1− bq,Bq)

], (2)

in terms of the more general Fox H-function. Denote

ρ =

p∏i=1

A−Aii

q∏j=1

BBj

j , ∆ =

j∑k=1

Bj −p∑

i=1

Ai .

If ∆ > −1, the pΨq-function is an entire function of z , z ∈ C,and if ∆ = −1, this series is absolutely convergent in the disk

|z |<ρ, and for |z |=ρ if

<(µ) = <

q∑

j=1bj −

p∑i=1

ai + p−q2

>1/2.

6 / 40

For A1 = · · · = Ap = 1,B1 = · · · = Bq = 1 in (1) and (2),the Wright g.h.f. reduces to a generalized hypergeometric

pFq-function, and so also to a Meijer’s G -function

pΨq

[(a1, 1),. . ., (ap, 1)(b1, 1),. . ., (bq, 1)

∣∣∣∣ z] = c pFq(a1,. . ., ap; b1,. . ., bq; z)

= c∞∑k=0

(a1)k . . . (ap)k(b1)k . . . (bq)k

zk

k!(3)

= c G 1,pp,q+1

[−z∣∣∣∣ 1− a1, . . . , 1− ap

0, 1− b1, . . . , 1− bq

];

with c =

p∏i=1

Γ(ai ) /

q∏j=1

Γ(bj)

, (a)k := Γ(a + k)/Γ(a).

7 / 40

Operators of Generalized Fractional Calculus

Generalized Fractional Calculus (GFC): first, generalized fractionalintegration operators are considered by Kalla as a common notion,in the form:

I f (z) =

1∫0

Φ(σ)σγ f (zσ)dσ = z−γ−1z∫

0

Φ(ξ

z)ξγf (ξ)dξ, (4)

where Φ(σ) is suitably chosen continuous/ analytical function, sothat above integral makes sense and satisfies the main axioms ofthe classical FC. To have a good theory of GFC, the adequatechoice of such kernel-function is indeed crucial : NOT to be toogeneral, neither to be too particular, and so that one candetermine what is the fractional order of integration.

For too general kernel SF, NOT too much formal propertieswere derived, as a formal (only) inversion formula (by means of theMellin transform) and NO full theory and applications weredeveloped. So, it was necessary to make a suitable choice of theG - and H-kernel function, so to develop a full theory of GFC withdetermined fractional order (multi-order) and its applications. 8 / 40

Kiryakova’s generalized fractional integrals

The multiple E-K integral (of multiplicity m > 1), is defined bymeans of the real parameters’ sets: (δ1, ..., δm) – multi-order ofintegration, (γ1, ..., γm) – multi-weight; and (β1 > 0, ..., βm > 0)– additional multi-parameter, as:

I(γk ),(δk )(βk ),m

f (z) :=

1∫0

Hm,0m,m

[σ

∣∣∣∣∣ (γk + δk + 1− 1βk, 1βk

)m1(γk + 1− 1

βk, 1βk

)m1

]f (zσ)dσ,

(5)if

m∑k=1

δk > 0; and as the identity operator:

I(γk ),(0,...,0)(βk ),m

f (z) = f (z), if δ1 = δ2 = · · · = δm = 0. Note that

the kernel Hm,0m,m-function (with 3 singularities: 0, 1, ∞) is analytic

function in the unit disk and Hm,0m,m(σ) ≡ 0 for |σ| > 1.

If all β’s are equal: β1 = β2 = ... = βm = β > 0, (5) reduces toa simpler representation by means of Meijer’s G -function,

I(γk ),(δk )(β,...,β),mf (z) := I

(γk ),(δk )β,m f (z)=

1∫0

Gm,0m,m

[σ

∣∣∣∣ (γk + δk)m1(γk)m1

]f (zσ1/β)dσ.

(6)9 / 40

We call all the operators of the form

I f (z)=zδ0 I(γk ),(δk )(βk ),m

f (z), I f (z)=zδ0 I(γk ),(βk )β,m f (z), with some δ0≥0,

(7)as generalized fractional integrals of multi-order (δ1, ..., δm).

The corresponding generalized fractional derivatives (of R-Ltype) are defined by means of the differ-integral operator:

D(γk ),(δk )(βk ),m

f (z) := Dη I(γk+δk ),(ηk−δk )(βk ),m

f (z) (8)

= Dη

1∫0

Hm,0m,m

[σ

∣∣∣∣∣ (γk + ηk + 1− 1βk, 1βk

)m1(γk + 1− 1

βk, 1βk

)m1

]f (zσ) dσ,

where the auxiliary (integer order) differential operator Dη isdefined as

Dη =

m∏r=1

ηr∏j=1

(1

βrzd

dz+ γr + j

) , (9)

by means of integers ηk =[δk ] + 1, or ηk =δk if integer, k =1,...,m.

