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In the case of a function f C∞( δ, δ), its Taylor series is introduced by letting n go to + in the formula (3):∈ − ∞ +∞ (j) f(x)= aj hj+1(x), aj = f (0). (5) j=0 X To illustrate the main objects of this paper, let us represent the power function hj+1 from the formulas (1), (3), and (5) as a convolution power. We start with the following important formula that is a direct consequence from the well-known representation of the Euler Beta-function in terms of the Gamma-function:

(hα hβ)(x) = hα β(x), α,β> 0, x> 0, (6) ∗ + where is the Laplace convolution ∗ x (f g)(x)= f(x ξ)g(ξ) dξ. (7) ∗ − Z0 Then, by repeatedly applying the formula (6), we get the representation h (x)= hnα(x), n N, (8) α ∈ where the expression g stands for a convolution power 1, n =0, g(x), n =1, g(x) :=  (9) (g ... g)(x), n =2, 3,....  ∗ ∗ n times  Setting α = 1 in the formula (8) leads| {z to the} representation h1 (x)= hn(x), j =0, 1, 2 .... (10)

Denoting h1 with κ, the formulas (1), (3), and (5) can be rewritten as follows: +∞ Σκ(x)= aj κ (x), (11) j=0 X n−1 f(x)= aj κ (x) + rn(x), aj = a(f, κ, j), (12) j=0 X +∞ f(x)= aj κ (x), aj = a(f, κ, j). (13) j=0 X The series in form (11) with the functions κ that are continuous on the positive real semi-axis and can have an integrable singularity at the origin were recently introduced and studied in [1] in the framework of an operational calculus for the general fractional derivatives with the Sonine kernels. In [1], these series were called convolution series. In the case, the function κ is a Sonine kernel, solutions to the initial-value problems for the linear single- and multi-term fractional differential equations with the n-fold general fractional derivatives were derived in [1] in terms of the convolution series (11). The main focus of this paper is on the generalized convolution Taylor formula in form (12) and the generalized convolution Taylor series in form (13) with the Sonine CONVOLUTION SERIES 3 kernels κ. It is worth mentioning that some generalizations of the Taylor formula (3) and the Taylor series (5) to the case of the power series with the fractional exponents have been already considered in the literature (see [2] and the references therein). In this paper, we obtain some formulas of this kind as a particular case of our general results while specifying them for the Sonine kernel κ(x)= hα(x), 0 <α< 1. The rest of the paper is organized as follows. In the 2nd Section, we introduce the main tools needed for derivation of our results including the suitable spaces of functions, the Sonine kernels, and the general fractional integrals and derivatives with the Sonine kernels. The 3rd Section is devoted to the generalized Taylor series in form of a convolution series generated by the Sonine kernels. In particular, the formulas for the coefficients of the generalized convolution Taylor series in terms of the generalized fractional integrals and derivatives are derived and some particular cases are discussed in details. In the 4th Section, the convolution Taylor polyno- mials, approximations of the functions from some special spaces in terms of these polynomials (generalized convolution Taylor formula), as well as the formulas for a remainder in the generalized convolution Taylor formula are presented.

2. Preliminary results Recently, the general fractional integrals (GFIs) and the general fractional deriva- tives (GFDs) with the Sonine kernels became a subject of active research in Frac- tional Calculus (FC), see [1, 3–10]. In this paper, we first derive the second funda- mental theorem for the n-fold general sequential fractional derivatives of both the Riemann-Liouville and the Caputo types and then employ them for derivation of the generalized convolution Taylor formulas in two different forms. The GFD in the Riemann-Liouville sense is defined in form of the following integro-differential operator of convolution type d d t (D f)(x)= (k f)(x)= k(x ξ)f(ξ) dξ, (14) (k) dx ∗ dx − Z0 whereas in the definition of the GFD in the Caputo sense the order of differentiation and integration is interchanged:

t ′ ′ ( D k f)(x)=(k f )(x)= k(x ξ)f (ξ) dξ. (15) ∗ ( ) ∗ − Z0 The prominent particular cases of the GFDs (14) and (15) are the Riemann- Liouville and the Caputo fractional derivatives of order α, 0 <α< 1 (k(x) = h1−α(x), 0 <α < 1), the multi-term Riemann-Liouville and Caputo fractional n R derivatives (k(x) = k=1 ak h1−αk (x), 0 < α1 < < αn < 1, ak , k = 1,...,n), and the Riemann-Liouville and Caputo fractional··· derivatives of dis∈ tributed 1 P order (k(x)= 0 h1−α(x) dρ(α) with a Borel measure ρ defined on the interval [0, 1]). For other particular cases see [7, 8] and the references therein. For applications of the GFDs in generalR fractional dynamics and in general non-Markovian quantum dynamics we refer the interested readers to the recent publications [11, 12]. It is worth mentioning that the Laplace convolution of the kernel κ(x)= hα(x), α> 0 with a function f is known as the Riemann-Liouville fractional integral of order 4 YURI LUCHKO

