Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators

Cent. Eur. J. Phys. () 23 34-336 DOI:.2478/s534-3-27- Central European Journal of Physics Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators Research Article Virginia Kiryakova, Yuri Luchko 2 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bontchev Str., Block 8, 3 Sofia, Bulgaria 2 Department of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences, Luemburger Str., D 3353 Berlin, Germany Received 3 January 23; accepted 25 March 23 Abstract: In this paper some generalized operators of Fractional Calculus (FC) are investigated that are useful in modeling various phenomena and systems in the natural and human sciences, including physics, engineering, chemistry, control theory, etc., by means of fractional order (FO) differential equations. We start, as a background, with an overview of the Riemann-Liouville and Caputo derivatives and the Erdélyi-Kober operators. Then the multiple Erdélyi-Kober fractional integrals and derivatives of R-L type of multi-order (δ,..., δ m ) are introduced as their generalizations. Further, we define and investigate in detail the Caputotype multiple Erdélyi-Kober derivatives. Several eamples and both known and new applications of the FC operators introduced in this paper are discussed. In particular, the hyper-bessel differential operators of arbitrary order m > are shown as their cases of integer multi-order. The role of the so-called special functions of FC is emphasized both as kernel-functions and solutions of related FO differential equations. PACS (28): 2.3.Gp, 2.3.Hq, 2.3.Jr, 2.3.Uu Keywords: fractional calculus operators of Riemann-Liouville and Caputo type Erdélyi-Kober operators special functions integral transforms Cauchy problems Versita sp. z o.o.. Introduction During the last few decades, both ordinary and partial differential equations of fractional order have begun to play an important role in modeling of many phenomena and systems, especially in the natural sciences, medicine, and engineering. We refer e.g. to 3 for different applica- E-mail: virginia@diogenes.bg (Corresponding author) E-mail: luchko@beuth-hochschule.de tions of the integrals and derivatives of the fractional order in physics, chemistry, engineering, self-similar processes, astrophysics, etc. and to 4 for applications of the fractional differential equations in classical mechanics, quantum mechanics, nuclear physics, hadron spectroscopy, and quantum field theory. For other models in the form of fractional differential equations we refer the reader to 5 2, and 3 to mention only a few of many recent publications. In this paper, we first give an overview of the most used operators of the fractional integration and differentiation and their properties as well as of the suitable functional spaces and the related special functions. Then we in- 34

Virginia Kiryakova, Yuri Luchko troduce the new fractional differential operators of the Caputo type and study them in detail. These fractional derivatives are in fact some generalizations of the Caputo fractional derivative that nowadays plays a very important role in applications of the fractional calculus, thus we hope that these new operators will be useful for applications, too. Finally, we mention some particular cases of our new operators and their applications... Classical FC operators In Fractional Calculus (FC), the most often used definition of integration of an arbitrary (fractional) order comes from the etension of the known formula for the n-fold integration and is known as the Riemann-Liouville (R-L) operator of fractional integration (left-hand sided) of order δ, defined by the formula I δ f() = I δ + f() := D δ f() = Γ(δ) ( t) δ f(t)dt, I f() := f(). () The corresponding R-L fractional derivative of order δ is defined as a composition of the n-th order derivative (n <δ n, n N) and the fractional order integral () if δ >, and as the identity operator if δ = : ( ) n d D δ f() = D+ δ := Dn I n δ f(t) dt f() = d Γ(n δ) ( t) δ n+, D f() := f(). (2) For other kinds of definitions of fractional integrals and derivatives we refer the readers to the books on FC, as Samko- Kilbas-Marichev 4, Podlubny 5, Kilbas-Srivastava-Trujillo 6, Baleanu-Diethelm-Scalas-Trujillo 7, etc. The R-L operators () and (2) are defined in such a way that some basic aioms including the semigroup property I δ I σ = I δ+σ, δ, σ, the inversion property D δ I δ = Id, δ, and the coincidence of I n and D n with the n-fold integration and differentiation, respectively, are satisfied. To compare with the case of classical Calculus, having in mind the FC formula D δ { p Γ(p + ) } = Γ(p δ + ) p δ, δ >, p >, (3) we obtain for p = the strange" (for classical analysts) fact that D δ {const} = const δ, unless if δ = n =, 2, 3,... Γ( δ) To avoid this seeming controversy and other difficulties when considering initial value problems for the fractional order differential equations, another alternative definition of the fractional order derivative is also considered. This derivative is usually named after Caputo, who applied such operators in seismographic waves, see e.g. 8. The Caputo derivative is defined by the formula: D δ f() := I n δ D n f() = Γ(n δ) f (n) (t) dt, n < δ n. ( t) δ n+ (4) Since in general D n I n α I n α D n, the two definitions (2) and (4) are different. When the passage of the n-th derivative in (2) under the sign of integral in (4) is legitimate, we get n D δ f() = D δ f() + f (j) jδ (+) Γ(j δ + ). (5) j= Usually, the R-L definition (2) is preferred in mathematically oriented papers, while the Caputo one (4) is more restrictive, but allows consideration of the conventional initial conditions that are epressed in terms of integer order derivatives 35

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators with well known physical interpretations. For eample, the Laplace transform of the Caputo derivative is given by the formula n L{ D δ f(); s} = s δ L{f(); s} f (k) (+) s n k, n < δ n. (6) k= Moreover, like in the classical Calculus, the Caputo derivative of a constant is zero: D δ {const} =. These properties and other parallels and distinctions for the R-L and Caputo derivatives, are discussed in details, e.g. in 5 7, 9. Comparing the R-L and the Caputo fractional derivatives, we need to attract the attention of analysts and applied scientists to a new definition recently introduced by Hilfer, see e.g. in, Ch.2. This is the so-called generalized R-L fractional derivative or order α and type, defined for α >, and n < α n N, as follows: D α, f() := I (n α) d n d n I( )(n α) f(). (7) The additional type allows one to interpolate continuously from the R-L derivative D α, D α to the Caputo derivative D α, D α. A master equation with fractional time derivative of the form (7) was used to model continuous random walks and the M-L function appeared there as the waiting time density, see Hilfer, Ch.2. In this paper, we deal with the Erdélyi-Kober (E-K) fractional integrals and derivatives that are introduced in the net section and focus on the R-L and Caputo variants of the E-K fractional derivatives. Net, we consider the compositions of a finite number of such operators known as the multiple E-K fractional integrals and derivatives. Finally we introduce the Caputo-type modifications of the multiple E-K derivatives and study the relations between all these E-K type operators of the generalized FC. A natural open problem remains to study multiple analogues of the Hilfer derivative (7)..2. Basic functional spaces In this paper, we basically work in the weighted spaces C µ (, )) of continuous or smooth functions of the real variable > introduced by Dimovski in 2 (see also 2 and 22). Definition. The space of functions C µ, µ R consists of the functions f(), > that can be represented in the form f() = q f() with q > µ and continuous f C(, )). The functional space C n µ, n N consists of the functions f(), > representable in the form f() = q f() with q > µ and f C n (, )). For the properties of these spaces see e.g. 2 or 22. Evidently, Cµ n Cν m, for µ ν, n m. However, similar constructions and results are also valid for the weighted spaces Lµ(, p ), p < of Lebesgue integrable functions with the norm f µ,p = µ f() p d /p < and for the spaces Hµ (Ω) of analytic functions with power weights in the comple z-plane or in disks { z < R} or in domains Ω that are starlike with respect to the origin (see 2)..3. Some FC special functions Let us recall the definitions of the basic special functions used in FC and in particular, in this paper. These are etensions of the classical Special Functions (SF) with fractional (in fact, arbitrary) indices. For details regarding the SF of FC we refer e.g. to the handbooks by Prudnikov-Brychkov-Marichev 23, Srivastava and Buschmann 24, Kilbas-Srivastava-Trujillo 6, and Mainardi 2. Definition 2. The Fo H-function is defined by means of the Mellin-Barnes type contour integral Hp,q m,n σ (a j, A j ) p (b k, B k ) q = H m,n p,q (s) σ s ds, σ, (8) 2πi L 36