And the Caputo-type g.f.d. is: ∗D(γk ),(δk )(βk ),m

= I(γk+δk ),(ηk−δk )(βk ),m

Dη.

10 / 40

Basics of GFC theory

For α ≥ maxk

[−βk(γk + 1)], say in Cα, up to a constant multiplier,

I(γk ),(δk )(βk ),m

zp = cpzp, p > α, where cp =

m∏k=1

Γ(γk + pβk

+ 1)

Γ(γk + δk + pβk

+ 1),

and it is an invertible mapping I(γk ),(δk )(βk ),m

: Cα 7→ C(η1+···+ηm)α ⊂ Cα.

Analogously, under the same conditions, (5) maps the class ofweighted analytic functions Hα(Ω) into itself, preserving the powerfunctions (up to constant multipliers like above) and the image ofa power series has the same radius of convergence.

It is also shown that (5) has a Mellin type convolutionalrepresentation, based on its Mellin image.

Another well expected result, for Lebesgue integrable functions isthe following: (5) is a bounded linear operator from the Banachspace Lα,p(0,∞) into itself:∥∥∥I (γk ),(δk )(βk ),m

f∥∥∥α,p≤ hα,p ‖f ‖α,p, i.e.

∥∥∥I (γk ),(δk )(βk ),mf∥∥∥ ≤ hα,p

with hα,p = ....

11 / 40

The following operational rules confirm that the operators of ourGFC satisfy the axioms of FC:

I(γk ),(δk )(βk ),m

λf (cz)+ηg(cz)=λI(γk ),(δk )(βk ),m

f

(cz)+ηI(γk ),(δk )(βk ),m

g

(cz)

(bilinearity of (5));

I(γ1,...,γs ,γs+1,...,γm),(0,...,0,δs+1,...,δm)(β1,...,βm),m

f (z)= I(γs+1,...,γm)(δs+1,...,δm)(βs+1,...,βm),m−s f (z)

(i.e., if δ1 =δ2 = . . .=δs =0, then the multiplicity reduces to(m−s));

I(γk ),(δk )(βk ),m

zλf (z) = zλI(γk+

λβk

),(δk )

(βk ),mf (z), λ ∈ R

(generalized commutability with power functions);

I(γk ),(δk )(βk ),m

I(τj ),(αj )

(εj ),nf (z) = I

(τj ),(αj )

(εj ),nI(γk ),(δk )(βk ),m

f (z)

(commutability of operators of form (5));

the left-hand side of above = I((γk )

m1 ,(τj )

n1)((δk )

m1 ,(αj )

n1)

((βk )m1 ,(εj )

n1),m+n f (z)

(compositions of m-tuple and n-tuple integrals (5) give(m+n)-tuple integrals of same form);

12 / 40

I(γk+δk ),(σk )(βk ),m

I(γk ),(δk )(βk ),m

f (z)= I(γk ),(σk+δk )(βk ),m

f (z), δk > 0, σk > 0,

(law of indices, product rule or semigroup property);I(γk ),(δk )(βk ),m

−1f (z) = I

(γk+δk ),(−δk )(βk ),m

f (z)

(formal inversion formula).This inversion follows from the above index law for σk =−δk < 0,

k = 1, ...,m and definition for zero multi-order of integration:

I(γk+δk ),(−δk )(βk ),m

I(γk ),(δk )(βk ),m

f (z) = I(γk ),(0,...,0)βk ,m

f (z) = f (z).

But the symbols (5) are not defined for negative multi-orders ofintegration −δk < 0, k = 1, ...,m. The problem was to propose anappropriate meaning for them and so to avoid the appearance ofdivergent integrals, similarly to the classical case when the R-L andE-K integrals have to be inverted by additional differentiation ofsuitable integer order η = [δ] + 1, thus defining the R-L or CaputoFDs. In GFC, the problem was more difficult and resolved by

auxiliary differential operator Dη, polynomial of zd

dz, using a set of

integers η = (η1, ..., ηm) related to δ = (δ1, ..., δm).13 / 40

Some special cases of GFC operators

m = 1: Erdelyi-Kober operators. E-K integrals:

I γ,δβ f (z) =1

Γ(δ)

1∫0

σγ (1− σ)δ−1 f (zσ1β ) dσ, (10)

and E-K type integrals: I f (z) = zδ0 I γ,δβ f (z), δ0 ≥ 0.

E-K derivatives: Dγ,δβ f (z) = Dn I

γ+δ,n−δβ f (z) (11)

=n∏

j=1

(1

βzd

dz+ γ + j

)I γ+δ,n−δβ f (z), n − 1 < δ ≤ n, n ∈ N.