α: x α 1 α−1 I f (x)=(hα f)(x)= (x ξ) f(ξ) dξ, x> 0. (16) 0+ ∗ Γ(α) − Z
0 Thus, the Riemann-Liouville fractional derivative of order α, 0 <α< 1 can be represented as a composition of the first-order derivative and the Riemann-Liouville fractional integral of order 1 α: − α d d 1−α (D f)(x)= (h α f)(x)= (I f)(x), x> 0. (17) 0+ dx 1− ∗ dx 0+ The Caputo fractional derivative is the Riemann-Liouville fractional integral of order 1 α applied to the first order derivative of a function f: − α ′ 1−α ′ ( D f)(x)=(h α f )(x)=(I f )(x), x> 0. (18) ∗ 0+ 1− ∗ 0+ Applying the formula (6) to the kernels hα(x) and h1−α(x) of the Riemann- Liouville fractional integral and the Riemann-Liouville or Caputo fractional deriva- tives, respectively, we get the relation

(hα h α)(x)= h (x)= 1 , 0 <α< 1, x> 0, (19) ∗ 1− 1 { } where by 1 we denote the function that is identically equal to 1. In [13],{ the} general kernels κ, k that satisfy the condition (κ k)(x)= h (x)= 1 , x> 0 (20) ∗ 1 { } have been introduced and studied. Nowadays, the functions that satisfy the condition (20) (Sonine condition) are called the Sonine kernels. For a Sonine kernel κ, the kernel k that satisfies the Sonine condition (20) is called its associated kernel. Of course, κ is then an associated kernel to k. The set of the Sonine kernels is denoted by . It was known already to Sonine ( [13]) that theS GFD (14) with the Sonine kernel k and the GFI with its associated kernel κ in form t (I κ f)(x)=(κ f)(x)= κ(x ξ)f(ξ) dξ (21) ( ) ∗ − Z0 satisfy the so called 1st fundamental theorem of FC (see [14]), i.e., d d (D I f)(x)= (k (κ f))(x)= ( 1 f)(x)= f(x) (22) (k) (κ) dx ∗ ∗ dx { } ∗ on the corresponding spaces of functions. In the publications [1, 3, 4, 8, 9], some important classes of the Sonine kernels as well as the GFDs and GFIs with these kernels on the appropriate spaces of functions have been introduced and studied. In [3], the case of the Sonine kernels in form of the singular (unbounded in a neighborhood of the point zero) locally integrable completely monotone functions was discussed. A typical example of a pair of the Sonine kernels of this sort is as follows ( [3]):

κ(x)= h1−β+α(x) + h1−β(x), 0 <α<β< 1, (23) β−1 α k(x)= x Eα,β( x ), (24) − CONVOLUTION SERIES 5 where Eα,β stands for the two-parameters Mittag–Leffler function that is defined by the following convergent series: +∞ zk Eα,β(z)= , α> 0, β,z C. (25) Γ(αk + β) ∈ k X=0 In the publications [1, 8, 9], the Sonine kernels continuous on R+ that have an integrable singularity at the origin were treated. In this paper, we mainly deal with the kernels from this space of functions that is defined as follows: p−1 C , (0, + ) = f : f(x)= x f (x), x> 0, 0 0. (30) 1 ∗ 0 1 In its turn, the convolution formula (30) suggests the following natural definitions of the GFDs with the kernels k = h1(t) and k(t)= h0(t), respectively: d d (D f)(x) := (I f)(x)= (I0 f)(x)= f ′(x), (31) (h1) dx (h0) dx 0+ d d (D f)(x) := (I f)(x)= (I1 f)(x)= f(x), (32) (h0) dx (h1) dx 0+ D I ′ 0 ′ ′ ( ∗ (h1) f)(x) := ( (h0) f )(x)=(I0+ f )(x)= f (x), (33) ′ 1 ′ ( D h0 f)(x) := (I h1 f )(x)=(I f )(x)= f(x) f(0). (34) ∗ ( ) ( ) 0+ − 6 YURI LUCHKO

Several other particular cases of the GFI (21) can be easily constructed using the known Sonine kernels (see [8]). Here, we mention just one of them: I 1−β+α 1−β ( (κ) f)(x)=(I0+ f)(x)+(I0+ f)(x), x> 0 (35) with the Sonine kernel κ defined by (23). The GFD of the Riemann-Liouville type that corresponds to the GFI (35) has the Mittag-Leffler function (24) in the kernel: x d β−1 α (D f)(x)= (x ξ) Eα,β( (x ξ) ) f(ξ) dξ, 0 <α<β< 1, x> 0. (36) (k) dx − − − Z0 The GFD of the Caputo type with the Mittag-Leffler function (24) in the kernel takes the form x β−1 α ′ ( D k f)(x)= (x ξ) Eα,β( (x ξ) ) f (ξ) dξ, 0 <α<β< 1, x> 0. (37) ∗ ( ) − − − Z0 Several important properties of the GFI (21) on the space C−1(0, + ) can be easily derived from the known properties of the Laplace convolution including∞ the mapping property I : C (0, + ) C (0, + ), (38) (κ) −1 ∞ → −1 ∞ the commutativity law I I = I I , κ , κ , (39) (κ1) (κ2) (κ2) (κ1) 1 2 ∈L1 and the index law

I κ1 I κ2 = I κ1 κ2 , κ , κ . (40) ( ) ( ) ( ∗ ) 1 2 ∈L1 The first fundamental theorem of FC for the GFI (21) and the GFDs (14) and (15) of the Riemann-Liouville and the Caputo types, respectively, has been proved in [8].

