Dirichlet Averages of Generalized Mittag-Lefﬂer Type Function

: Since Gösta Magus Mittag-Lefﬂer introduced the so-called Mittag-Lefﬂer function in 1903 and studied its features in ﬁve subsequent notes, passing the ﬁrst half of the 20th century during which the majority of scientists remained almost unaware of the function, the Mittag-Lefﬂer function and its various extensions (referred to as Mittag-Lefﬂer type functions) have been researched and applied to a wide range of problems in physics, biology, chemistry, and engineering. In the context of fractional calculus, Mittag-Lefﬂer type functions have been widely studied. Since Carlson established the notion of Dirichlet average and its different variations, these averages have been explored and used in a variety of ﬁelds. This paper aims to investigate the Dirichlet and modiﬁed Dirichlet averages of the R -function (an extended Mittag-Lefﬂer type function), which are provided in terms of Riemann-Liouville integrals and hypergeometric functions of several variables. Principal ﬁndings in this article are (possibly) applicable. This article concludes by addressing an open problem.


Introduction and Preliminaries
The Mittag-Leffler function E α (z) (see [1]) Γ being the familiar Gamma function (see, for example, Section 1.1 in [2]), is named after the eminent Swedish mathematician Gösta Magus Mittag-Leffler (1846-1927), who explored its features in 1902-1905 in five notes (consult, for instance, [1]) related to his summation technique for divergent series (see also Chapter 1, [3]). Because Γ( + 1) = ! ( ∈ N 0 ) and therefore E 1 (z) = e z , this function gives a straightforward extension of the exponential function. Here and elsewhere, let N, Z − 0 , R, R + , and C be the sets of positive integers, non-positive integers, real numbers, positive real numbers, and complex numbers, respectively, and put N 0 := N ∪ {0}. Passing the first half of the 20th century during which the majority of scientists remained almost unaware of the function, the Mittag-Leffler function and its various extensions (referred to as Mittag-Leffler type functions) have been studied and applied to a wide range of problems in physics, biology, chemistry, engineering, etc. This function's most significant features are described in Chapter XVIII [4], which is dedicated to so-called miscellaneous functions. The Mittag-Leffler function was categorized as miscellaneous because it was not until the 1960s that it was discovered as belonging to a broader class of higher transcendental functions known as Fox H-functions, thus the term "miscellaneous" (consult, for instance, [5]). In reality, this class was not well-established until Fox's landmark study (see [6]). The simplest (and most crucial for applications) extension of the Mittag-Leffler function, notably the two-parametric Mittag-Leffler function was separately studied by Humbert and Agarwal in 1953 (see, for example, [7]) and by Dzherbashyan in 1954 (see, for example, [8]). However, it first appeared formally in Wiman's article [9]. Prabhakar [10] introduced the following three-parametric Mittag-Leffler function: where (λ) ν denotes the Pochhammer symbol defined (for λ, ν ∈ C) by it being accepted conventionally that (0) 0 := 1. This Function (3) is being used for a variety of applicable issues. Scientists, engineers, and statisticians recognize the significance of the aforementioned H-function due to its great potential for applications in several scientific and technical domains. In addition to the Mittag-Leffler Functions (1)-(3), the H-function includes a variety of functions (see, for example, [5]). Among several monographs on the H-function, monograph [5] discusses the theory of the H-function with a focus on its applications. The H-function (or Fox's H-function [6]) is defined by means of a Mellin-Barnes type integral in the following manner (consult also [5]): where ω = √ −1, and We also assume the following: z −s = exp[−s{ln |z| + i arg z}], where ln |z| is the natural logarithm, and η < arg z < η + 2π for some η ∈ R. The integration path L = L iγ∞ (γ ∈ R) extends from γ − i∞ to γ + i∞ with indentations, if necessary, so that the poles of Γ(1 − a j − α j s) (1 ≤ j ≤ n ∈ N 0 ) can be separated from those of Γ(b j + β j s) (1 ≤ j ≤ m ∈ N 0 ) and has no those poles on it. The parameters p, q ∈ N 0 satisfy the conditions 0 ≤ n ≤ p, 0 ≤ m ≤ q; the parameters α j , β j ∈ R + and a j , b j ∈ C. The empty product in (6) (and elsewhere) is (as usual) understood to be unity.