If γ = 0, β = 1, the E-K operators reduce to Riemann-Liouville(R-L) operators:

Rδf (z)=zδ I 0,δ1 f (z), Dδf (z)=z−δD−δ,δ1 f (z); (12)

conversely, I γ,δ1 f (z)=z−γ−δRδzγ f (z).Also, the corresponding Caputo-type derivatives come as special

cases.

14 / 40

Important property: Composition/ decomposition:

I(γk ),(δk )(βk ),m

f (z) =

[m∏

k=1

I γk ,δkβk

]f (z) (13)

=

1∫0

· · ·1∫

0

[m∏

k=1

(1−σk)δk−1σγkkΓ(δk)

]f(z σ

1/β11 . . .σ

1/βmm

)dσ1. . . dσm,

and similarly (sequential derivatives),

D(γk ),(δk )(βk ),m

f (z) = Dγ1,δ1β1

...Dγm,δm

βm

f (z).

m = 2: Hypergeometric fractional integrals with kernel H2,02,2 =

2F1, the Gauss function. We have several examples, like the mostgeneral Hohlov operators F f (z)=2F1(a, b; c ; z) f (z) (inunivalent function theory), and in particular the Saigo operatorswith “our” β1 = β2 = 1:

Iα,β,ηf (z) =z−β

Γ(α)

1∫0

(1− σ)α−1 2F1(α + β,−η;α; 1− σ) f (zσ)dσ.

(14)

15 / 40

But these GFC operators with m = 2 are also compositions of twoE-K operators, or of two weighted R-L integrals:

Iα,β,ηf (z) = z−β I(η−β,0),(−η,α+η)(1,1),2 f (z) = z−β I η−β,−η1 I 0,α+η1 f (z)

= I η,−η1 I β,α+η1 z−βf (z) = I(η,β),(−η,α+η)(1,1),2 z−βf (z) (15)

= R−η z−α−β Rα+ηf (z).

The corresponding Saigo fractional derivative has explicitdiffer-integral expression (Kiryakova, 1994) as

Dα,β,ηf (z) = (d/dz)n Iα+n,β−n,η−nf (z) with n = [−<α] + 1,

It is also a GFC operator and composition of 2 E-K derivatives:

Dα,β,ηf (z) = zβD(η,β),(−η,α+η)(1,1),2 f (z) = zβDη,−η

1 Dβ,α+η1 f (z).

16 / 40

m = 3: Marichev (1974)-Saigo-Maeda (1985, 1996) operators(M-S-M)

Let a, a′, b, b′, c be complex parameters, <c > 0,

I a,a′,b,b′,c f (z)=

z−a

Γ(c)

z∫0

(z−ξ)c−1ξ−a′F3(a, a′, b, b′; c ; 1−ξ

z, 1−z

ξ)f (ξ)dξ

= zc−a−a′

1∫0

(1−σ)c−1

Γ(c)σ−a

′F3(a, a′, b, b′; c; 1−σ, 1− 1

σ) f (zσ)dσ.

(16)The kernel-function, the so-called Appel F3 function (Hornfunction), |z | < 1, |ξ| < 1:

F3(a, a′, b, b′, c , z , ξ

)=∞∑

m,n=0

(a)m(a′)n(b)m(b′)n(c)m+n

zmξn

m!n!,

is however a H3,03,3 , and even the simpler G 3,0

3,3 , the kernel of GFCintegral with m = 3:

G 3,03,3

[σ

∣∣∣∣ ......]

=(1− σ)c−1

Γ(c)F3

(a, a′, b, b′, c , 1− 1

σ, 1− σ

).

17 / 40

Therefore, the M-S-M integrals are GFC integrals with m = 3, thatis, compositions of 3 commutable classical Erdelyi-Kober integrals(see Kiryakova, 1994), a fact seemingly unknown to many otherauthors (or just ignored):

I a,a′,b,b′,c f (z) = zc−a−a

′I(a−a′,b−a′,c−2a′−b′),(b,c−a′−b,a′)(1,1,1),3 f (z)

= zc−a−a′I a−a

′,b1 I b−a

′,c−a′−b1 I c−2a

′−b′,a′1 f (z). (17)

Saigo and Maeda, and authors after them, considered alsogeneralized M-S-M fractional derivatives Da,a′,b,b′,c , and originallydefined analogously to the Saigo derivatives Dα,β,η. But these arespecial cases of 3-tuple generalized fractional derivatives in oursense (8), namely:

Da,a′,b,b′,c f (z) = D(a−a′,b−a′,c−2a′−b′),(b,c−a′−b,a′)(1,1,1),3 z−c f (z).

