Certain Generating Relations Involving the Generalized Multi-Index Bessel–Maitland Function

Shilpi Jain, Juan J. Nieto , Gurmej Singh, and Junesang Choi Department of Mathematics, Poornima College of Engineering, Jaipur 302022, India Instituto de Matematicas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain Department of Mathematics, Mata Sahib Kaur Girls College, Talwandi Sabo, Bathinda 151103, India Department of Mathematics, Singhania University, Pacheri Bari, Jhunjhunu, India Department of Mathematics, Dongguk University, Gyeongju 38066, Republic of Korea

If q � 1, c � 1, ] is replaced by ] − 1, and z is replaced by − z in (2), then generalized Bessel-Maitland function reduces to the Mittag-Leffler function which was studied by Wiman [3] as follows: If ] is replaced by ] − 1 and z is replaced by − z in (2), then the generalized Bessel-Maitland function reduces to the well-known generalized Mittag-Leffler function E c,q λ,] (z) which was introduced by Shukla and Prajapati [4] as follows: Jain and Agarwal [5] generalized Bessel-Maitland function J λ ] (z) (1) as follows: Choi and Agarwal [6] investigated the following generalized multi-index Bessel function: where m ∈ N and λ j , ] j , c, q, and z ∈ C (j � 1, . . . , m) such that Remark 2. It is easily found that generalized multi-index Bessel-Maitland function (9) is equivalent to the generalized multi-index Mittag-Leffler function defined and studied by Saxena and Nishimoto [7] (see also [8]). Pohlen [9] introduced the Hadamard product (or the convolution) f * g of two analytic functions f and g as follows: (11) where R ≥ R f · R g . Here, f(z) and g(z) are analytic at z � 0 whose Maclaurin series with their respective radii of convergence R f and R g are e concept of the Hadamard product has turned out to be useful, particularly, in factorizing a newborn function, which is usually expressed as a Maclaurin series, into two known functions (see, e.g., [10][11][12][13]). e k-th derivative of the function f(p) � p − λ− nξ (λ, ξ ∈ C, n ∈ N) is easily found to be given in terms of gamma function as follows: Generating functions have been widely used in exploring certain properties and formulas involving sequences and polynomials in a wide range of research subjects. Many researchers have developed a remarkably large number of generating functions associated with a variety of special functions. For some works on this subject, one may refer, for example, to an extensive monograph [14][15][16][17][18][19][20][21][22][23][24][25] and the literature cited therein. In this search, we aim to provide some presumably new generating relations in connection with generalized multi-index Bessel-Maitland function (9). e main results developed here, being very general, can be reduced to produce a large number of presumably new and potentially useful generating relations for other known functions, some of which are demonstrated.

Generating Relations
We give two generating relations involving generalized multi-index Bessel-Maitland function (9) asserted by the following theorems.
Also, let |t| < 1. en, Proof. We replace 1 + t by s in the left-hand side of (15) and denote the resulting expression by g(s). en, using form (9), on expanding the function in series, gives Differentiating k times both sides of (16) with respect to s with the aid of (13) (term-by-term differentiation can be verified under the given conditions), we find 2 Mathematical Problems in Engineering which is simplified to yield Decomposing series (18) into Hadamard product (11), we obtain Expanding g(s + t) as the Taylor series gives Combining (16), (19), and (20), we obtain Finally, setting s � 1 yields desired result (15).
Also, let |t| < 1. en, Proof. Let J be the left-hand side of (23). Using (9), on expanding the function in series, gives Interchanging the order of summations in (24) and using the known identity (see, e.g., [26, p. 5 we have Using the generalized binomial expansion, we find that the inner sum in (26) gives Finally, interpreting (26) with the help of (27) yields desired result (23).

Further Remarks
Here, we choose to give some equivalent identities and particular cases of the results in eorems 1 and 2. As noted in Remark 2, setting ] j by ] j − 1 and z by − z in (15) and (23) gives two corresponding generating relations involving the generalized multi-index Mittag-Leffler function E c,q (λ j ,] j ) m (z), which are asserted, respectively, in Corollaries 1 and 2.