Theorem 2.1 ( [8]). Let κ 1 and k be its associated Sonine kernel. Then, the GFD (14) is∈L a left-inverse operator to the GFI (21) on the space C (0, + ): −1 ∞ (D I f)(t)= f(t), f C (0, + ), t> 0, (41) (k) (κ) ∈ −1 ∞ and the GFD (15) is a left inverse operator to the GFI (21) on the space C1 (0, + ): −1,(k) ∞ 1 ( D k I κ f)(t)= f(t), f C (0, + ), t> 0, (42) ∗ ( ) ( ) ∈ −1,(k) ∞ where C1 (0, + ) := f : f(t)=(I φ)(t), φ C (0, + ) . −1,(k) ∞ { (k) ∈ −1 ∞ } In the rest of this section, we consider the compositions of the GFIs(n-fold GFIs) and construct the sequential GFDs.

Definition 2.2 ( [8]). Let κ 1. The n-fold GFI (n N) is defined as a compo- sition of n GFIs with the kernel∈Lκ: ∈ I I I ( (κ) f)(x) := ( (κ) ... (κ) f)(x), x> 0. (43) n times | {z } CONVOLUTION SERIES 7

By employing the index law (40), we can represent the n-fold GFI (43) as a GFI with the kernel κ: (I f)(x)=(κ f)(x)=(I f)(x), x> 0. (44) (κ) ∗ (κ) It is worth mentioning that the kernel κ , n N belongs to the space C−1(0, + ), but it is not always a Sonine kernel. ∈ ∞

Definition 2.3. Let κ 1 and k be its associated Sonine kernel. The n-fold sequential GFDs in the Riemann-Liouville∈ L and the Caputo senses, respectively, are defined as follows: D D D ( (k) f)(x) := ( (k) ... (k) f)(x), x> 0, (45) n times D D D ( ∗ (k) f)(x) := ( ∗| (k){z... ∗ }(k) f)(x), x> 0. (46) n times Please note that in [1, 8], the n-fold| GFDs{z (n} N) were defined in a different form: ∈ dn (Dn f)(x) := (k f)(x), x> 0, (47) (k) dxn ∗ ( Dn f)(x) := (k f (n))(x), x> 0. (48) ∗ (k) ∗ The n-fold sequential GFDs (45) and (46) are generalizations of the Riemann- Liouville and the Caputo sequential fractional derivatives to the case of the Sonine kernels from 1. RepeatedlyL applying the first fundamental theorem of FC for the GFI (21) and the GFDs (14) and (15) of the Riemann-Liouville and the Caputo type, respectively (Theorem 2.1), we arrive at the following result: Theorem 2.2 (First Fundamental Theorem of FC for the n-fold sequential GFDs). Let κ 1 and k be its associated Sonine kernel. Then,∈L the n-fold sequential GFD (45) in the Riemann-Liouville sense is a left inverse operator to the n-fold GFI (43) on the space C (0, + ): −1 ∞ (D I f)(x)= f(x), f C (0, + ), x> 0 (49) (k) (κ) ∈ −1 ∞ and the n-fold sequential GFD (46) in the Caputo sense is a left inverse operator to the n-fold GFI (43) on the space Cn (0, + ): −1,(k) ∞ ( D I f)(x)= f(x), f Cn (0, + ), x> 0, (50) ∗ (k) (κ) ∈ −1,(k) ∞ where Cn (0, + ) := f : f(x)=(I φ)(x), φ C (0, + ) . −1,(k) ∞ { (k) ∈ −1 ∞ } 3. Generalized convolution Taylor series

For a Sonine kernel κ 1, a convolution series in form (11) was introduced in [9] in the framework of∈ an L operational calculus for the GFDs of the Caputo type with the Sonine kernels. It is worth mentioning that a part of the results regarding convolution series that were presented in [9] is valid for any function κ C−1(0, + ) (that is not necessarily a Sonine kernel). In particular, this applies to∈ Theorem∞ 4.4 from [9] that we now formulated and prove for a larger class of the kernels and in a slightly modified form. 8 YURI LUCHKO