Then, all properties the other authors derive follow at once fromour GFC theory!

18 / 40

m ≥ 2, arbitrary: the hyper-Bessel operators (Dimovski (1966))the Gel’fond-Leont’ev operators (w.r.t. multi-index M-L function,Kiryakova (1996)), etc.

The hyper-Bessel differential operators are interesting exampleof generalized “fractional” derivatives but of integer multi-order(δ1, δ2, ..., δm) = (1, 1, ..., 1). These are singular differentialoperators of higher integer order m ≥ 1 with variable coefficients

B = zα0d

dzzα1

d

dz· · · zαm−1

d

dzzαm , 0 < z <∞ , (18)

and some real parameters αk , k = 1, ...,m andβ = m − (α0 + α1 + · · ·+ αm) > 0. They have also alternativerepresentations in the following equivalent forms, either as:

B = z−βm∏

k=1

(zd

dz+ βγk

)= z−βQm(z

d

dz), 0 < z <∞, (19)

symmetric with respect to the zeros µk = −βγk , k = 1, . . . ,m ofthe m-th degree polynomial Qm(µ), whereγk = 1

β (αk + · · ·+ αm −m + k), k = 1, ...,m;

19 / 40

or as

B = z−β[zm

dm

dzm+ a1z

m−1 dm−1

dzm−1+ · · ·+ am

].

The latter gives better impression on the nature of the h.-B.differential operators, appearing often in problems of mathematicalphysics, and extending the Bessel differential operator Bν of 2ndorder (m = 2) with γ1,2 = ±ν/2, β = 2.

However, (31) and its linear right inverse integral operator L(hyper-Bessel integral operator) are examples of GFC operators:

B = z−1D(γk−1),(1,1,...,1)(β,β,...,β),m , L = z I

(γk ),(1,1,...,1)(β,β,...,β),m . (20)

In 1968, Dimovski considered also fractional powers of theintegral operator L, namely the operators Lλ, λ > 0. To expressthem, he used the notion of convolution (the basic one in his“Convolutional Calculus”, 1990). He represented the fractionalpowers of L as convolutional products, where (∗) had a verycomplicated form:

Lλf = lλ∗f , with lλ = tβ(λ−δ−1)

m∏k=1

Γ(λ−δ+γk)

, δ ≥ max

kγk . (21)

20 / 40

And it was proved that under this definition, the FC semigroupproperty is satisfied: Lλ Lµ = Lλ+µ, λ > 0, µ > 0, Ln = L · L · · · Lfor n ∈ N.

However, from our point of view of SF, based on theG -functions, we found (1983) a representation similar to its formin (20) (where λ = 1), for example

Lλf (z) =zβλ

(βλ)m

1∫0

Gm,0m,m

[σ

∣∣∣∣ (γk + λ)m1(γk)m1

]f (zσ1/β)dσ (22)

= zβλ I(γk ),(λ,λ,...,λ)(β,β,...,β),m f (z), m-tuple Erdelyi-Kober integral.

Then our hint was to think about: what was to be if we replacethe parameters in the upper row of the above kernel G -function

(γ1 + λ, γ2 + λ, ..., γm + λ) by (γ1 + δ1, γ2 + δ2, ..., γm + δm)

with arbitrary and different δ1 > 0, δ2 > 0, ..., δm > 0 ? This led usto the idea of the operators of fractional integration of multi-order(vector order) (δ1, δ2, ..., δm), whose theory (GFC) was developedin details in Kiryakova (1986, 1988, 1994).

21 / 40

In Kiryakova (1994, Ch.3): fundamental system solutions of theh-B. diff. equation of m-th order (i.e. of multi-order (1, 1, ..., 1))

B y(z) = λ y(z), λ 6= 0, 0 < z <∞, (23)

was found. Namely, for j = 1, ...,m:

yj(z) = G 1,00,m

[−λz

∣∣∣∣ −−−γj ,−γ1, ...,−γj−1,−γj+1, ...,−γm−1, 0

]=

(λt)−γjm∏

k=1

Γ(γk + 1)0Fm−1 ((1 + γi − γj)i 6=j ;λz) (24)

:= J(m−1)(1+γi−γj )i 6=j

(λz), are the hyper-Bessel functions.

These SF are interesting special case of the multi-index M-Lfunctions we shall further consider, as these appear as multi-indexBessel functions !

22 / 40

With respect to relations and use of the kernel H- and G -functionsas pdf-functions in probabilty, let us mention the following:

- For m = 1 and m = 2 these are well known, for example, Gm,0m,m

for σ < 1 are (note always Gm,0m,m(σ) ≡ 0 for |σ| > 1:

G 1,01,1

[σ

∣∣∣∣ γ + δγ

]= (1− σ)δ−1σγ/Γ(δ), the beta distribution !

