Fundamental solutions for semidiscrete evolution equations via Banach algebras

We give representations for solutions of time-fractional differential equations that involve operators on Lebesgue spaces of sequences defined by discrete convolutions involving kernels through the discrete Fourier transform. We consider finite difference operators of first and second orders, which are generators of uniformly continuous semigroups and cosine functions. We present the linear and algebraic structures (in particular, factorization properties) and their norms and spectra in the Lebesgue space of summable sequences. We identify fractional powers of these generators and apply to them the subordination principle. We also give some applications and consequences of our results.


Introduction
In this work, we study the following semidiscrete Cauchy problem: ⎧ ⎨ ⎩ ∂ t u(n, t) = Bu(n, t) + g(n, t), n ∈ Z, t > 0, u(n, 0) = ϕ(n), n ∈ Z, where B is the convolution operator in the discrete variable, that is, with b belonging to the Banach algebra 1 (Z). A typical example is the one-dimensional discrete Laplacian d , which can be obtained by taking b = δ -1 -2δ 0 + δ 1 , where δ i (j) denotes the Kronecker delta (or discrete Dirac measure). In such a case, equation (1.1) corresponds to the nonhomogeneous semidiscrete diffusion equation (also known as the semidiscrete heat equation or the lattice diffusion equation). The analytical study of such equations has received an increasing interest in the last decade, mainly due to many their applications in diverse areas of knowledge. For instance, in probability theory, the value u(n, t) in (1.1) with B = d describes the probability that a continuous-time symmetric random walk on Z visits a point n at time t; see [25,Sect. 4]. In chemistry, (1.1) describes the flow of a chemical in an infinite system of tanks arranged in a row, where each two neighbors are connected by pipes [42,Sect. 3], and in transport theory, (1.1) describes the dynamics of an infinite chain of cars, each being coupled to its two neighbors. The value u(n; t) is the displacement of car n at time t from its equilibrium position; see [24,Example 1]. From an analytical point of view, quite recently, Slavik [43] studied the asymptotic behavior of solutions of (1.1) when B = d , showing that a bounded solution approaches the average of the initial values if the average exists. Note that choosing b = δ -1δ 0 in (1.2), we obtain the forward difference operator B = , and hence (1.2) corresponds to the semidiscrete transport equation, studied recently by Abadias et al. [1].
It is interesting that in [22] and [37] the authors studied the fundamental solutions of (1.1) and the second-order semidiscrete equation ⎧ ⎨ ⎩ ∂ tt u(n, t) = Bu(n, t) + g(n, t), n ∈ Z, t > 0, u(n, 0) = ϕ(n), u t (n, 0) = φ(n), n ∈ Z, when B = -(d ) α is the discrete fractional Laplacian. Particularly, in [37] the authors combined operator theory techniques with the properties of the Bessel functions to develop a theory of analytic semigroups and cosine operators generated by d and -(d ) α . Also note that the fractional forward difference operator B = -(-) α was studied in [1], where the maximum and comparison principles in the context of harmonic analysis are proved. However, to our knowledge, to date, there is no attempt to investigate the fundamental solutions of the general equation (1.1) in a unified way. Our goal in this paper is to propose a solution to this problem.
Our key observation concerning this issue is that the discrete fractional Laplacian can be obtained from (1.2) by allowing the fractional powers of b to be an element of the Banach algebra 1 (Z). This original approach, which we provide in this paper, allows us to obtain new insights by introducing a completely new method to analyze both qualitative behavior and fundamental solutions of (1.1) in a unified way.