Theorem 3.1. Let a function κ C (0, + ) be represented in the form ∈ −1 ∞ κ(x)= hp(x)κ (x), x> 0, p> 0, κ C[0, + ) (51) 1 1 ∈ ∞ and the convergence radius of the power series +∞ j Σ(z)= aj z , aj C, z C (52) ∈ ∈ j=0 X be non-zero. Then the convolution series +∞ Σκ(x)= aj κ (x) (53) j=0 X is convergent for all x > 0 and defines a function from the space C−1(0, + ). Moreover, the series ∞ +∞ 1−α 1−α x Σκ(x)= aj x κ (x), α = min p, 1 (54) { } j=0 X is uniformly convergent for x [0, X] for any X > 0. ∈ Proof. First we mention that in the case p 1 the function κ is continuous on ≥ [0, + ). Then the representation (51) with κ1(x)= κ(x) and p = 1 is valid. Thus, without∞ any loss of generality, the representation (51) holds valid with the parameter p restricted to the interval (0, 1]. Now we introduce an arbitrary but fixed interval [0, X] with X > 0. The function 1−p κ1(x)=Γ(p)x κ(x) from the representation (51) is continuous on [0, + ) and the estimate ∞ 1−p MX > 0 : κ (x) = Γ(p)x κ(x) MX , x [0, X] (55) ∃ | 1 | | | ≤ ∈ holds valid. We proceed with derivation of a suitable estimate for the convolution powers κ, j 1 on the interval (0,X]. For j = 1, we get ≥ x <2> κ (x) = (κ κ)(x) hp(x ξ) κ (x ξ) hp(ξ) κ (ξ) dξ | | | ∗ | ≤ − | 1 − | | 1 | ≤ Z0 2 2 M (hp hp)(x)= M h p(x), 0 < x X. X ∗ X 2 ≤ The same arguments easily lead to the inequalities (j+1)p−1 j+1 j+1 x κ (x) M h j p(x)= M , 0 < x X, j N (56) | | ≤ X ( +1) X Γ((j + 1)p) ≤ ∈ 0 that can be rewritten as follows: xjp x1−p κ(x) M j+1 , 0 x X, j N . (57) | | ≤ X Γ((j + 1)p) ≤ ≤ ∈ 0

For the further estimates, we choose and fix any point z0 = 0 from the convergence interval of the power series (52). Because the series is absolutely6 convergent at the point z0, the following inequalities hold true:

j M M > 0 : aj z M j N aj j N . (58) ∃ | 0| ≤ ∀ ∈ 0 ⇒ | | ≤ z j ∀ ∈ 0 | 0| CONVOLUTION SERIES 9

Combining the inequalities (57) and (58), we arrive at the following estimate:

j M Xp jp X 1−p M j+1 x |z0| x aj κ (x) M M MX , j N (59) | | ≤ z j X Γ((j + 1)p) ≤ Γ(( j + 1)p) ∈ 0 | 0| that is valid for all x [0, X]. The number series ∈ p j +∞ MX X |z0| M M X Γ(( j + 1)p) j=0 X is absolutely convergent because of the Stirling asymptotic formula x x Γ(x + 1) √2πx , x + . ∼ e → ∞   This fact and the estimate (59) let us to apply the Weierstrass M-test that says that the series +∞ 1−p x ajκ (x) (60) j=0 X is absolutely and uniformly convergent on the interval [0, X]. Because the functions 1−p x ajκ (x), j N0 are all continuous on [0, X] (see the inequality (56) and remember that p ∈(0, 1]), the uniform limit theorem ensures that the series (60) is a function continuous∈ on the interval [0, X]. Because X can be chosen arbitrary large, the convolution series (53) is convergent for all x > 0 and defines a function from the space C (0, + ).  −1 ∞ Now we proceed with analysis of the convolution series in form (53) with the kernel functions κ 1. In what follows, we always assume that the convergence radius of the power seres∈L (52) is non-zero. As proved in Theorem (3.1), the convolution series (53) defines a function from the space C (0, + ) that we denote by f: −1 ∞ +∞ f(x)= aj κ (x). (61) j=0 X The problem that we now deal with is to determine the coefficients aj, j N0 in terms of the function f. The series at the right-hand side of (61) is uniformly∈ convergent on any closed interval [δ, X], 0 <δ 0 we can apply the GFI (21) with the kernel k (that is the Sonine kernel associated to κ) to this series term by term:

+∞ +∞ (I k f)(x)=(k f)(x)= k aj κ (x)= aj k κ (x). (62) ( ) ∗ ∗ ∗ j=0 ! j=0 X X
Due to the Sonine condition (20), the last formula can be represented in the form

+∞ (I k f)(x)= a + 1 aj κ (x)= a +( 1 f )(x). (63) ( ) 0 { } ∗ +1 0 { } ∗ 1 j=0 ! X 10 YURI LUCHKO

According to Theorem 3.1, the inclusion f C (0, + ) holds valid. As have been 1 ∈ −1 ∞ shown in [16], the definite integral of a function from C−1(0, + ) is a continuous function on the whole interval [0, + ) that takes the value zero∞ at the point zero: ∞ ( 1 f )(x)=(I1 f )(x) C[0, + ), (I1 f )(0) = 0. (64) { } ∗ 1 0+ 1 ∈ ∞ 0+ 1 Substituting the point x = 0 into the equation (63), we arrive at the formula