Virginia Kiryakova, Yuri Luchko where L is a suitable contour in C, the orders (m, n, p, q) are non negative integers so that m q, n p, the parameters A j >, B k > are positive, and a j, b k, j =,..., p; k =,..., q can be arbitrary comple numbers such that A j (b k + l) B k (a j l ), l, l =,, 2,... ; j =,..., n; k =,..., m, and the integrand in (8) has the form H m,n p,q (s) = m Γ(b k + B k s) n Γ( a j A j s) q k=m+ j= Γ( b k B k s) p j=n+. (9) Γ(a j + A j s) Depending on the parameters, the H-function is an analytic function of σ in a circle domain σ < ρ or in some sectors of this domain or in the whole comple plane C (for the details, see the books mentioned above). When A = = A p =, B = = B q =, (8) reduces to the Meijer G-function G m,n p,q, whose theory is represented also in the classical handbook by A. Erdélyi et al. 25. A typical eample of a special function that is a particular case of the Fo H-function but is not reducible to the Meijer G-function (as a classical SF) in the case of irrational A j, B k is as follows. Definition 3. The Wright generalized hypergeometric function p Ψ q (σ), called also the Fo-Wright function, is defined by the series (see 6, 23): (a, A ),..., (a p, A p ) pψ q (b, B ),..., (b q, B q ) σ = Γ(a + ka )... Γ(a p + ka p ) σ k () Γ(b + kb )... Γ(b q + kb q ) k! = H,p p,q+ σ k= ( a, A ),..., ( a p, A p ) (, ), ( b, B ),..., ( b q, B q ). () Obviously, for A = = A p =, B = = B q =, () becomes a generalized hypergeometric p F q -function (see 25): pf q (a,..., a p ; b,..., b q ; σ) = Γ (a, ),..., (a p, ) pψ q (b, ),..., (b q, ) σ, q p (2) Γ = Γ(b j )/ Γ(a i ). One of the most popular SFs of FC is the following. Definition 4. The Mittag-Leffler function, entitled as the queen -function of FC, is defined by the series E α, (σ) = k= σ k (, ) Γ(αk + ) = Ψ (, α) j= i= σ, E α (σ) := E α, (σ), α >, R. (3) It is an entire function of order ρ = /α and type. The Mittag-Leffler function is an obvious etension of the eponential, trigonometric, hyperbolic, error, and incomplete gamma functions. For the details we refer the reader e.g. to 5, 6, and 2. Another important particular case of the Wright generalized hypergeometric function is the multi-inde Mittag-Leffler function that is a generalization of the Mittag-Leffler function. Definition 5. Let m > be an integer, α,..., α m > and,..., m be arbitrary real numbers. The multi-inde Mittag-Leffler function (vector inde M-L function) is defined as follows (Kiryakova 26, 27, Luchko et al. 22, 28): E (αi ),( i )(σ) := E (m) (α i ),( i ) (σ) = σ k Γ(α k + )... Γ(α m k + m ) = k= 37

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators Ψ m (, ) ( i, α i ) m σ = H,,m+ σ (, ) (, ), ( i, α i ) m. (4) As proved in Kiryakova 29, it is an entire function of order ρ = (α + + α m ) and type ( ) α ρ ( ) αmρ... > if m >. α ρ α m ρ The multi-inde Mittag-Leffler function (4) plays a very important role in FC and has been applied e.g. in operational calculus for the multiple Erdélyi-Kober fractional integration and differentiation operators, for solving IVPs involving these operators in 3 and 22, and for deriving the scale-invariant solutions of some PDEs of fractional order in 3 33, and 34. Analytical properties of the multi-inde Mittag-Leffler function were studied in 27, 29, and 35. Many particular cases of this function that appear as solutions of some fractional order integral and differential equations can be found in 35 and 36. Special attention has been given to the case m = 2 that is related to the weighted compositions of two R-L or E-K operators. Two prominent eamples of this kind are the generalized Wright function φ(µ, ν + ; σ) = Ψ (ν +, µ) σ and the Dzrbashjan M-L type function (37) with 2 2 arbitrary parameters. 2. Erdélyi-Kober fractional integrals and derivatives Definition 6. The (left-hand sided) Erdélyi-Kober (E-K) fractional integral of the order δ > is defined by I γ,δ f() = Γ(δ) (γ+δ) ( t ) δ t (γ+) f(t)dt = Γ(δ) ( σ) δ σ γ f(σ / )dσ, (5) where > and γ R. For δ =, the E-K fractional integral is defined as the identity operator. Evidently, for γ =, =, (5) reduces to the R-L integral () with a power weight. There is also a right-hand sided E-K fractional integral, although here we concentrate only on the left-hand sided one. These E-K operators have been used by many authors, in particular, to obtain solutions of single, dual, and triple integral equations involving special functions of mathematical physics as their kernels. For the theory and applications of the Erdélyi-Kober fractional integrals, see Sneddon 38, also e.g. 4, 6, 2, 22, 28, 39 46, to mention only few of many relevant publications. In particular, in 2, 4 the following important properties of the operator (5) have been proved: The Mellin integral transform (see e.g. 47, 48) I γ,δ λ f() = λ I γ+λ,δ f(), (6) I γ,δ I γ,δ Iγ+δ,α f() = I γ,δ+α f(), (7) Iα,η f() = Iα,η Iγ,δ f(). (8) M {f(); s} := + of the E-K fractional integral is given by the formula (6, 2, 22) f(t) t s dt (9) { } M I γ,δ f(); (s) Γ( + γ s/) = M{f(); s}. (2) Γ( + γ + δ s/) 38