Carlson developed the notion of the Dirichlet average in his work [13] (see also [14][15][16][17][18]). Carlson also provided a full and thorough analysis of the numerous varieties of Dirichlet averages. A function's so-called Dirichlet average is the integral mean of the function with regard to the Dirichlet measure. Subsequently and more recently, this study topic has been explored in publications such as [19][20][21][22][23][24][25][26][27][28]. Neuman and Van Fleet [19] defined Dirichlet averages of multivariate functions and demonstrated their recurrence formula. Daiya and Kumar [20] researched the double Dirichlet averages of S-functions. Saxena and Daiya [29] proposed and explored the S-functions. Kilbas and Kattuveettill [22] investigated Dirichlet averages of the three-parametric Mittag-Leffler Function (3), whose representations are provided in terms of the Riemann-Liouville fractional integrals and the hypergeometric functions with multiple variables. Saxena et al. [25] explored Dirichlet averages of the generalized multi-index Mittag-Leffler functions (see, for instance, [30]), whose representations are expressed in terms of Riemann-Liouville integrals and hypergeometric functions of several variables. Using Riemann-Liouville fractional integral operators, Vyas [31] investigated the solution of the Euler-Darboux equation in terms of Dirichlet averages of boundary conditions on Hölder space and weighted Hölder spaces of continuous functions. For further Dirichlet averages in connection with fractional calculus, one may consult [21,24,[32][33][34][35][36]. These Dirichlet averages were used in a number of studies, in particular, Dirichlet splines (see [19]), B-splines (see [18,23]), and Stolarsky means (see [37]).
In this work, we propose to investigate the Dirichlet and modified Dirichlet averages of the R-function (an extended Mittag-Leffler type function) (see, for details, Section 2). Main results stated in this paper, which are presented in terms of Riemann-Liouville integrals and hypergeometric functions of several variables, are (potentially) useful.
Let Ω be a convex set in C and z := (z 1 , . . . , z n ) ∈ Ω n (n ∈ N \ {1}). Suppose that f is a measurable function on Ω. Then the general Dirichlet average of the function f is defined as follows (see [15]): where b and u denote the arrays of n parameters b 1 , . . . , b n and u 1 , . . . , u n , respectively, and dµ b (u) is the Dirichlet measure defined by and E n−1 is the Euclidean simplex in R n−1 (n ∈ N \ {1, 2}) given by and B(b) is the multivariate Beta-function defined by Here and throughout this paper, the notation 1, p := {1, . . . , p} (p ∈ N) is used. The special case of (11) when n = 2 reduces to the following form: Carlson [15] investigated the average (10) for the function f (z) = z k (k ∈ R) in the following form: whose special case n = 2 was given as follows (see [13,15]): where β, β ∈ C with min{ (β), (β )} > 0, and x, y ∈ R, B(β, β ) is the familiar Beta function (consult, for instance, Chapter 1, [2]). The Riemann-Liouville fractional integral of a function f is defined as follows (consult, for instance, (p. 69) [38]): For α ∈ C with (α) > 0 and a ∈ R, The Srivastava-Daoust generalization F A:B (1) ;...;B (n) C:D (1) ;...;D (n) of the Lauricella hypergeometric function F D in n variables is defined by (see (p. 454) [39]; see also (p. 37) [40], (p. 209) [5]) where the coefficients, for all k ∈ 1, n, are real and positive, and (a) abbreviates the array of A parameters a 1 , . . . , a A , b (k) abbreviates the array of B (k) parameters b (k) j j ∈ 1, B (k) for all k ∈ 1, n, with similar interpretations for (c) and d (k) (k ∈ 1, n); et cetera. One may refer to Srivastava and Daoust [41] for the specific convergence requirements of the multiple series (17).

The Generalized Mittag-Leffler Type Function (the R-Function)
The R-function, which Kumar and Kumar [42] proposed and Kumar and Purohit [43] studied, is defined as follows: where a j n (j ∈ 1, p) and b j n (j ∈ 1, q) are the Pochhammer symbols in (4).
The series (18) is defined when If any parameter a j is a negative integer or zero, then the series (18) terminates to become a polynomial in z.