G 2,02,2

[σ

∣∣∣∣ γ1 + δ1, γ2 + δ2γ1, γ2

]=

(1− σ)δ1+δ2−1σγ2/Γ(δ1 + δ2) 2F1(γ2 + δ2 − γ1, ; δ1 + δ2; 1− σ),

related to the hypergeometric distribution ... !

23 / 40

Although for m > 2 the kernel Gm,0m,m- and Hm,0

m,m- functions are notamong well known but in particular cases can be identified withmany SF used in Math. Physics, Engineering, etc.,

an interesting fact is that for arbitrary m > 1, the Gm,0m,m-functions

were used by Kabe in statistics as density functions of a randomvariable, yet long ago:

D.G. Kabe, Some applications of Meijer’s G -function todistribution problems in statistics, Biometrica 45 (1958), 578-580.

There he also distinguished the cases m = 1 and m = 2, related toE-K and hypergeometric case. We found also other papers by sameauthor, where SF (especially generalized hypergeometric functions)were used, to express the exact distributions from the knownexpressions for the moments, for example:

D.G. Kabe, On the exact distribution of a class of multivariate testcriteria, The Annals of Mathematical Statistics 33, Number 3(1962), 1197-1200.

24 / 40

Other particular cases of GFC operators (R-L type)

– Hardy-Littlewood / Cesaro integral operator (m = 1);– Uspensky integral transform (m = 1);– Bessel and Bessel-Clifford differential operators (m = 2);– Poisson and Sonine integral transforms (m = 1);– Poisson-Sonine-Dimovski transmutation operators (arb. m > 1);– Rusheweyh derivative (m = 1);– Weinstein differential operator (m = 2);– Tricomi / Gelersted differntial operators (arb. m > 1);– Ditkin-Prudnikov and Botashev differential operators (arb.m > 1):

d

dzzd

dzz ...

d

dzzd

dz;

– Kraetzel differential operator (arb. m > 1);– In GFT: the operators of Biernacki, Komatu, Libera,Owa-Srivastava, Obradovic, Carlson and Shaffer, Alexander,Hohlov, etc.

25 / 40

Mittag-Leffler type functions – specific SF of FC.

Mittag-Leffler type functions, α > 0, β > 0, γ ∈ R

Eα(z) =∞∑k=0

zk

Γ(αk + 1), Eα,β(z) =

∞∑k=0

zk

Γ(αk + β)=1Ψ1(z),

Eγα,β(z) =∞∑k=0

(γ)k zk

k! Γ(αk + β)(Prabhakar f .,

(25)are “fractional index” (α > 0) extensions of the exponential andtrigonometric functions exp(z) and cos z (when α = 1, resp.α = 2), satisfying ODEs Dny(λz) = λny(λz) of order n = 1, 2.

However, the classical M-L functions (25) and the recentlyintroduced multi-index (vector index) M-L functions (Kiryakova,Luchko et al., Kilbas) as their extensions, appear in the solutions offractional order problems (of order α, or multi-order (αi )

m1 , m ≥ 1):

E(αi ),(βi )(z) :=E(m)(αi ),(βi )

(z)=∞∑k=0

zk

Γ(α1k+β1) . . . Γ(αmk+βm)

(26)=1Ψm

[(1, 1)

(βi , αi )m1

∣∣∣∣ z]=H1,11,m+1

[−z∣∣∣∣ (0, 1)

(0, 1), (1−βi , αi )m1

].

26 / 40

Many of the elementary and special functions are particular casesof Eα,β, and much more – of the multi-index M-L functions(31)-(32).

Examples of the M-L functions:

• α > 0, β = 1: E0,1(z) =1

1− z; E1,1(z) = exp(z) ;

E2,1(z2) = coshz , E2,1(−z2) = cosz ; E1/2,1(z1/2) =

exp(z)[1 + erf(z1/2)

]= exp(z) erfc(−z1/2) =

exp(z)

[1 +

1√πγ(

1

2, z)

](the error functions, or incomplete

gamma functions);

• β 6= 1: E1,2(z) =ez − 1

z; E1/2,2(z) =

sh√z

z;

E2,2(z) =sh√z√z

, etc.

• β = α: the α-exponential (Rabotnov) functionyα(z) = zα−1Eα,α(λzα), α > 0.

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Examples of multi-index M-L functions E(αi ),(βi )(z)• For m = 1 - the classical M-L function Eα,β(z) with all itsspecial cases (including also the so-called Rabotnov function, orα-exponential function yα(z) = zα−1Eα,α(λz).