Virginia Kiryakova, Yuri Luchko In the case =, this formula was proved by Kober in 49. Consider the functional space C µ (Def. ) with the inde µ (γ + ). Then the Erdélyi-Kober fractional integration operator (5) is a linear map of the space C µ into itself, i.e., I γ,δ : C µ C µ. (2) For a proof, see e.g. Luchko and Trujillo 43. The corresponding fractional differentiation operator, as analogue of the R-L derivative (2), can be defined by means of an auiliary differential operator D n that is a polynomial of the Euler differential operator d as follows (see 2, 3, d and 22): Definition 7. Let n < δ n, n N. The differ-integral operator D γ,δ f() := D n I γ+δ,n δ f() = n j= d ) d + γ + j I γ+δ,n δ f() (22) is called the (left-hand sided) Erdélyi-Kober (E-K) fractional derivative of order δ of the Riemann-Lioville type. The following result holds true (see e.g. Kiryakova 2 or Luchko and Trujillo 43). Lemma 8. For the functions from the space C µ, µ (γ + ), the E-K fractional derivative (22) is a left-inverse operator to the E-K fractional integral (5), i.e., D γ,δ I γ,δ f() f(), f C µ. (23) As a matter of fact, in terms of the spaces from Def., the more precise mappings are as follows: C µ I γ,δ C n µ D γ,δ C µ and C n µ D γ,δ C µ I γ,δ C n µ. (24) In the general case, the E-K fractional derivative is not a right-inverse operator to the E-K fractional integration operator (5). The following result was derived in Luchko and Trujillo 43. Theorem 9. Let n < δ n, n N, µ (γ + ) and f Cµ n. Then the following relation between the R-L type E-K fractional derivative and the E-K fractional integral of order δ holds true: n f() = f() c k (+γ+k), (25) with the coefficients c k = I γ,δ D γ,δ Γ(n k) Γ(δ k) lim k= (+γ+k) n d ) d + γ + i + i=k+ I γ+δ,n δ f(). (26) The formula (25) is a more complicated analogue of the corresponding formula for the R-L fractional integral () and derivative (2) (see e.g. 6 or 4): I δ + D δ +f() = f() n k= δ k Γ(δ k) lim (D δ k + f)(). (27) The epressions lim (D δ k + f)(), k =,..., n, whose physical interpretation is unknown, appear also as initial conditions in the Cauchy problems for the fractional differential equations with the R-L fractional derivatives. This is one of the 39

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators reasons, especially for modeling of applied and physically consistent problems, to introduce the Caputo fractional derivative (4), for which the Laplace transform epression (6) and the Cauchy problems include the reasonable initial values f(),..., f (n ) (). For the Caputo derivative, the relation corresponding to (27) reads as (5, 6): n I+ δ D+f() δ k = f() k! lim f (k) (). (28) In Luchko and Trujillo 43, a Caputo-type modification of the E-K fractional derivative is introduced and studied in analogy with the Caputo derivative (4). k= Definition. Let n < δ n, n N. The integro-differential operator D γ,δ f() := Iγ+δ,n δ D n f() = I γ+δ,n δ n j= d ) d + γ + j f() (29) is called the (left-hand sided) Caputo-type modification of the Erdélyi-Kober fractional derivative of order δ. For a composition of the E-K fractional integral and the Caputo-type E-K fractional derivative, the following result was derived in 43: Theorem. Let n < δ n, n N, µ (γ + δ + ) and f Cµ n. Then the following relation between the Caputo-type Erdélyi-Kober fractional derivative (29) and the Erdélyi-Kober fractional integral (5) of order δ holds true: n f() = f() p k (+γ+k), (3) I γ,δ with the coefficients D γ,δ p k = lim (+γ+k) k= n i=k+ including only ordinary derivatives of the function f with some weights. d ) d + γ + i + f(), (3) In the same paper 43, the conditions under which the Caputo-type and the R-L-type E-K derivatives coincide on C n µ, µ (γ + δ + ) were also derived. Theorem 4.2 from 43 reads that the Caputo-type and the R-L-type E-K derivatives coincide if and only if the coefficients c k and p k are equal for all k =,,..., n. Moreover, Theorem 4.3 from 43 states that the Caputo-type E-K derivative is a left-inverse to the E-K integral: ( D γ,δ I γ,δ f)() f(), f C µ, µ (γ + ). (32) 3. Multiple Erdélyi-Kober fractional integrals Since the 96s, various generalizations of the R-L and E-K operators with some special functions like the Gauss hypergeometric function 2 F, the Bessel function J ν, and the G- and H-functions in the kernel have been considered. These generalized operators of FC can be often written in the form proposed by Kalla (see 5) If() = γ Φ( t ) tγ f(t)dt, (33) where the kernel Φ is an arbitrary continuous function such that the above integral makes sense in sufficiently large functional spaces. Some of their operational properties analogous to the aioms of the classical FC have been studied. In the general case, the kernel Φ is an arbitrary Fo H-function, i.e., Φ(σ) = H m,n p,q and details see e.g. Kiryakova 2. a( t )r (a j, A j ) p (b k, B k ) q. For references 32

Virginia Kiryakova, Yuri Luchko To insure that the transform of type (33) possesses some nice properties, the kernel Φ has to be suitably chosen. For eample, in Samko-Kilbas-Marichev 4, Sect. power-weighted compositions of a finite number of the R-L integrals, i.e., compositions of E-K integrals (5) are considered and represented in the form (33). For these compositions, the kernels are the Fo H-functions of the form H m, (or Hm+n,m+n m,n if compositions of both left- and right-hand sided operators are taken). In Kiryakova 2, a theory of Generalized Fractional Calculus has been developed that deals with generalized fractional integrals and derivatives defined as compositions of the E-K operators that can be represented in the form (33) with the kernel-functions G m, and H. m, The developed theory has been applied in 2 to different topics like special functions, integral transforms, operational calculus, classes of integral and differential equations, and geometric function theory. Definition 2. Let m be an integer and δ k, γ k, >, k =,..., m be arbitrary real numbers. Consider δ = (δ,..., δ m ) as a multi-order of integration, γ = (γ,..., γ m ) as a multi-weight, and = (,..., m ) as an additional multi-parameter. The m integral operator defined for δ k > by the formula ( ),m f() := = H m, σ (γ k + δ k + /, / ) m (γ k + /, / ) m f(σ) dσ (34) t H m, (γ k + δ k + /, / ) m (γ k + /, / ) m f(t)dt (34 ) is said to be a generalized (multiple, m-tuple) Erdélyi-Kober (E-K) operator of integration of the multi-order δ. For δ =... = δ m =, the operator is defined as the identity operator. The operator of the form If(z) = δ ( ),m f() with an arbitrary δ, (35) is then called a generalized (m-tuple) operator of fractional integration of the Riemann-Liouville type, or briefly a generalized (R-L) fractional integral (see 2, Ch.5). In the case of equal s, i.e., when = >, k =,..., m, the multiple E-K fractional integral (34) has a simpler representation by means of the Meijer G-function of the form (see 2, Ch.) (,...,),m f() := I(γ k ),(δ k ),m f() = G m, σ (γ k + δ k ) m (γ k ) m f(σ ) dσ. (36) Some basic eamples: Evidently, for m = we get the classical (single) E-K fractional integral, since the kernel function in (34), (36) becomes H,, σ γ + δ + /, / γ + /, / = σ G,, σ γ + δ γ = σ ( σ ) δ Γ(δ) σ γ, that is the kernel of (5). For m = 2, the operators (34), (36) are the so-called hypergeometric fractional integrals (see e.g. 5, 2, Ch.), since the G 2, 2,2- and H2, 2,2-kernels include the Gauss hypergeometric function. For eample, if = 2 :=, then we get the representation Hf() = H 2, 2,2 ( σ) δ +δ 2 Γ(δ + δ 2 ) σ σ γ 2 (γ k + δ k + /, /) 2 (γ k + /, /) 2 f(σ) dσ = 2 F ( γ2 γ + δ 2, δ ; δ + δ 2 ; σ ) f(σ)dσ, (37) 32