The Gelfand transform associated with ( 1 (Z), * ) is the discrete Fourier transform F : We recall that the spectrum of f , denoted σ 1 (Z) (f ), is defined by In what follows, we consider the general theory of commutative Banach algebras as a framework. We collect the results that will be of our interest in the following theorem.
The algebra 1 (Z) is a semisimple regular Banach algebra, and the discrete Fourier transform F is injective.
Proof The first claim follows from the fact that the algebra 1 (Z) has an identity; see, for example, [35], and the second one can be found in [35, p. 116 We observe that the range of the Gelfand transform is the Wiener algebra A(T), the pointwise algebra of absolutely convergent Fourier series, that is, F(e iθ ) = n∈Z f (n)e iθn , (θ ∈ T) with f ∈ 1 (Z). For F ∈ A(T), we also write F(z) = n∈Z f (n)z n for |z| ≤ 1.
The inverse discrete Fourier transform is given by the expressions for F ∈ A(T) (and for other functions in larger sets). The classical formulation of Wiener's lemma characterizes the functions F ∈ A(T) that are invertible in A(T) as follows. For F ∈ A(T) where F(e iθ ) = n∈Z f (n)e iθn for θ ∈ T, F(e iθ ) = 0 for all θ ∈ T if and only if 1/F ∈ A(T), that is, (1/F)(e iθ ) = n∈Z g(n)e iθn with (g(n)) n∈Z ∈ 1 (Z); in this case, f * g = δ 0 [32,Theorem 5.5].
Recall the definition of the classical Mittag-Leffler function (see (A.3)). We now introduce the following definition.
Note that The set exp( 1 (Z)) := {e a ; a ∈ 1 (Z)} is the connected component of δ 0 in the set of regular elements in 1 (Z) [35,Theorem 6.4.1]. We follow the usual terminology in semigroup theory: the element a is called the generator of the entire group (e za ) z∈C ; the cosine and sine functions are defined as Cos(z, a) := E 2,1 (z 2 a) and Sin(z, a) := zE 2,2 (z 2 a). We have Cos(s, a) ds, z ∈ C, for a ∈ 1 (Z); see [10,Sects. 3.1 and 3.14]. Moreover, the Laplace transform of an entire group or a cosine function is connected with the resolvent of its generator as follows: In particular, e zδ 1 = ∞ j=0 z j δ j j! and Cos(z, δ 1 ) = ∞ j=0 z 2j δ j (2j)! are generated by δ 1 .
Considering generalized versions of the Mittag-Leffler function, as well as of other hypergeometric series, as presented, for example, in [3][4][5], more examples can be easily derived.
In the next proposition, we collect some basic properties of these vector-valued Mittag-Leffler functions. As usual, we consider Bochner vector-valued integration in the Banach space 1 (Z); see, for example, [41,Sect. 1.2]. For the definition of the Wright function γ , see the Appendix, formula (A.1).
A nice application of the classical Wiener lemma is the invariance of spectrum for convolution operators defined on p (Z) for 1 ≤ p ≤ ∞. This issue is contained in the following theorem that is the key abstract result in this paper. Theorem 2.6 For a ∈ 1 (Z), we define (2.7) Then A ∈ B( p (Z)) for all 1 ≤ p ≤ ∞. Moreover, A = a 1 , and for all 1 ≤ p ≤ ∞, we have the following identities: For all a ∈ 1 (Z), we have that e za is an entire group in p (Z) with generator a, and for all 1 ≤ p ≤ ∞, we have the following identities: Proof From Young's inequality it follows that A ∈ B( p (Z)). Since the algebra p (Z) has the identity δ 0 , the property of the norm follows. The element a in the theorem is also called the symbol of the operator A.
Remark 2.7 It is also straightforward to check that the adjoint operator of A is again a convolution operator given by A (g)(n) := (ã * g)(n), wherẽ a(n) = a(-n), n ∈ Z.