• For m = 2, the function (31) denoted as Φρ1,ρ2(z ;µ1, µ2) =

E(2)

( 1ρ 1, 1ρ 2

),(µ1,µ2)(z), is Dzrbashjan’s M-L type function (1960).

Some of its particular cases are:the M-L function Eα,β(z);the Bessel function Jν(z)=(z/2)νE(1,1),(ν+1,1)(−z2/4);Struve and Lommel functions sµ,ν(z), Hν(z) = const.sν,ν(z).

The so-called Wright function from the studies of Fox (1928),Wright (1933), Humbert and Agarwal (1953), extended also forα > −1, happens is a case of multi-M-L function with m = 2:

ϕ(α, β; z)=∞∑k=0

1

Γ(αk + β)

zk

k!=0Ψ1

[−

(β, α)

∣∣∣∣ z]=E(2)(α,1),(β,1)(z).

(27)

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This Wright f. plays important role in the solution of linear PFDEsas the fractional diffusion-wave equation studied by Nigmatullin(1984-1986, to describe the diffusion process in media with fractalgeometry, 0 < α < 1) and by Mainardi et al. (1994-), forpropagation of mechanical diffusive waves in viscoelastic media,1 < α < 2). In the form M(z ;β) = ϕ(−β, 1− β;−z), β := α/2, itis recently called also as the Mainardi function. In our denotations,

it is: M(z ;β) = E(2)(−β,1),(1−β)(−z) and has examples like:

M(z ; 1/2) = 1/√π exp(−z2/4) and the Airy function:

M(z ; 1/3) = 32/3 Ai(z/31/3).In other form and denotation, the Wright function is known as

Wright-Bessel, or misnamed as Bessel-Maitland function:

Jµν (z) = ϕ(µ, ν+1;−z) = 0Ψ1

[−

(ν + 1, µ)

∣∣∣∣− z

]=∞∑k=0

(−z)k

Γ(ν + kµ+ 1) k!= E

(2)(1/µ,1),(ν+1,1)(−z) , (28)

again an example of the Dzrbashjan function. It is an obvious”fractional index” analogue of the classical Bessel function.

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Nowadays, several further “fractional-indices” generalizations ofJν(z) are exploited, and we can present them as multi-M-Lfunctions! Such one is the so-called generalizedWright-Bessel (-Lommel) functions, due to Pathak (1966-1967):

Jµν,λ(z) = (z/2)ν+2λ∞∑k=0

(−1)k(z/2)2k

Γ(ν+kµ+λ+1)Γ(λ+k+1)

(29)= (z/2)ν+2λ E

(2)(1/µ,1),(ν+λ+1,λ+1)

(−(z/2)2

), µ > 0,

including, for µ = 1, the Lommel (and thus, also the Struve)functions J1ν,λ(z) = constS2λ+ν−1,ν(z). Next one, is thegeneralized Lommel-Wright function with 4 indices, introduced byde Oteiza, Kalla and Conde, r > 0, n ∈ N, ν, λ ∈ C:

J r ,nν,λ(z) = (z/2)ν+2λ∞∑k=0

(−1)k(z/2)k

Γ(ν+kr+λ+1)Γ(λ+k+1)n

(30)= (z/2)ν+2λ E

(n+1)(1/r ,1,...,1),(ν+λ+1,λ+1,...,λ+1)

(−(z/2)2

).

This is an interesting example of a multi-M-L function withm = n + 1.

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Other special cases:• For arbitrary m ≥ 2: let ∀ρi =∞ (1/ρi = 0) and ∀µi = 1,i = 1, ...,m. Then, from definition of (31),

E(0,0,...,0),(1,1,...,1)(z) =∞∑k=0

zk =1

1− z.

• Consider the case m ≥ 2, ∀ρi = 1, i = 1, . . . ,m, when:

E(m)(1,1,...,1),(µi+1)(z)=1Ψm

[(1, 1)

(µi , 1)m1

∣∣∣∣ z]=const 1Fm (1;µ1, µ2, ..., µm; z)

reduces to 1Fm- and to a Meijer’s G 1,11,m+1-function. Denote

µi = γi +1, i =1, . . . ,m, and let additionally one of the µi to be 1,e.g.: µm =1, i.e. γm =0. Then the multi-M-L function becomes ahyper-Bessel function (29) (mentioned in earlier slide):

J(m−1)γi ,...,γm−1(z)=

( z

m

)m−1∑i=1

γiE(m)(1,1,...,1),(γ1+1,γ2+1,...,γm−1+1,1)

(−(

z

m)m).

(31)In view of the above relation, the multi-index M-L functions witharbitrary (α1, . . . , αm) 6= (1, . . . , 1) can be seen asfractional-indices analogues of the hyper-Bessel functions.