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators see e.g. 2 or 4. A long list of eamples of the operators in the form (34), (36) along with their applications to various problems of analysis and mathematical physics can be found in our previous works. For eample, if = >, δ k =, k =,..., m, we get the so-called hyper-bessel integral operators introduced in Dimovski 2 (see also 2, 22, Ch.3). These operators are the right-inverse operators of the hyper-bessel differential operators that are defined as higher order (m > ) generalizations of the 2nd order Bessel differential operators that appear in many equations of mathematical physics (see details in Sections 4, 5). And another, more general case with an arbitrary m > is for the multiple E-K fractional integrals of the form L = I (λ k ),(δ k ) (/δ k ),m with = /δ k so that the products (δ k ) (/δ k ) = > are fied. These operators have been studied in their composition form (4) as repeated E-K integrals (see Theorem 4) and applied for solving some of different problems in 3 and 22. Some more details regarding these operators are given in Section 6. Now let us consider the Mellin transform image and the convolutional structure of the multiple E-K fractional integrals. In what follows, we assume that the following conditions are satisfied for the parameters of the operators and the indices of (γ k + ) > µ (for f L p µ,p (, )), and (γ k + ) > µ (for f C µ, ), f H µ (Ω)); (38) δ k, k =,..., m. Theorem 3. (Th.5..5, 2) In terms of the Mellin convolution (k f)() = k( dt )f(t) t t, the multiple E-K fractional integral (34) has the following convolutional type representation in L µ,p (, ) ( ),m f() = Hm, (γ k + δ k +, / ) m (γ k +, / ) m f() := k() f(), (39) and for conditions (38) and Rs < min k (γ k + ) (say, if Rs < µ/p), its Mellin transformation satisfies the relation { } m M ( ),m f(); s Γ(γ k + s/ ) = M{f(); s}. (4) Γ(γ k + δ k + s/ ) Indeed, it follows from (39), the { Mellin convolution } theorem, and the Mellin transform formula (9) of the H-function that M ( ),m f(); s = M {(k f)(); s} = M{k(); s} M{f(); s} and we get (4). For more information regarding the role of the Mellin transform theory in the fractional calculus we refer the reader to Luchko 47 and Luchko and Kiryakova 48. Theorem 4. (Composition/Decomposition theorem) Let the conditions (38) be satisfied. The classical E-K fractional integrals, k =,..., m of the form (5) commute in the space L µ,p (resp. in C µ, H µ ) and their composition I γ k,δ k I γm,δm m I γ m,δ m m... { ( I γ,δ )} m f() = I γ k,δ k f() =... (m) m ( σ k ) δ k σ γ k k Γ(δ k ) ( f σ m... σ m ) dσ... dσ m can be represented as an m-tuple E-K operator (34), i.e., by means of a single integral involving the H-function: (4) m I γ k,δ k f() = ( ),m f() = H m, σ (γ k + δ k +, ) m (γ k +, ) m f(σ)dσ. (42) Conversely, under the same conditions, each multiple E-K operator of the form (34) can be represented as a product (4). 322

Virginia Kiryakova, Yuri Luchko Proof. One idea for the proof would be to compare the Mellin transforms (2) of the classical E-K integrals, k =,..., m and the relation (4). The coincidence of the Mellin images I γ k,δ k {( { } m ) } M ( ),m f(); s = M I γ k,δ k f(); s suggests the coincidence of the considered operators. Another way is using the principle of mathematical induction. The eamples given after Def. 2 and an easy check of the formula I γ 2,δ 2 2 I γ,δ f() = (37) confirm the statement for m = and m = 2. Further, suppose (42) holds valid for some m >. Then, consider a commutative composition of the (m + ) operators (5) that after the variables substitution t = σ and change of the order of integrations can be represented as follows: m+ { } I γ k,δ k f() = I γ m+,δ m+ m+ ( ),m f() = = H,, t/ f(σ)dσ σ γ m+ +δ m+ + / m+, / m+ γ m+ + / m+, / m+ t H,, (t/)hm, (σ/t)dt = t t dt H m+, m+,m+ H m, σ/ σ/t (γ k +δ k + /, / ) m (γ k + /, / ) m (γ k +δ k + /, / ) m+ (γ k + /, / ) m+ f(σ)dσ f(σ)dσ, where in the form (34 ) we have used that H,, (t/) for t >, Hm, (σ/t) for < t < σ and the known formula for the integral from to of a product of two H-functions (see e.g. (E.2 ), 2). We thus have proved that (42) holds true for (m + ) and therefore for every m. For f L µ,p and when the conditions (38) (resp. for f C µ and (38) with p = ) are satisfied, all integrals from the above formulas make sense and the changes of the integration order are justified by the Fubini theorem. The resulting functions belong to the corresponding space, too. Thus, the equivalence of the operators (4) and (42) is proved. By means of Theorem 3 and a modification of the Hardy-Littlewood-Polya theorem (see 4, Th..5), one easily proves that the multiple E-K fractional integrals are bounded linear operators from L µ,p into itself under the conditions (38) (Th.5..3, 2). The mapping properties of (34) in the spaces C µ (Def. ) can be characterized as follows: Theorem 5. The multiple E-K fractional integral (34) preserves the power functions from C µ, µ ma k (γ k + ) up to a constant multiplier: ( ),m {q } = c q q, q > µ, m Γ(γ k + q where c q = k + ) Γ(γ k + δ k + q + ), (43) and maps C µ isomorphically into itself: ( ),m : C µ C µ. (44) Under the conditions (38), the E-K integral (34) preserves also the class H µ (Ω) of weighted analytic functions, say, in Ω := R = { < R} of the form { } f() = µ a n n = µ n (a + a +... ) H µ ( R ), with R = lim sup an, (45) n n= mapping them into the functions ( ),m f() = µ { n= a n m } Γ(γ k + n+µ + ) Γ(γ k + δ k + n+µ n H µ ( R ) (46) + ) that have the same radius of convergence R > and the same signs of the coefficients in their series epansions. 323