Some finite difference operators in 1 (Z)
An important case of finite difference operators is given by sequences in the set c c (Z) := a ∈ 1 (Z) : ∃m ∈ Z + : a(n) = 0, ∀|n| > m) .
In such a case, the discrete Fourier transform of a ∈ c c (Z) is the trigonometric polynomial It is interesting to observe that if m j=-m a(j) = 0, then 0 ∈ σ 1 (Z) (a). This immediately follows from (2.8).
In this paper, we concentrate our study on the operators that appear in the seminal paper of Bateman [16]. Definition 3.1 For f ∈ p (Z) with 1 ≤ p ≤ ∞, we define the following operators: ( We remark that when considering the above-defined operators in the context of numerical analysis, the operators -and ∇ are related to the Euler scheme of approximation, and the operator d corresponds to the second-order central difference approximation for the second-order derivative. The operator dd appears in Bateman's paper [16, p. 506] in connection with the equations of Born and Karman on crystal lattices in vibration.

The operator -
The forward difference operator f (n) := f (n + 1)f (n) is a classical operator used in approximation theory and in the theory of difference equations. Considering it as an operator from p (Z) to p (Z), our main result is as follows.

Theorem 3.2
The operatorf = a * f , where a := δ 0δ -1 , possesses the following properties: (1) The norm is given by The norm of the group is given by e -ta 1 = 1, t > 0; Proof (1) The Minkowski inequality shows that ≤ 2. Then observe that δ 0 ∈ p (Z) with δ 0 p = 1 satisfies δ 0 p = 2, proving the claim. To prove (4), we apply (2.1) to get for |λ + 1| > 1. We show (5) directly: for z ∈ C and n ∈ Z. The norm e -ta 1 = 1 for t > 0 is straightforward from (5). Finally, to show (7), we apply the Laplace transform and formula (A.10) to get for n ≤ 0. By (4) we have that and we apply (2.3) to conclude the claimed equality and identify the generator of the cosine function with -a.
We remark that groups generated by are treated in [1, Sect. 2] and cosine functions in [16,Introduction].
We observe that when considering the Fourier transform in the context of signal processing, the conversion from continuous-time systems to discrete-time systems is done through the Euler transformation 1 -1 z . In such a context, it is important to remark that z -1 represents a delay in time.

The operator d
Theorem 3. 4 The operator d f = a * f , where a := δ -1 -2δ 0 + δ 1 , possesses the following properties: The group e za (n) = e -2z I n (2z), z ∈ C, n ∈ Z, and its generator is a; Proof Statements (1) and (2) follow as in the previous theorems. To prove (3), observe that To show (4), we proceed as in the previous theorems, obtaining where we have used (A.7) in the last identity. To prove (5), we use (4) and Appendix A.2(4).
To prove (6), we apply the Laplace transform and formula (A.8) to get for λ > 0 and n ∈ Z. By the principle of analytic continuation we can extend the equality to the set λ ∈ C \ [-4, 0]. Finally, to show (7), we apply again the Laplace transform and where we have applied (6) for λ > 0 and n ∈ Z.
We observe that groups generated by the discrete Laplacian d are treated in [21, Sect. 2] and cosine functions in [37, Theorem 1.2]. Here we have presented a complete and alternative approach using the framework of Banach algebras combined with the Laplace transform method.
Some simple computations show linear, algebraic, and dual relations between the operators defined previously, which are presented in the following result. (ii) For 1 ≤ p < ∞, we have the following identities on p (Z): (-) = ∇; (∇) = -; Proof The proofs are straightforward and left to the reader.
In the next theorem, we present a decomposition for the Bessel function, which seems to be new. For simplicity, for n ∈ Z and z ∈ C, we define

Fractional powers of generators of uniformly bounded semigroups in 1 (Z)
As we have commented in the introduction, to define fractional powers in a Banach algebra (and in operator theory) is, in general, a difficult task. Not every element in 1 (Z) has fractional powers. For example, δ 1 does not have square root in 1 (Z). In contrast, there may be a continuous function f ∈ C(T) such that (f (z)) 2 = z for z ∈ T. When σ 1 (Z) (a) ⊂ C + and α ∈ R, we may consider the function F α (z) = z α , which is holomorphic in a neighborhood of σ 1 (Z) (a). By the analytic functional calculus, the element As the next definition shows, we may follow a general methodology to treat fractional powers of elements in 1 (Z), analogously to the case of operators in Banach spaces; see [47, p. 265]. Definition 4.1 Let 0 < α < 1, and let a ∈ 1 (Z) be such that (e ta ) t≥0 is a uniformly bounded semigroup, that is, sup s>0 e as 1 < ∞. Then we write by (-a) α the fractional power of a given by the following integral representation: Remark 4.2 In fact, Definition 4.1 is an analogous formula in 1 (Z) of the well-known equality As an immediate consequence of this definition, we have that for 0 < α < 1, where we have applied (2.8).
It is well known that the uniformly bounded semigroup (e -t(-a) α ) t≥0 is subordinated to (e ta ) t≥0 (principle of Lévy subordination) by the formula Now we present the fractional powers of the four elements in 1 (Z) given in Definition 3.1. For a ∈ 1 (Z), note that A ∈ B( p (Z)), where A(f ) := a * f for f ∈ p (Z) and 1 ≤ p ≤ ∞. In the case that the fractional power a α ∈ 1 (Z) for α > 0, we have A α ∈ B( p (Z)), where A α (f ) := a α * f for f ∈ p (Z) and 1 ≤ p ≤ ∞.