a0 =(I(k) f)(0) (65) for the coefficient a0 of the convolution series (61). To proceed with the next coefficient, we first differentiate the representation (63) and arrive at the formula d +∞ (I f)(x)= a κ(x). (66) dx (k) j+1 j=0 X The convolution series at the right-hand side of (66) corresponds to the power series with the same convergence radius as the series (52) and thus we can apply exactly same arguments as before to determine the coefficient a1: d a = I (I f) (0) = I D f (0). (67) 1 (k) dx (k) (k) (k)  
Repeating the same reasoning as for derivation of the coefficient a1 again and again, we get the formula I D aj = (k) (k) f (0), j =2, 3,..., (68) D   where (k) stands for the j-fold sequential GFD in the Riemann-Liouville sense defined by (45). Evidently, the formula (65) is a particular case of the formula (68) with j = 0. Summarizing the arguments presented above, we get a proof of the following theorem: Theorem 3.2. Any function f in form of a convolution series (61) with the Sonine kernel κ 1 can be represented as the following generalized convolution Taylor series: ∈ L +∞ I D f(x)= aj κ (x), aj = (k) (k) f (0), (69) j=0 X   where I(k) is the GFI (21), k is the Sonine kernel associated to the kernel κ and D (k) is the j-fold sequential GFD in the Riemann-Liouville sense defined by (45). Example 3.1. Let us illustrate the statement of Theorem 3.2 on an example and consider the Sonine kernel κ(x) = hα(x), 0 <α< 1 with the associated kernel k(x)= h α(x). Evidently, the inclusion κ is valid and we can apply the result 1− ∈L1 of Theorem 3.2. For the kernel k(x)= h1−α(x), 0 <α< 1, the GFI I(k) is reduced to the Riemann-Lioville fractional integral of order 1 α and the sequential GFD D − (k) is the well-known sequential Riemann-Lioville fractional derivative. As already mentioned, the convolution power κ can be determined in explicit form: xα(j+1)−1 κ(x)= h(x)= h (x)= , x> 0. (70) α α(j+1) Γ(α(j + 1)) CONVOLUTION SERIES 11

The generalized convolution Taylor series (69) takes then the following form:

+∞ αj x f(x)= xα−1 a , a = I1−α Dα f (0), (71) j Γ(α j + α) j 0+ 0+ j=0 X 
 1−α where I0+ is the Riemann-Lioville fractional integral (16) of order 1 α and α − D0+ is the sequential Riemann-Lioville fractional derivative in form of a com- position of j Riemann-Lioville fractional derivatives (17) of order α. The formula
(71) can be interpreted as a representation of a function f from the space C−1(0, + ) in form of a power series with non-integer exponents. For other forms of such rep-∞ resentations see e.g. [2] and the references therein. It is worth mentioning that the representation (71) is a particular case of a more general formula derived in [17] for the case of the Dzherbashyan-Nersesyan fractional derivative. The general con- volution Taylor series (69) provides a far reaching generalizations of the results mentioned in this example. Example 3.2. It is very instructive to look at the limiting case of the formula (71) as α 1. As already mentioned, the function h1(x) = 1 is not a Sonine kernel and thus→ in this case we cannot directly apply the theory{ pres} ented above. However, we can use the relation (30) in sense of generalized functions and derive the following conventional Taylor series:

+∞ j x f(x)= a , a = I0 D1 f (0). (72) j j! j 0+ 0+ j=0 X 
 Let us note that the coefficients aj coincide with those known in the theory of the conventional Taylor series (see the definitions (28), (29), (31), and (32)):

0 1 (j) (j) aj = I D f (0) = (δ f )(0) = f (0), j N , 0+ 0+ ∗ ∈ 0 δ being the Diracδ-function.
 It is worth mentioning that in [9] some important convolution series were intro- duced and employed for analytical treatment of the initial-value problems for the single- and multi-term fractional differential equations with the GFDs. If we start with the geometric series +∞ Σ(z)= λjzj, λ C, z C (73) ∈ ∈ j=0 X that for λ = 0 has the convergence radius r =1/ λ , then Theorem 3.1 ensures that the convolution6 series (κ ) | | ∈L1 +∞ j lκ,λ(x)= λ κ (x), λ C (74) ∈ j=0 X is convergent for all x> 0 and defines a function from the space C−1(0, + ). For the kernel function κ = 1 , we immediately get the formula κ(x) = { } 1 (x)= hj (x). Then the convolution series (74) becomes a familiar power { } +1 12 YURI LUCHKO series for the exponential function: +∞ +∞ (λ x)j l (x)= λjh (x)= = eλx. (75) κ,λ j+1 j! j=0 j=0 X X In the case of the kernel κ(x)= hα(x) of the Riemann-Liouville fractional integral, the formula κ (x)= hα (x)= h(j+1)α(x) is valid and the convolution series (74) takes the form +∞ +∞ λj xjα l (x)= λjh (x)= xα−1 = xα−1E (λ xα), (76) κ,λ (j+1)α Γ(jα + α) α,α j=0 j=0 X X where the two-parameters Mittag-Leffler function Eα,α is defined by (25). Another interesting case is the kernel κ(x)= h1−β+α(x)+ h1−β(x), 0 <α<β< 1 (see the formula (23)). As shown in [9], the convolution series (74) takes in this case the following form: 1 +∞ j! (λx1−β+α)l1 (λx1−β)l2 lκ,λ(x)= = λx l1!l2! Γ(l1(1 β + α)+ l2(1 β)) j=0 l1 l2 j X X+ = − − 1 E (λx1−β, λx1−β+α), λx (1−β,1−β+α),0 where E(1−β,1−β+α),0 is a particular case of the multinomial Mittag-Leffler function

+∞ m li j! i=1 zi E(α1,...,αm),β(z1,...,zm) := m (77) l1! lm! Γ(β + αili) j=0 l1 l j i=1 X +···X+ m= × · × Q introduced for the first time in [19] (see also [20]). P In [9], the convolution series of type (74) and their generalizations were employed for derivation of analytical solutions to the initial-value problems for the fractional differential equation with the GFDs of Caputo type.