31 / 40

• One may consider multi-index analogues of the Rabotnov(α-exponential function), with all µi = 1/ρi = α > 0, i = 1, . . . ,m:

y (m)α (z)=zα−1E

(m)(α,...,α),(α,...,α)(z

α)=zα−1∞∑k=0

zαk

[Γ(α+αk)]m, (32)

α = 1 :∞∑k=0

zk

[k!]m.

• In general, for rational values of ∀αi , i = 1, . . . ,m, themulti-index M-L functions (31) are reducible to Meijer’sG -functions (that is, to classical special functions).

Further extensions of the multi-M-L functions (31), are:

• The multivariate M-L type functions by Luchko:

E(a1,...,an),b(z1,..., zn) :=∞∑k=0

∑l1+···+ln=kl1≥0,...,ln≥0

(k ; l1, . . . , ln)

∏ni=1 z

lii

Γ(b +∑n

i=1 ai li ),

where (k; l1, . . . , ln) are the so-called multinomial coefficients).• The M-L type generalization by Kilbas-Srivastava-Trujillo:

Eρ((αj , βj)1,m; z) =∞∑k=0

(ρ)km∏j=1

Γ(αjk + βj)

zk

k!.

32 / 40

Basic theory of the multi-index M-L functions (31), seeKiryakova 1999, 2000, 2002, 2010:

Theorem. The multi-index M-L functions (31) are entire functions

of order ρ with1

ρ=

1

ρ1+ · · ·+ 1

ρm, and type

σ =

(ρ1ρ

) ρρ1

. . .

(ρmρ

) ρρm

> 1 (for m > 1). Setting

µ = µ1 + · · ·+ µm, we have an asymptotic estimate, as:

|E(αi ),(βi )(z)| ≤ C |z |ρ((1/2)+µ−(m/2)) exp (σ|z |ρ) , |z | → ∞.

We have shown also that the multi-index ML functions areeigen-functions of these G-L operators of multi-order (α1, ..., αm):

D(αi ),(βi ) E(αi ),(βi )(λz) = λE(αi ),(βi )(λz), λ 6= 0.

where D(αi ),(βi ) is a generalized fractional differentiation operator

of the form z−1D(βi−αi−1),(αi )(1/αi ),m

and of multi-order (1/α1, ..., 1/αm).

33 / 40

Special Functions of Fractional Calculusas GFC-operators of basic functions

In several our works (Kiryakova, 1994, 1997, 2010, etc.) we haveshown the following basic proposition, first for the classical SF,because all of them practically are pFq-functions.

Theorem. All the generalized hypergeometric functions pFq,that is all the classical special functions, can be considered as

generalized (q-multiple) fractional integrals (11): I(γk ,δk )(β,...,β),m, or /

and their respective generalized fractional derivatives D(γk ),(δk )(β,...,β),m,

of one of the following 3 basic elementary functions, dependingon whether p < q, p = q or p = q + 1, resp.:

cosq−p+1(z) (if p<q) , zα exp z (if p=q) , zα (1−z)β (if p=q+1).(37)

Note that these 3 basic functions are also closely related topdf-functions of known distributions.

34 / 40

Here, the cosm-function is the generalized cosine function of m-thorder :

y(z) = cosm(z) =∞∑k=0

(−1)k zkm

(km)!,

which is the solution of the m-th order IVP

y (m)(z) = −y(z); y(0) = 1, y′(0) = · · · = y (m−1)(0) = 0.

When m = 2, naturally we have the classical cos(z) = cos2(z).Then the corresponding result for the hyper-Bessel functionspresents our a generalization of the Poisson integral for the Besselfunctions.

Similar kind of results are also proven for the Special Functionsof Fractional Calculus, namely for arbitrary Wright generalizedhypergeometric functions pΨq, by using the operators of GFC with

Hm,0m,m-kernel function, see Kiryakova (2010, Computers and Math.

with Appl.)

35 / 40

Based on this mentioned result, we have introduced the followingclassifications of the classical SF pFq, as well as of the SF ofFC, represented generally by pΨq:

• If p < q, these functions form the class of so-called“trigonometric”, or “Bessel”-type SF functions, starting from 0F1,cos(z) etc.• If p = q, these are the so-called “confluent”-type SF, withsimplest example 1F1, exp(Z ), etc.• If p = q + 1, these are the so-called “hypergeometric”-type SF,with simplest example 2F1 (and the geometric series 1/1−z).

The suitability of such possible classification can be in easiercomprehension and interpretation of the SF as somewhat similar tothe basic mentioned functions. Note the fractional order integralsmay preserve in great extent the asymptotic (in general) and othercharacteristic behaviour of the treated special or elementaryfunction, so this can be helpful for applied scientists to have ideaabout.