Bivariate Dirichlet Averages
The Dirichlet average of the generalized Mittag-Leffler type Function (18) is denoted and defined as follows: where (a) 1,n and (b) 1,n (n ∈ N) denote the horizontal arrays a 1 , . . . , a n and b 1 , . . . , b n , respectively; z = (x, y) ∈ R 2 and min{ (β), (β )} > 0. In fact, it is shown that the Dirich-let average of the R-function (18) is stated in terms of the Riemann-Liouville fractional integrals (16) claimed by Theorems 1 and 2. Theorem 1. Let z, α, β, β , δ, γ, κ ∈ C such that (α) > max{0, (κ) − 1} and min{ (κ), (β), (β )} > 0. Also let x, y ∈ R with x > y and I β 0+ be the Riemann-Liouville fractional integral given in (16). Then the Dirichlet average of the generalized Mittag-Leffler type function (18) is given by the following formula: where the function f is given by Proof. With the aid of (10) to (13), by applying the R-function (18) to (23), we find that By changing the order of integration and summation, which is verified under the stated conditions, we get Setting t := u(x − y), we find that Then, using (16) and (18), we arrive at the desired result in (24). This completes the proof.
We take into account the following modification to the Dirichlet average in (23): where λ ∈ C with (λ) > 0 and z = (x, y).

Dirichlet Average Expressed in Terms of Srivastava-Daoust Function
This section discusses an alternative formulation of the modified Dirichlet averages of the R-function. Theorem 3. Let β, β , δ, λ ∈ C with min{ (β), (β ), (λ)} > 0 and x, y, κ, α, γ ∈ R with x > y and min{κ, α, −γ} > 0. The convergence conditions of the R-function are supposed to be satisfied. Then the following formula holds true: where a ( ) , here and throughout this paper, abbreviates the array of times repetition of the same parameter a's, a,. . .,a, and (a) and (b) abbreviate the arrays of p and q parameters a 1 , . . . , a p and b 1 , . . . , b q , respectively.
Proof. In view of (28) and (15), we have Exchanging the order of integral and summation and using the generalized binomial series (a) n z n n! (|z| < 1; a ∈ C) and the Beta function, we obtain Applying Γ(λ + ν) = Γ(λ) (λ) ν (λ, ν ∈ C) and which, in view of (17), leads to the right-hand side of (30). This completes the proof.
The Lauricella function F D defined for complex parameters d = (d 1 , . . . , d n ) ∈ C n is defined as follows (consult, for example, Section 1.4 in [40]): The series (32) converges for all variables inside unit circle max 1≤j≤n z j < 1.

Concluding Remarks
The Dirichlet and modified Dirichlet averages of the R-function in (18) (a generalized Mittag-Leffler type function) were explored. In Theorems 1 and 2, the bivariate Dirichlet averages of the R-function (18) were expressed in terms of the Riemann-Liouville fractional integrals whose kernel functions are products of some elementary functions and the R-function (18). In Theorem 3, the bivariate Dirichlet average of the R-function (18) (see Theorem 2) was shown to be expressed in terms of the Srivastava-Daoust generalization (17) of the Lauricella hypergeometric function. In Theorem 4, the multivariate Dirichlet average of the R-function (18) was proven to be expressed in terms of the Srivastava-Daoust generalization (17) of the Lauricella hypergeometric function. The main results in Theorems 1-4 are believed to be useful.
The Mittag-Leffler function E α (z) in (1), the two-parametric Mittag-Leffler function E α,β (z) in (2), the three-parametric Mittag-Leffler function E γ α,β (z) in (3), and the R-function in (18) are obviously contained as special cases in the well-known Fox-Wright function p Ψ q (see, for details, p. 21 [40]; see also p. 56 [38]). Because the R-function in (18) is of general character, all results in Theorems 1-4 are seen to be able to yield a large number of particular instances. The following corollary demonstrates just a particular instance of Theorem 1: Corollary 1. Let the conditions in Theorem 1 be satisfied and set p = q = 1 and a j = b j = 1 in (24). Then the Dirichlet average for the generalized Mittag-Leffler function holds true: κ 1 M α,δ;γ 1 β, β ; x, y = Γ(β + β ) where E γ,κ α,δ is given in (21).
As with the H-function of the single variable in (5), the H-function of multiple variables is generated using multiple contour integrals of the Mellin-Barnes type (see pp. 205-207, Appendix A.1 in [5]). This article concludes with the questions posed: Like (8) Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.