4. Generalized convolution Taylor formula In this section, we derive two different forms of the generalized convolution Taylor formula for representation of the functions from a certain space in form of the convolution polynomials with a remainder in terms of the GFIs and GFDs. We start with the case of the GFD defined in the Riemann-Liouville sense and first prove the following result formulated for the functions from the space C(1) (0, + )= −1,(k) ∞ f C−1(0, + ): (D(k) f) C−1(0, + ) (please note that this space does not coincide{ ∈ with the∞ space C1 ∈(0, + ) introduced∞ } in the previous section and the −1,(k) ∞ inclusion C1 (0, + ) C(1) (0, + ) is valid). −1,(k) ∞ ⊂ −1,(k) ∞ Theorem 4.1 (Second Fundamental Theorem of FC for the GFD in the R-L sense). Let κ and k be its associated Sonine kernel. For a function f C(1) (0, + ), ∈L1 ∈ −1,(k) ∞ the formula (I D f)(x)= f(x) (k f)(0)κ(x)= f(x) (I f)(0)κ(x), x> 0 (78) (κ) (k) − ∗ − (k) holds valid. CONVOLUTION SERIES 13

Proof. First we determine the kernel of the GFD D on the space C(1) (0, + ): (k) −1,(k) ∞ d (D f)(x)= ((I f)(x)=0 (I f)(x)= C, C R (k) dx (k) ⇔ (k) ∈ ⇔ (k f)(x)= C (κ (k f))(x)=(κ C )(x)= C(κ 1 )(x) ∗ ⇔ ∗ ∗ ∗ { } ∗ { } ⇔ ( 1 f)(x)=( 1 Cκ)(x) f(x)= C κ(x). { } ∗ { } ∗ ⇔ Thus, the kernel of D(k) is as follows: Ker D = C κ(x): C R . (79) (k) { ∈ } Let us now introduce an auxiliary function

F (x) := (I(κ) D(k) f)(x). (80) Because the GFD (14) is a left inverse operator to the GFI (21) on the space C (0, + ) (see the formula (41)), we get the relation −1 ∞ (D(k) F )(x)=(D(k) I(κ) D(k) f)(x)=(D(k) f)(x) and thus the function ψ(x) = F (x) f(x) belongs to the kernel of the GFD D(k), i.e., − ψ(x)= F (x) f(x)= C κ(x). (81) − To determine the constant C, we act on the last relation with the GFI I(k) and get the formula (I (F f))(x)=(k Cκ)(x)= C 1 = C. (k) − ∗ { } Otherwise,

(I k F )(x)=(I k (I κ D k f))(x)=((I k I κ ) D k f)(x)=( 1 D k f)(x). ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } ∗ ( ) Combining the last two relations, we arrive at the formula

( 1 D k f)(x) (I k f)(x)= C. (82) { } ∗ ( ) − ( ) (1) For a function f C (0, + ), the inclusion D k f C (0, + ) holds valid. ∈ −1,(k) ∞ ( ) ∈ −1 ∞ Thus 1 D(k) f C[0, + ) and ( 1 D(k) f)(0) = 0. Substituting x = 0 into the formula{ } ∗ (82) leads∈ to the∞ relation{ } ∗

C = (I k f)(0) − ( ) that together with the formulas (80) and (81) finalizes the proof of the theorem. 

For the Sonine kernel κ(x) = hα(x), 0 <α< 1, the representation (78) is well- known (see e.g. [18]): xα−1 (Iα Dα f)(x)= f(x) (I1−α f)(0) , x> 0, (83) 0+ 0+ − 0+ Γ(α) α α where I0+ and D0+ are the Riemann-Liouville fractional integral and derivative, respectively. It is also worth mentioning that in the case κ(x) = h1(x) the space of functions C(1) (0, + ) corresponds to the space of continuously differentiable functions and −1,(k) ∞ the formula (78) reads x f ′(ξ) dξ = f(x) f(0). − Z0 14 YURI LUCHKO

Now we generalize Theorem 4.1 to the case of the n-fold GFIs and the n-fold sequential GFDs in the Riemann-Liouville sense. This time, the result is formulated for the functions from the space C(n) (0, + ) = f C (0, + ): (D f) −1,(k) ∞ { ∈ −1 ∞ (k) ∈ C−1(0, + ), j = 1,...,n that in the case κ(x) = h1(x) corresponds to the space of n-times∞ continuously differentiable} functions. Theorem 4.2 (Second Fundamental Theorem of FC for the sequential GFD in the R-L sense). Let κ and k be its associated Sonine kernel. For a function ∈ L1 f C(n) (0, + ), the formula ∈ −1,(k) ∞ n−1 (I D f)(x)= f(x) k D f (0)κ(x)= (84) (κ) (k) − ∗ (k) j=0 X   n−1 f(x) I D f (0)κ(x), x> 0 − (k) (k) j=0 X   I D holds valid, where (κ) is the n-fold GFI (43) and (k) is the n-fold sequential GFD (45) in the Riemann-Liouville sense. Proof. To prove the formula (84), we repeatedly employ Theorem 4.1. For n = 2, we first get the representation I<2> D<2> I I D D ( (κ) (k) f)(x)=( (κ) (κ) (k) (k) f)(x)=

(I(κ)(I(κ) D(k) (D(k)f)))(x).