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators Proof. To evaluate the image (43) of the function f() = q, the known properties and formulas for the H-function (as a kernel in (34)) are employed and in particular the integral formula (E.2) from 2 is used. Then, the image of the power series (45) is obtained by using a legitime term-by-term fractional integration. In view of (42), the mapping (44) is a result of the successive applications of the formula (2). Alternatively, we can use the representation from Def. 2 for a function f() = q f () C µ with f () < const A in each interval X, X > as a function continuous on, ), and the properties of the kernel H-function: { } ( ) ( ),m q f () = q σ q H m, (γ σ k + δ k + /, / ) m f (γ k + /, / ) m (σ) dσ = q f2 (), with f 2 C µ, since f 2 () = < A ( ) H m, (γ k + δ k + + (q )/, / ) m σ f (γ k + + (q )/, / ) m (σ)dσ (γ σ k + δ k + + (q )/, / ) m m (γ k + + (q )/, / ) m dσ = A H m, Γ(γ k + + q/ ) Γ(γ k + δ k + + q/ ) := A 2 <, again according to the integral formula (E.2) in Kiryakova 2 for the above integral of the H-function. To prove that (34) is an injective and surjective mapping, we observe that the unique solution of the convolution integral equation ( ),m f() = H m, σ (γ k + δ k + /, / ) m (γ k + /, / ) m f(σ) dσ = is the function f(), see e.g. 24. To this end, by a suitable substitution, a theorem of Mikusinski and Ryll- Nardzewski from 5 can be used. Lemma 6. For f C µ (N), µ ma k (γ k + ), N N, the following relations between the initial conditions for the function f and for its multiple E-K fractional integral (34) hold valid: { } (j) ( ),m f() () = cj f (j) (), j =,, 2,..., N, with c j = m Γ(γ k ++j/ ). (47) Γ(γ k +δ k ++j/ ) Proof. The formula (47) easily follows from the properties of the H-function (see e.g. (E.9) in 2, App.) and by applying j-th differentiation of f(σ) under the integral sign in (34). The coefficients c j can be found by calculation of the integral of the H-function m, from to which is the same as the integral from to. Theorem 7. (Th.5..6, Kiryakova 2) Suppose that the conditions (38) for C µ, resp. for H µ or L µ,p hold true. Then, the following basic operational rules for the multiple E-K fractional integrals are valid in these spaces of functions: { } { } ( ),m {af(c) + bg(c)} = a ( ),m f (cz) + b ( ),m g (c) (48) (bilinearity); I (γ,...,γ s,γ s+,...,γ m),(,...,,δ s+,...,δ m) (,..., m),m f() = I (γ s+,...,γ m)(δ s+,...,δ m) ( s+,..., m),m s f() (49) (i.e., if δ = δ 2 = = δ s =, then the multiplicity reduces to (m s)); ( ),m λ f() = λ I (γ k + λ k ),(δ k ) ( ),m f(), λ R (5) 324

Virginia Kiryakova, Yuri Luchko (generalized commutability with power functions); (commutability); ( ),m I(τ j ),(α j ) (ε j ),n f() = I(τ j ),(α j ) (ε j ),n I(γ k ),(δ k ) ( ),m f() (5) ( ),m I(τ j ),(α j ) (ε j ),n f() = I((γ k ) m,(τ j ) n )((δ k ) m,(α j ) n ) (( ) m,(ε j ) n ),m+n f() (52) (compositions of the m-tuple and the n-tuple integrals (34) give the (m + n)-tuple integrals of the same form); I (γ k +δ k ),(σ k ) ( ),m ( ),m f() = I(γ k ),(σ k +δ k ) ( ),m f(), if δ k >, σ k >, k =,..., m, (53) (indices law, product rule, or semigroup property); (formal inversion formula). { ( ),m } (γ f() = I k +δ k ),( δ k ) ( ),m f() (54) The proof of the above operational properties follows by using the definition (34) and the known properties of the Fo H-function (see e.g. 2, Appendi, 23, or 24). Note that the above formal inversion formula (54) follows from the semigroup property (53) with σ k = δ k, k =,..., m and the definition of (34) in the case of the zero multi-order of integration: I (γ k +δ k ),( δ k ) ( ),m ( ),m f() = I(γ k ),(,...,) ( ),m f() = f(). However, the symbols I (γ k +δ k ),( δ k ) ( ),m with the negative multi-orders of integration δ k <, k =,..., m as in (54) are not yet defined. Our net aim is to assign some correct meaning to the objects of the form I (γ k +δ k ),( δ k ) ( ),m to avoid the divergent integrals in the representation (34). This is the subject of the net section. 4. Multiple Erdélyi-Kober fractional derivatives in the Riemann-Liouville sense The situation with a suitable definition of the generalized fractional derivative that corresponds to the multiple E-K integral (34) is very similar to the case of the R-L and the E-K fractional integrals of the order δ > that can be inverted using a derivative of the order n that satisfies the condition n < δ n. Say, for the R-L integral (), the idea is to use the differential relation ( t) δ Γ(δ) = ( d d ) n ( t) δ+n Γ(δ + n) ( ) n d, i.e. Φ δ (, t) = Φ δ+n(, t) d for the kernel Φ δ (, t) = ( t) δ /Γ(δ) of () that allows to increase the eponent of the power function Φ δ and to make it nonnegative even for the negative values of δ. In the case of the multiple E-K operators, the following result for the kernel function H m, plays an important role for their inversion. Lemma 8. Let n k, k =,..., m be arbitrary integers. Then the following differential relation holds true: t H m, (a k, / ) m (b k, / ) m t = D n H m, (a k + n k, / ) m (b k, / ) m, (55) where the differential operator D n stands for a polynomial of d d of the degree n = n + + n m, defined by the formula ( D n = P n d ) = d m n r r= j= ) r d + a r + j. (56) 325