Theorem 4.3
For all 0 < α < 1, we have for z ∈ T and In particular, σ (K α Proof Using the first identity in (4.1), we obtain proving the first identity in (4.5). On the other hand, note that [ ∞ j=0 k -α (j)δ -j ](n) = k -α (-n). Therefore, taking into account (4.3), we obtain Then by the uniqueness of the Fourier transform we conclude the first identity in (4.6). The proof of the second identities in (4.5) and (4.6) is analogous. The property of the spectrum follows from the second identity in (4.1). Finally, we have that and we conclude the proof.
Now we consider the groups generated by the fractional powers K α + and K α as elements of the Banach algebra 1 (Z).
We summarize the main properties of the kernel K α d in the following result.

Theorem 4.6
For 0 < α < 1, we have and In particular, where k -α -(n) := k -α (-n). Moreover, σ (K α d ) = [0, 4 α ], and Proof Identity (4.7) follows from (4.1). To show (4.8), we apply the discrete Fourier transform to obtain that for z ∈ T. Since the discrete Fourier transform is one-to-one, we obtain the equality. To prove (4.9), we note that the right-hand side evaluated at n ∈ Z is equal to (k -α - * k α )(n), and we have and the result follows from (4.8). The spectrum is given in [ An interesting consequence is the following corollary, which seems to be a new formula for binomials of noninteger entries.

Corollary 4.7
Let α ∈ (0, 1) and n ∈ N 0 . We have the following equality: Proof The combinatorial equality is a straightforward consequence of the explicit expression of the kernel convolutions K α + , K α -, and K α d .
The following result collects the main results on the fractional discrete semigroup. For other results, see also [37]. Theorem 4.8 For any 0 < α < 1, we have that the fractional discrete semigroup generated by -K α d is given by for n ∈ Z and z ∈ C. Moreover: (i) The discrete Fourier transform of e -zK α d is given by (ii) e -tK α d (n) ≥ 0, and e -tK α d 1 = 1 for n ∈ Z and t ≥ 0, that is, it is a Markovian semigroup.
Proof The fractional discrete semigroup generated by -K α d is given in [37, Theorem 1.3]. There the entire group (e -zK α d ) z∈C is written as L α z . Statement (i) follows from Proposition 2.4(ii) combined with (4.7) in Theorem 4.6. The proof of (ii) is contained in [37, Theorem 1. 3(v)]. Finally, to prove (iii), we use (2.9) in Theorem 2.6 and (4.7) in Theorem 4.6.
Now we consider the entire group (e -zK α dd ) z∈C generated by -K α dd .

Fundamental solutions for semidiscrete evolution equations
In this section, we consider the operator Bf (n) : , and n ∈ Z. Our objective is obtaining a fundamental representation of solutions for the following semidiscrete fractional evolution equation: where β ∈ (0, 2]. For a sufficiently regular function v, we denote by D β t the Caputo derivative of order β given by for 1 < β < 2. For β = 1 and β = 2, we consider the usual first-and second-order derivatives. Note that see, for example, [17,30]. To begin with, we consider the semidiscrete Cauchy problem (1.1) given in the introduction, ⎧ ⎨ ⎩ ∂ t u(n, t) = Bu(n, t) + g(n, t), n ∈ Z, t > 0, and its fundamental solution, which is obviously given by Duhamel's formula u(n, t) = e Bt ϕ(n) + t 0 e B(t-s) g(n, s) ds, n ∈ Z, t ≥ 0.
Analogously, in the case of the second-order semidiscrete Cauchy problem we have that the fundamental solution is given by D' Alembert formula where Cos(t, B) and Sin(t, B) are generated by B.
We now consider fractional in time generalizations. Given 0 < β ≤ 1, we first consider the equation n ∈ Z.