Then we apply Theorem (4.1) to the inner composition I(κ) D(k) acting on the func- tion (D(k) f) and get the formula

<2> <2> (I D f)(x)= I κ (D k f)(x) k D k f (0) κ(x) (x)= (κ) (k) ( ) ( ) − ∗ ( ) 

(I D f)(x) k D f (0) κ<2>(x). (κ) (k) − ∗ (k) I D Now we apply Theorem (4.1) once again, this time
to the composition (κ) (k) f at the right-hand side of the last formula and get the final result: (I<2> D<2> f)(x)= f(x) (k f)(0)κ(x) k D f (0) κ<2>(x). (κ) (k) − ∗ − ∗ (k) To proceed with the general case, we employ the following recurren
t formula: I D I I D D ( (κ) (k) f)(x)=( (κ) ( (κ) (k) ( (k) f)))(x)=

I (Df)(x) k Df (0)κ(x) (x), n =3, 4,.... (κ) (k) − ∗ (k) The representation h (84) easily follows from this recurrenti formula and the principle of the mathematical induction. 

Now we are ready to formulate and prove one of the main results of this section in form of the following theorem: CONVOLUTION SERIES 15

Theorem 4.3 (Generalized convolution Taylor formula for the GFD in the R-L sense). Let κ and k be its associated Sonine kernel. For a function f ∈ L1 ∈ C(n) (0, + ), the generalized convolution Taylor formula −1,(k) ∞ n−1 f(x)= aj κ (x)+ rn(x), x> 0 (85) j=0 X holds valid. The coefficients aj are given by the expression

aj = k D f (0) = I D f (0), j =0, 1 ...,n 1 (86) ∗ (k) (k) (k) − and the remainder can be represented  as follows: rn(x)=(I D f)(x)=(D f)(ξ)( 1 κ )(x), 0 <ξ x, (87) (κ) (k) (k) { } ∗ ≤ I D where (κ) is the n-fold GFI (43) and (k) is the n-fold sequential GFD (45) in the Riemann-Liouville sense. Proof. First we mention that the generalized convolution Taylor formula (85) with I D the coefficients (86) and the remainder in form rn(x)=( (κ) (k) f)(x) immedi- ately follows from the second Fundamental Theorem of FC for the sequential GFD in the Riemann-Liouville sense (Theorem 4.2). The second form of the remainder is obtained by application of the integral mean value theorem: x I D D ( (κ) (k) f)(x)= κ (x ξ)( (k) f)(ξ) dξ = 0 − x Z D D ( (k) f)(ξ) κ (x ξ) dξ = (k) f)(ξ)( 1 κ )(x), 0 <ξ x. 0 − { } ∗ ≤ Z 

In the case κ(x) = hα(x), 0 <α< 1 (the kernel of the Riemann-Liouville fractional integral), the generalized convolution Taylor formula (85) is reduced to the following known form ( [17]): n −1 xαj f(x)= xα−1 a + r (x), x> 0, (88) j Γ(α j + α) n j=0 X 1−α α aj = I D f (0), j =0, 1 ...,n 1 (89) 0+ 0+ − with the remainder 
 αn n α α α x rn(x)= I D f (x)= D f (ξ) , 0 <ξ x, (90) 0+ 0+ 0+ Γ(α n + α) ≤     α

α where I0+ is the Riemann-Liouville fractional integral and D0+ is the n-fold sequential Riemann-Liouville fractional derivative. The conventional Taylor formula is obtained from the formula (88) by letting α go to 1.
Now we proceed with the case of the GFD in the Caputo sense. This time, the results will be formulated for the functions from the space Cn (0, + )= f : f (n) −1 ∞ { ∈ C−1(0, + ) , n N. This space of functions was introduced in [16] and employed in [1, 8, 9]∞ for} derivation∈ of several results regarding the GFD in the Caputo sense. In particular, the following result was proved in [8]: 16 YURI LUCHKO

Theorem 4.4 (Second Fundamental Theorem of FC for the GFD in the Caputo sense). Let κ 1 and k be its associated Sonine kernel. Then, the relation∈L

(I κ D k f)(x)= f(x) f(0), x> 0 (91) ( ) ∗ ( ) − holds valid for the functions f C1 (0, + ). ∈ −1 ∞ For the Sonine kernel κ(x) = hα(x), 0 <α< 1 that generates the Caputo α fractional derivative ∗D0+ defined by (18), the formula (91) is well-known (see e.g. [16]): (Iα Dα f)(x)= f(x) f(0), x> 0, (92) 0+ ∗ 0+ − α where I0+ is the Riemann-Liouville fractional integral. As in the case of the GFD in the Riemann-Liouville sense, we generalize Theorem 4.4 to the case of the n-fold GFI and the n-fold sequential GFD in the Caputo sense. Theorem 4.5 (Second Fundamental Theorem of FC for the sequential GFD in the Caputo sense). Let κ 1 and k be its associated Sonine kernel. For a function f Cn (0, + ), the formula∈ L ∈ −1 ∞ n−1 (I D f)(x)= f(x) f(0) D f (0) 1 κ (x) (93) (κ) ∗ (k) − − ∗ (k) { } ∗ j=1 X  
I D holds valid, where (κ) is the n-fold GFI (43) and ∗ (k) is the n-fold sequential GFD (46). Proof. The formula (93) immediately follows from Theorem 4.4. Indeed, for n = 2, we first get the representation I<2> D<2> I I D D ( (κ) ∗ (k) f)(x)=( (κ) (κ) ∗ (k) ∗ (k) f)(x)=