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators Proof. To prove (55), the differentiation formula (see (2) from 8.3.2 in 23) η r j= ( z r ) d + c j dz r z α r H M,N P,Q z σ r (a i, A i ) P (b i, B i ) Q = z α r H M,N+ηr P+η r,q+η r z σ r ( c i α, σ) ηr, (a i, A i ) P (b i, B i ) Q, ( c i α +, σ) ηr is applied m-times (r =,..., m) for z r := (/t) r. We note that this formula can be deduced from the formula (corollary of (5) and (7) from 8.3.2 in 23) ( z d + c) H M,N dz P,Q = z c+ z c H M,N+ P+,Q+ (a i, A i ) P z (b i, B i ) Q z = ( z c+ d zc) H M,N dz P,Q (, ), (a i, A i ) P (b i, B i ) Q, (, ) = H M,N P,Q (a i, A i ) P z (b i, B i ) Q (a i, A i ) P z (b i, B i ) Q. For the details we refer the interested reader to the proof of Corollary B.6 in 2, Appendi that is presented for the case of the G-function of the same form. The formula (55) means that the parameters a k, k =,..., m of the H-function can be increased by suitable integers, by applying the differential operator D n. Let us take a k = γ k + γ k + δ k + = b k, k =,..., m and introduce the auiliary integer numbers η,..., η m as follows: η k < δ k η k, that is, η k := { δk +, for noninteger δ k, k =,..., m. δ k, for integer δ k, (57) { Now we are ready to give an appropriate meaning to the symbols differ-integral epressions. ( ),m } (γ = I k +δ k ),( δ k ) ( ),m in (54) by means of suitable Definition 9. Let η m r D η = r= j= d ) r d + γ r + j (58) be a differential operator with the same parameters as in Def. 2 and the integers as in the formula (57). Then the multiple (m-tuple) Erdélyi-Kober fractional derivative of multi-order δ = (δ,..., δ m ) is defined by means of the differ-integral operator of the R-L type: D (γ k ),(δ k ) ( ),m f() := D η I (γ k +δ k ),(η k δ k ) ( ),m f() = D η H m, σ (γ k + η k +, ) m (γ k +, ) m f(σ) dσ. (59) In the case =... = m = >, we obtain a simpler representation involving the Meijer G-function that corresponds to the generalized fractional integral (36) : D (γ k ),(δ k ),m f(z) = D η I (γ k +δ k ),(η k δ k ),m = More generally, the differ-integral operators of the form η m r r= j= d ) d + γ r + j I (γ k +δ k ),(η k δ k ),m f(). (6) D f() = D (γ k ),(δ k ) ( ),m δ f() = δ D (γ δ k ),(δ k ) ( ),m f() with δ (6) are called generalized (multiple, multi-order) fractional derivatives of the R-L type. 326

Virginia Kiryakova, Yuri Luchko The generalized derivatives (59) and (6) are the counterparts of the generalized fractional integrals (34) and (36) and the analogues of the R-L type fractional derivatives (2) and (22). Theorem 2. The multiple E-K fractional derivative (59) is a left-inverse operator to the multiple E-K fractional integral (34) on the functional space C µ, µ ma k (γ k + ), i.e., D (γ k ),(δ k ) ( ),m I(γ k ),(δ k ) ( ),m f() = f(), f C µ. (62) The same relation is valid also for the generalized fractional derivatives and integrals of the form (35), (6): D If() = f(). Proof. The formula (62) can be checked symbolically by using the Def. 9, the operational rules (5) and (53), and the same relation for the integer multi-order operators: D (γ k ),(δ k ) ( ),m I(γ k ),(δ k ) ( ),m = D η I (γ k +δ k ),(η k δ k ) ( ),m ( ),m = D η ( ),m I(γ k +δ k ),(η k δ k ) ( ),m = D (γ k ),(η k ) ( ),m I(γ k ),(η k ) ( ),m = Id. These symbolical manipulations can be justified with the help of the differential formula (55) that in fact suggested the form of the multiple E-K derivative. Having in mind the semigroup property (53) and the representation (34 ), then putting the differential operator D η under the integral sign and using the differential relation (55), we obtain: = D η = H m, t D η H m, t = D (γ k ),(δ k ) ( ),m I(γ k ),(δ k ) ( ),m f() = D η I (γ k +δ k ),(η k δ k ) ( ),m ( ),m f() = D η I (γ k ),(η k ) ( ),m f() = D η H m, (γ k + η k +, / ) m (γ k +, / ) m (γ k + η k +, / ) m (γ k +, / ) m H m, t t (γ k + η k + /, / ) m (γ k + /, / ) m f(t) t dt = D η f(t) dt = t (γ k + + /, / ) m (γ k + /, / ) m H m, = t H m, t f(t)dt (γ k + η k +, / ) m (γ k +, / ) m (γ k +, / ) m (γ k +, / ) m f(t)dt = I (γ k ),(,...,) ( ),m f() = f() f(t) t f(t) dt t that is the relation (62). Note that in the formulas above, we replaced the integration interval from to with the interval from to because the kernel function H(t/) m, is identically equal to zero for t >. The conditions (38) and the asymptotics of the H-function H m, as its argument tends to + and to, respectively, ensure the convergence of the above integrals. dt Remark 2. The relation (62) holds true also in the functional spaces L µ,p (, ) and H µ (Ω) under the conditions (38). Remark 22. In the case of integer orders δ k = η k, k =,..., m (i.e., in the case of an integer multi-order), the operator I (γ k +δ k ),(η k δ k ) ( ),m becomes the identity operator and so the derivative D (γ k ),(δ k ) ( ),m = D(γ k ),(η k ) ( ),m = D η is a purely differential operator that is left-inverse to the integral operator I (γ k ),(η k ) ( ),m. 327

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators In the case δ =... = δ m =, =... = m = >, the multiple E-K derivative is reduced to the so-called hyper-bessel differential operator B that is left-inverse to the the hyper-bessel integration operator L in the space C µ, µ ma k (γ k + ): BLf() f(), where B := D (,...,) = D (γ,...,γ m),(,...,) (,...,),m, L:= I (γ,...,γ m),(,...,) (,...,),m. (63) Note that the hyper-bessel operators belong to the class of the operators in the form (36), (6) with the Meijer G- function G-function m, in the kernel. These linear singular differential operators of an arbitrary order m > with the variable coefficients have been introduced by Dimovski 2 in the form B = α d d α d d... α m d d αm, := m (α + α +... + α m ) >, >, and can be also represented as ( B = Q m d ) m ( ) = d d d + γ k = m dm d + a m dm m d + + a m d m d + a m. (64) The hyper-bessel differential operators are natural generalizations of the 2nd order Bessel differential operators and appear very often in differential equations modeling problems in mathematical physics, especially in the aially-symmetric cases. Their theory, including the operational calculi via a family of convolutions and via a Laplace-type integral transform (known as the Obrechkoff transform), has been proposed by Dimovski 2 and then etended by means of the fractional calculus and the special functions in Kiryakova 2, Ch.3 (see also Dimovski and Kiryakova 52, Luchko et al. 22, 3, 47). As a net step, in Luchko and Kiryakova 53, 54 a generalization of the Hankel integral transform that is related to the hyper-bessel operators, was introduced and investigated. Remark 23. We can assign a certain meaning to the symbols I (γ k ),(η k ) ( ),m for arbitrary real δ k, k =,..., m, that is, define a fractional integration/differentiation of an arbitrary multi-order. Let the conditions δ <,..., δ s < ; δ s+ = = δ r = ; δ r+ >,..., δ m > be fulfilled. Then the symbol I (γ k ),(η k ) ( ),m is understood as a composition of an s-tuple E-K fractional derivative (59), (r s) identity operators, and an (m r)-tuple E-K fractional integral (34), namely: s m ( ),m := D γ k +δ k, δ k I... I (r s) j=r+ I γ j,δ j j = D (γ k +δ k ),( δ k ) ( ),s I (γ j ),(δ j ) ( j ),m r. (65) Having in mind definition (65), both ( ),m and D (γ k ),(δ k ) ( ),m can be called by the common name multiple E-K fractional differ-integrals. Remark 24. The properties of the multiple E-K integrals (34) and Definition 9 easily lead to the corresponding operational properties of the multiple E-K derivatives (59) that are analogous to those formulated in Theorem 5 and Theorem 7: D (γ k ),(δ k ) ( ),m {q } = q m Γ(γ k + δ k + + q/ ) Γ(γ k + + q/ ), q > µ, (66) D (γ k ),(δ k ) ( ),m : C (η +...+η m) µ C µ, (67) D (γ k ),(δ k ) ( ),m λ f() = λ D (γ k +λ/ ),(δ k ) ( ),m f(). (68) 328