(5.4)
We recall that E α,β (b) (with b ∈ 1 (Z)) is the vector-valued Mittag-Leffler function given in Definition 2.2. The main result is the following theorem.
(i) For 0 < β < 1, the function is the unique solution of the initial value problem (5.4). Moreover, u(·, t) belong to p (Z) for t > 0.
(ii) For 1 < β < 2, the function is the unique solution of the initial value problem (1.5). Moreover, u(·, t) belong to p (Z) for t > 0.
Proof Since the algebra 1 (Z) is semisimple (see Theorem 2.1), the formulae in (i) and (ii) are direct consequences of the scalar identities, which in case 0 < α < 1 can be found in  (11)]. See also the references therein.
Remark 5.2 Now we consider the behavior of the solution as β tends to the integer parameter, that is, β = 1, 2. For simplicity, we consider the homogeneous case g = 0. As β → 1 -, the solution of equation (1.4) converges to semigroup family operators E 1,1 (tb), and as β → 2 -, the solution of equation (1.5), converges to unique mild solution of second-order Cauchy problem, that is, the sum of a cosine function and a sine function generated by b; see [10, Corollary 3.14.8].
Note that this function is a solution of the following first-order modified Cauchy problem: for φ, ϕ ∈ p (Z). This fact is in accordance with the interpolation property of the Caputo fractional derivative; see (5.1).
The fundamental solutions u β,1 for systems (1.4) and (1.5) are obtained by requiring that the initial value ψ and the initial velocity φ be the sequences ψ = δ 0 and φ = 0. In the case 1 < β ≤ 2 (including the wave equation), a second fundamental solution u β,2 is given by ψ = 0 and φ = δ 0 ; see [26,Remark 3.2]. A consequence of Theorems 5.1 and 2.5 is the following subordination theorem for fundamental solutions, which extends [26, Corollary 3.5]. (i) Let 0 < β < 1. Then can be expressed in terms of the Airy function Ai(z), that is, 1 3 (z) = 3 2 3 Ai z 3 1 3 , z ∈ C; see, for example, [28]. A integral representation of the Airy function is given by the improper Riemann integral: This function appears in several applied problems, in particular, in the Schrödinger equation of quantum physics and in optics (study of caustics); see more details in [44]. By Corollary 5.3(i) we conclude that The particular case of Theorem 5.1 with B = -(-A) α , where A is the infinitesimal generator of an uniformly bounded C 0 -semigroup in B( p (Z)), has received a special attention.
. As a consequence of the results in Sect. 4, we can easily give a general version, which extends both results.

Applications
We study some concrete examples that appear in various applied fields.

The discrete Nagumo equation
Let us consider the linear part of the discrete Nagumo equation, which can be written as follows: ⎧ ⎨ ⎩ ∂ t u(n, t) = d u(n, t)ku(n, t), n ∈ Z, t > 0, u(n, 0) = ϕ(n), n ∈ Z, where 0 < k < 1/2. The discrete Nagumo equation is used as a model for the spread of genetic traits and for the propagation of nerve pulses in a nerve axon, neglecting recovery; see [48] and references therein. Using Theorem 3.4(3), we obtain σ e t( d -kI) = e tσ ( d -kI) = e ts : t ≥ 0, -4k ≤ s ≤ -k .
This implies that the unique solution of equation (6.1) is uniformly asymptotically stable, that is, Moreover, using Theorem 3.4(4) and the semigroup property, we can obtain the following representation of the fundamental solution:

The semidiscrete transport equation associated with the r-difference operator
Let us consider the semidiscrete transport equation where r > 0, and r is the r-forward difference operator defined by r f (n) := f (n + 1)rf (n); see [7,Sect. 5.5]. Observe that r = + (1r)I, where I is the identity operator. Then by perturbation semigroup theory the unique solution of (6.2) has the form u(n, t) = e t( +(1-r)I) ϕ(n), n ∈ Z. By the spectral mapping (2.9) in Theorem 2.6 we obtain that σ e t( +(1-r)I) = e tσ ( +(1-r)I)) .
Hence by Theorem 3.2(3) we deduce that σ ( + (1r)I) = {z ∈ T : |z + r| = 1}. Therefore for any r > 1, we have that the upper bound of the spectrum of B = + (1r)I is negative, that is, ω σ + (1r)I := sup z : z ∈ σ + (1r)I < 0, and, consequently, we obtain that for any r > 1, the unique solution of equation (6.2) is uniformly asymptotically stable, that is, uniformly with respect to ϕ ≤ 1. Of course, this result can be also directly deduced from Theorem 3.2 (5). Analogously, using the fact that re it > 0 implies re it α > 0 for any 0 < α < 1 and r > 1, we can deduce from Theorem 4.4 that the same property of asymptotic stability remains true for the unique solution of the fractional semidiscrete transport equation

The De Juhasz equation
We consider the following semidiscrete equation: In particular, this implies that on the Hilbert space 2 (Z), we have Cos(t) ≤ 1, and, consequently, the unique solution of (6.4) when ψ ≡ 0 must be bounded. This extends the previous result of Bateman [16,Sect. 5], who studied (6.4) with the initial conditions ψ ≡ 0 and ϕ(n) = δ 0 (n).