36 / 40

Images of SF under operators of FC and GFC

Recently many authors are spending lot of time and efforts on thetask to evaluate various operators of FC of quite particular SF or ofclasses of SF. The list of such works, that can be considered also asevaluation improper integrals of products of some elementary andspecial functions, is rather long and is yet growing daily. Since thespecial functions present indeed a great variety, and the operatorsof fractional calculus do as well, the mentioned job produces ahuge flood of publications. Most of them use same formal andstandard procedures, and besides, often the results sound not ofpractical use, with except to increase authors’ publication activities.

Our approach, based on the ideas of GFC provides a simple andunified way to do such task at once, in the rather general case:both for operators of generalized fractional calculus and for thegeneralized hypergeometric functions pΨq (as the general SF ofFC). Thus, great part of the results in the mentioned publicationsare well predicted and fall just as particular cases of our generalscheme.

37 / 40

See for example,V. Kiryakova, Fractional calculus operators of special functions?The result is well predictable! Chaos, Solitons and Fractals 102(2017), 2-15.and other 3 recent publications (AIP Conf. Proc. 2017, and toappear: AIP Conf. Proc. 2019, Front. Phys. 2019). There wehave shown that a very large number of papers on the subjectinclude ONLY specification of the results below.

Our extension for E-K operators of Wright g.h.f. pΨq

Lemma. For <δ > 0, <γ > −1, µ > 0, λ 6= 0, and if p = q + 1,we require |λzµ| < 1:

I γ,δβ

pΨq

[(a1,A1), . . . , (ap,Ap)(b1,B1), . . . , (bq,Bq)

∣∣∣∣λzµ]= p+1Ψq+1

[(ai ,Ai )

p1 , (γ + 1, 1/β)

(bj ,Bj)q1 , (γ + δ + 1, 1/β)

∣∣∣∣λzµ] . (33)

In particular, Rδzν−1pΨq(λzµ)

= zν+δp+1Ψq+1(λzµ).

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Results for the GFC operators of SF

Because, by m-steps: pΨq 7→ p+1Ψq+1 7→ p+2Ψq+2 7→ ..., wehave:

Theorem. For m-tuple (m = 1, 2, 3, ...) GFC integral withδk ≥ 0, γk > −1, k = 1, ...,m, and µ > 0, λ 6= 0, c ∈ R,

I(γk ),(δk )(βk ),m

zc pΨq

[(a1,A1), . . . , (ap,Ap)(b1,B1), . . . , (bq,Bq)

∣∣∣∣λzµ]= zc p+mΨq+m

[(ai ,Ai )

p1 , (γi + 1 + c

βi, µβi )

m1

(bj ,Bj)q1 , (γi + δi + 1 + c

βi, µβi )

m1

∣∣∣∣∣λzµ]. (34)

Corollary. When all βk =β=1, k =1, ...,m, the image of a pFqis another function of the same kind with indices increased by themultiplicity m:

I(γk ),(δk )1,m pFq (a1, ..., ap; b1, ..., bq;λz)

=p+mFq+m (a1, ..., ap, (γi +1)m1 ; (b1, ..., bq, (γi +δi +1)m1 ;λz) . (35)

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Few basic references: (V. Kiryakova)[1] Generalized Fractional Calculus and Applications, Longman–J. Wiley,Harlow–N. York, 1994;

[2] All the special functions are fractional differintegrals of elementaryfunctions. J. Phys. A: Math. & General 30 (1997), 5085-5103,doi:10.1088/0305-4470/30/14/019;

[3] Multiple (multiindex) Mittag-Leffler functions and relations to generalizedfractional calculus. J. Comput. Appl. Math. 118 (2000), 241–259,doi:10.1016/S0377-0427(00)00292-2;

[4] The special functions of fractional calculus as generalized fractional calculusoperators of some basic functions. Computers and Math. with Appl. 59(2010), 1128-1141, doi:10.1016/j.camwa.2009.05.014;

[5] The multi-index Mittag-Leffler functions as an important class of specialfunctions of fractional calculus. Computers and Math. with Appl. 59 (2010),1885-1895, doi:10.1016/j.camwa.2009.08.025;

[6] Fractional calculus operators of special functions?–The result is wellpredictable!. Chaos Solitons and Fractals 102 (2017), 2-15,doi:10.1016/j.chaos.2017.03.006.

[7] Use of fractional calculus to evaluate some improper integrals of specialfunctions. AIP Conf. Proc. 1910 (2017), doi:10.1063/1.5013994.

[8] Commentary: “A remark on the fractional integral operators and the imageformulas of generalized Lommel-Wright function”, Front. Phys. (2019), Toappear, doi: 10.3389/fphy.2019.00145. 40 / 40