Virginia Kiryakova, Yuri Luchko Like in the classical Calculus, the fractional derivative is in general not a right-inverse operator to the fractional integral unless some initial conditions are all equal to zero (see e.g. Theorem 9 for the E-K operators). The difference built by the identity operator and the composition of an operator and its left-inverse operator is called the projector of the operator, or its operator of the initial conditions. The form of the projector for the multiple E-K integral operator is given in the following theorem. Theorem 25. Let the integers η k N : η k < δ k η k, k =,..., m be defined as in (57), µ ma k (γ k + ) as in (38), and f C (η +...+η m) µ. Then the projector of the multiple E-K integral (34) of the multi-order δ = (δ,..., δ m ) has the form Ff() := f() ( ),m D(γ k ),(δ k ) ( ),m f() = m η k j= A k,j (γ k +j), (69) where the coefficients A k,j are connected with the initial conditions (at = +) for the fractional differ-integrals of f() and are given by the formula η m Γ(η k + j) k A k,j = Γ(δ k + j) lim ( (γ k +j) d ) d + γ k + i I (γ k +δ k ),(η k δ k ) ( ),m f(). (7) i=j+ Proof. As mentioned in Remark 24, under the conditions of the theorem, D (γ k ),(δ k ) ( ),m : C (η +...+η m) µ C µ. The mapping property D γ k,δ k : C (η k ) µ C µ is an immediate consequence of the first part of the proof of Theorem 3. from 43. Let us now introduce an auiliary function defined by g() := f(). Using Th. 2 and (6), we then get that can be rewritten in the form ( ),m D(γ k ),(δ k ) ( ),m D (γ k ),(δ k ) ( ),m g() = D(γ k ),(δ k ) ( ),m I(γ k ),(δ k ) ( ),m D(γ k ),(δ k ) ( ),m f() = D(γ k ),(δ k ) ( ),m f(), D (γ k ),(δ k ) ( ),m f() g() = D(γ k ),(δ k ) ( ),m h(), > with the function h() := f() g(). To find the form of the projector Ff() means now to solve the last equation for the function h(). But it follows from the definition (59) that the kernel of this operator consists of all functions h that satisfy the relation D η y(), where we denoted I (γ k +η k ),(η k δ k ) ( ),m h() by y(). The operator D η defined by (58) is a linear homogeneous differential operator of the order η = η +... + η m (see (63) or (64)) and we get the equation η m r D η y() = r= j= d ) r d + γ r + j y() =. It is an easy eercise to check that the functions y k,j () = (γ k +j), k =,..., m, j =,..., η k deliver a system of η linearly independent solutions of this differential equation (fundamental system) and therefore its general solution can be written in the form η m k y() = B k,j (γ k +j). j= The coefficients B k,j were already deduced in Kiryakova 2, Ch.3 and Luchko et al. 22, 3, but in the case under consideration they depend on the initial conditions written in terms of the multiple E-K fractional differ-integrals I (γ k +η k ),(η k δ k ) ( ),m of f(). Then we can solve the equation I (γ k +η k ),(η k δ k ) ( ),m h() = m η k j= B k,j (γ k +j) 329

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators by applying the derivative D (γ k +η k ),(η k δ k ) ( ),m to both sides of this equation and having in mind the property (66): with A k,j = h() = m η k j= B k,j D (γ k +η k ),(η k δ k ) ( ),m (γ k +j) = m Γ(η k + j) Γ(δ k + j) B k,j; B k,j = lim (γ k +j) η k i=j+ m η k j= A k,j (γ k +j), d ) d + γ k + i I (γ k +δ k ),(η k δ k ) ( ),m f() More details regarding derivation of the last formula can be found in the proof of Theorem 3. from Luchko and Trujillo 43 for the case of the E-K operators. The case of the hyper-bessel operators was considered in Lemma 3.2.2 from Kiryakova 2, Ch.3, see also Luchko et al. 22, 3. The results presented above and in particular the formula for the projector operator are valuable for solving initial value problems and eigenvalue problems for fractional differential equations involving the multiple E-K derivatives or their particular cases. Such equations have been already considered by several authors using either operational calculus or the FC techniques, or some suitable integral transforms. In particular, we refer to the works of Luchko et al. 22, 3, 32 34 (operational calculus method), Kiryakova et al. 4, 55 (solutions in terms of the M-L functions (3) and the multi-inde M-L functions (4)), and Furati 56 (case m = 2) to mention only few of many relevant publications. The solutions of the hyper-bessel type differential equations of the form By() = λy(), By() = f(), and By() = λy() + f() have been derived in Kiryakova 2, Ch.3 in terms of the Meijer G-functions.. 5. Multiple Erdélyi-Kober fractional derivatives in the Caputo sense In the previous section, we considered multiple Erdélyi-Kober fractional derivatives in the R-L sense. Like in the case of the R-L and E-K operators, Caputo-type multiple Erdélyi-Kober fractional derivatives can be also introduced, e.g. to avoid the physically nonsense initial conditions as in Theorem 25. The definition of the multiple Erdélyi-Kober fractional derivatives in the Caputo sense has a form similar to (59), but the auiliary differential operator D η is moved under the integration sign by analogy with (4) and (29). Definition 26. The Caputo-type multiple Erdélyi-Kober fractional derivative is defined as the integro-differential operator D (γ k ),(δ k ) ( ),m f() := I(γ k +δ k ),(η k δ k ) ( ),m D η f() = = H m, σ (γ k + η k +, ) m (γ k +, ) m H m, σ η m r r= j= (γ k + η k +, ) m (γ k +, ) m d ) r d + γ r + j D η f(σ) dσ (7) f(σ) dσ, where the parameters are the same as in Def. 2, Def. 9, the integers η k are given by (57) and the differential operator D η by (58). In what follows, we consider multiple Erdélyi-Kober fractional derivative (7) of the Caputo type in the functional space f C (η) µ with µ ma k (γ k + δ k + ). Evidently, for m = (7) is reduced to the Caputo-type E-K operator (29). Remark 27. In the case of integer multi-order of differentiation, i.e., if δ k = η k, the R-L and the Caputo type (multiple) E-K derivatives coincide and are differential operators D η = D (γ k ),(η k ) ( ),m of integer order η = η +... + η m, namely: D (γ k ),(δ k ) ( ),m = D(γ k ),(δ k ) ( ),m = D(γ k ),(η k ) ( ),m = D η since I (γ k +δ k ),(η k δ k ) ( ),m = I (γ k +δ k ),(,,...,) ( ),m = Id. (72) If δ k = η k =, =, k =,..., m, the multiple E-K derivatives are reduced to the hyper-bessel differential operators (63), (64). 33