The Caputo-Fabrizio derivative
We recall that given a sufficiently regular function u and 0 < α < 1, the Caputo-Fabrizio derivative of order α is defined as [19] Note that the Caputo-Fabrizio derivative has been very recently used to propose a new mathematical modeling of human liver [12], HIV [13], parallel RCL circuits [8], the Rubella disease model [14], epidemic childhood diseases [11], and COVID-19 [15]. However, with the exception of the implicit solution for the linear model in the scalar case, proposed by Losada and Nieto in [38], so far no explicit formulas have been proposed for the solution of the fractional Cauchy problem in the context of Banach algebras.
We consider the equation where we recall that Bf (n) : Since B is bounded, assuming that 1 1-α ∈ ρ(B), we obtain the following representation for the solution of (6. (note that there is a small but important misprint in [38, p. 90, l. 16], where we must read σ (t) instead ofσ (t)). Using the identity and hence (6.6) follows by using the classical Duhamel formula.
In terms of Banach algebras, this result reads as follows. The proof is similar to that of Theorem 5.1 and is therefore omitted. Theorem 6.1 Let ϕ ∈ p (Z), and let g : Z × R + → C be such that for each t ∈ R + , g(·, t) ∈ p (Z), t → g(t, n) is differentiable and sup s∈[0,t] g(·, s) p < ∞ with 1 ≤ p ≤ ∞. For 0 < α < 1, is the unique solution of the initial value problem (6.5), where Moreover, u(·, t) belong to p (Z) for t > 0.

Applications to special functions
In this section, we present some new formulae obtained as applications of the results proved in this paper. We give expressions of generalized Mittag-Leffler functions for concrete fractional powers. We also interpret some known formulas in terms of the Weierstrass subordination formula. Finally, the application of subordination principle to Wright function of some concrete difference operators allows us to obtain some new integral formulae for Bessel and Wright functions. Theorem 7.1 The generalized Mittag-Leffler functions E γ ,β for K α + , K α -, K α d , and K α dd with γ , β > 0 and 0 < α < 1 are given as follows:

Weierstrass formula
A relation between cosine functions and semigroups generated by an element a ∈ 1 (Z) is established by the Weierstrass formula in (2.6). Now we apply this formula to concrete finite difference operators.
As it is commented in [37,Remark 3], this formula is a particular case of the general equality

Subordination principle on Wright function
In this subsection, we apply Corollary 5.3 to finite difference operators. We obtain some known formulae, but others seem to be new; see (7.1), (7.2), and (7.3). They give some interesting new connections between the Wright and Bessel functions. Take a = δ -1δ 0 or a = δ 1δ 0 in Corollary 5.3.
For n = 0, we obtain formula (A.4) and for t = 0, formula (A.2). For β = 1 3 , we obtain the following integral formula for the Airy function: where γ is a contour that starts and ends at -∞ and encircles the origin once counterclockwise; see, for example, [28, formula (28)]. These functions were initially studied by Wright [46] in connection with the asymptotic theory of partitions. For 0 < α < 1, we have the following properties: ∞ 0 α (t) dt = 1. It follows that α is a probability density function on R + . In fact, the Wright function has been used for models in stochastic processes; see, for example, [28,29]. In a general sense, as α → 1 -, we may interpret that α → δ 1 , where δ 1 is the Delta measure concentrated in t = 1 ( [28]).
The following formula is known for the moments of the Wright function: ∞ 0 x p α (x) dx = (p + 1) (αp + 1) , 0<α < 1, p > -1; (A.2) see [28]. The Mittag-Leffler function is a complex function depending on two complex parameters α and β. When the real part of α is strictly positive, it may be defined by the series where is the gamma function. When β = 1, it is abbreviated as E α (z) = E α,1 (z). An interesting fact is the following relationship between the Wright function and the Mittag-Leffler functions: For z ∈ C and 0 < α < 1, we also have see [30, formula 2.29]. For more detail on the Wright functions, we refer to [28-30, 39, 40, 46] and the references therein.