(I(κ)(I(κ) ∗D(k) ( ∗D(k)f)))(x). Then we apply Theorem 4.4 and get the formula

<2> <2> (I D f)(x)= I κ ( D k f)(x) D k f (0) (x)= (κ) ∗ (k) ( ) ∗ ( ) − ∗ ( ) 

(I κ D k f)(x) (I κ ( D k f)(0))(x)= ( ) ∗ ( ) − ( ) ∗ ( ) f(x) f(0) ( D k f)(0)( 1 κ)(x). − − ∗ ( ) { } ∗ In the general case, the representation (93) immediately follows from the recurrent formula I D I I D D ( (κ) ∗ (k) f)(x)=( (κ) ( (κ) ∗ (k) ( ∗ (k) f)))(x)=

I ( Df)(x) Df (0) (x), n =3, 4,.... (κ) ∗ (k) − ∗ (k) and the principle ofh the mathematical induction. i 

The representation (93) can be rewritten in form of a generalized convolution Taylor formula. CONVOLUTION SERIES 17

Theorem 4.6 (Generalized convolution Taylor formula for the GFD in the Caputo sense). Let κ 1 and k be its associated Sonine kernel. For a function f Cn (0, + ), the∈ generalized L convolution Taylor formula ∈ −1 ∞ n−1 f(x)= f(0) + D f (0) 1 κ (x)+ Rn(x), x> 0 (94) ∗ (k) { } ∗ j=1 X  
holds valid with the remainder in the form Rn(x)=(I D f)(x)=( D f)(ξ)( 1 κ )(x), 0 <ξ x, (95) (κ) ∗ (k) ∗ (k) { } ∗ ≤ I D where (κ) is the n-fold GFI (43) and ∗ (k) is the n-fold sequential GFD (46) in the Caputo sense. The second form of the remainder is obtained by application of the integral mean value theorem: x (I D f)(x)= κ(x ξ)( D f)(ξ) dξ = (κ) ∗ (k) − ∗ (k) Z0 x ( D f)(ξ) κ(x ξ) dξ =( D f)(ξ)( 1 κ)(x), 0 <ξ x. ∗ (k) − ∗ (k) { } ∗ ≤ Z0 In the case κ(x)= hα(x), 0 <α< 1 (the kernel of the Caputo fractional deriva- tive), κ (x) = hαn(x) and the generalized convolution Taylor formula (94) takes the form: n−1 αj x f(x)= f(0) + Dα f (0) + R (x), x> 0 (96) ∗ 0+ Γ(α j + 1) n j=1 X 
 with the remainder n α α Rn(x)= I0+ ∗D0+ f (x), (97) α   α where I0+ is the Riemann-Liouville fractional
integral and ∗D0+ is the n-fold sequential Caputo fractional derivative. As we see, the generalized convolution Taylor formula involving
the Riemann- Liouville fractional derivative (formula (88)) and the one involving the Caputo frac- tional derivative (formula (96)) are completely different. Whereas the generalized convolution Taylor polynomial at the right-hand side of the formula (88) has an integrable singularity of a power function type at the origin, the Taylor polynomial at right-hand side of the formula (96) is continuous at the point x = 0. Letting n goto + in the formula (94) leads to the generalized convolution Taylor series for the GFD∞ in the Caputo sense in the form +∞ f(x)= f(0) + D f (0) 1 κ (x), x> 0. (98) ∗ (k) { } ∗ j=1 X  
The representation (98) holds valid under the condition that the convolution series at its right-hand side is a convergent one (this is the case, say, if the sequence D f (0), j N is bounded). ∗ (k) ∈  The generalized convolution Taylor formulas and the generalized convolution Tay- lor series that were presented in this section can be applied among other things for 18 YURI LUCHKO derivation of analytical solutions to the fractional differential equations with the GFDs in the Riemann-Liouville sense. This type of equations was not yet treated in the FC literature and will be considered elsewhere.

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[19] Yu. Luchko, Theory of the Integral Transformations with the Fox H-function as a Kernel and Some of Its Applications Including Operational Calculus. PhD. thesis, Belorussian State University, Minsk (1993). [20] S.B. Hadid, Yu. Luchko, An operational method for solving fractional differ- ential equations of an arbitrary real order. Panamerican Math. J. 6 (1996), 57–73. Current address: Beuth Technical University of Applied Sciences Berlin, Department of Math- ematics, Physics, and Chemistry, Luxemburger Str. 10, 13353 Berlin, Germany Email address: [email protected]