Virginia Kiryakova, Yuri Luchko Theorem 28. Let the integers η k N : η k < δ k η k, k =,..., m be defined as in (57) and f C (η +...+η m) µ with µ ma k (γ k + δ k + ). Then the following relation between the Caputo-type multiple E-K fractional derivative (7) and the multiple E-K integral (34) of the multi-order δ = (δ,..., δ m ) holds true: η m k ( ),m D (γ k ),(δ k ) ( ),m f() = f() Ff(), with Ff() = C k,j (γ k +j), (73) j= where the coefficients C k,j depend only on the limit values (at = +) of the ordinary derivatives of the function f(), namely, C k,j = lim d ) d + γ k + i f(). (74) η k (γ k +j) i=j+ Proof. According to Def. 26, the composition of ( ),m : C µ C η +...η m µ and D (γ k ),(δ k ) ( ),m : C η +...η m µ C µ can be symbolically written in the form ( ),m D (γ k ),(δ k ) ( ),m f() = I(γ k ),(δ k ) ( ),m I(γ k +δ k ),(η k δ k ) ( ),m D η f() = I (γ k ),(η k ) ( ),m D(γ k ),(η k ) ( ),m f() = f() Ff(). To get the final formula, we used the semigroup property (53) for the composition of two multiple E-K integrals and Theorem 25 for the case of the integers δ k = η k. Then all multipliers of the form Γ(η k + j)/γ(δ k + j) are equal to and in the formula for A k,j their product is equal to, too. Because I (γ k +δ k ),(η k δ k ) ( ),m = I (γ k +δ k ),(,,...,) ( ),m = Id, all fractional integrals in the formula for A k,j disappear. Therefore, we get that proves the theorem. Ff() := Ff() = m η k η k lim (γ k +j) j= i=j+ d ) d + γ k + i The multiple analogue of Theorem 4.2 from Luchko and Trujillo 43 reads as follows. f() (γ k +j), Theorem 29. Let the parameters satisfy the conditions (38) and (57). The Caputo-type multiple E-K fractional derivative D (γ k ),(δ k ) ( ),m and the Riemann-Liouville type multiple E-K fractional derivative D (γ k ),(δ k ) ( ),m coincide for the functions f C η +...+η m µ if and only if the equalities C k,j = lim (γ k +j) η k (γ k +j) i=j+ η k i=j+ are fulfilled for all k =,..., m; j =,..., η k. d d + γ k + i d ) d + γ k + i ) m f() = I (γ k +δ k ),(η k δ k ) ( ),m f() Γ(η k + j) Γ(δ k + j) lim = A k,j (75) Proof. that In the space C η +...+η m µ, both the Caputo and the R-L type multiple E-K fractional derivatives eist. Suppose ( ),m f() D(γ k ),(δ k ) ( ),m f(), >. D (γ k ),(δ k ) Applying the multiple E-K fractional integral ( ),m to the last equality, we get ( ),m D (γ k ),(δ k ) ( ),m f() I(γ k ),(δ k ) ( ),m D(γ k ),(δ k ) ( ),m f(), >. (76) 33

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators Then it follows from Theorem 28 and Theorem 25 that m η k C k,j (γ k +j) = m η k j= j= A k,j (γ k +j), >. Because the functions { } (γ k +j), k =,..., m, j =,..., η k are linearly independent, the coefficients in (75) should coincide. Conversely, if (75) holds true, then the formula (76) follows from Theorem 28 and Theorem 25. Applying the R-L type derivative D (γ k ),(δ k ) ( ),m to the equation (76) and using Theorem 2, we get ( ),m f() D(γ k ),(δ k ) f(), >. (77) D (γ k ),(δ k ) ( ),m We show now that the Caputo type multiple E-K derivative satisfies an analogue of Theorem 2 for the R-L type multiple E-K derivative. Theorem 3. The Caputo-type multiple Erdélyi-Kober fractional derivative D (γ k ),(δ k ) ( ),m is a left-inverse operator to the multiple E-K fractional integral for the functions from the space C µ, µ ma k (γ k + ), that is, D (γ k ),(δ k ) ( ),m I(γ k ),(δ k ) ( ),m f() f(), f C µ. (78) Proof. First, an auiliary function g() = ( ),m f() C η +...+η m µ (79) is introduced. Now we show that the equalities C k,j = A k,j =, k =,..., m, j =,..., η k (8) for the coefficients defined in Theorem 29 are valid for this auiliary function g(). Indeed, the inclusion f C µ means that f() = q f (), q > µ that yields g() = ( ),m f() = q f 2 () C η +...+η m µ. Then for a polynomial P n ( d ) of the Euler operator d of any order n (η +... + η m ), we again get the chain of the equalities P n ( d )g() = P n ( d )I(γ k ),(δ k ) ( ),m f() = q f 3 () with q > µ, where f, f 2, f 3 C, ), that is, they are bounded on each finite interval, X. Then the coefficients C k,j can be determined as follows: C k,j = lim η k (γ k +j) i=j+ d ) d +γ k +i ( ),m q f () = lim η k (γ k +j) i=j+ d ) d +γ k +i q f 2 () = lim (γ k +j) q f 3 () = lim (γ k +j)+q f 3 () = lim Q f 3 () =, since Q > because of the inequalities q > µ ma k (γ k + ). To determine the coefficients A k,j for the function g(), we use the semi-group property (53) and again the mappings I (γ k ),(η k ) ( ),m : C µ C µ, P n ( d ) : C µ C µ and thus get the chain of the equalities: A k,j = m Γ(η k + j) Γ(δ k + j) lim = (γ k +j) m Γ(η k + j) Γ(δ k + j) lim η k i=j+ (γ k +j) d ) d + γ k + i I (γ k +δ k ),(η k δ k ) ( ),m η k i=j+ d ) d + γ k + i I (γ k ),(η k ) ( ),m f() ( ),m f() 332