Integral transforms of an extended generalized multi-index Bessel function

1 Department of Mathematics, University of Sargodha, Sargodha, Pakistan 2 Department of Mathematics, University of Lahore, Sargodha, Pakistan 3 Department of Mathematics, Shaheed Benazir Bhutto University, Sheringal, Upper Dir, 18000, Pakistan 4 Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, KSA 5 Department of Medical Research, China Medical University, Taichung 40402, Taiwan 6 Department of Mathematics, College of Arts and Sciences, Wadi Aldawser, 11991, Prince Sattam bin Abdulaziz University, Saudi Arabia


Introduction
The Bessel function [1][2][3][4][5][6][7][8] has great importance in the field of mathematics, physics and engineering due to its applications. Researchers and mathematicians developed a new class of Bessel functions in the sense of multi-index functions, which motivate the future research work in the field of special functions and fractional calculus. The theory of multi-index multivariate Bessel function discussed by Dattoli et al. [9] in 1997.
is defined in the following way: .
Remark 1.1. The E 1 GMBF can also be write as (1.18)

Particular special cases
In this section, we establish some particular special cases of E 1 GMBF as below • if we set p = 0, then E 1 GMBF reduce into extended multi-index Bessel function . (2.1) .

Results of E 1 GMBF
In this section, we investigate the E 1 GMBF, and studied some important observations. Moreover, we develop integral and differential of E 1 GMBF in the form of theorems.
Theorem 3.1. The E 1 GMBF can be able to represent with α j , β j , b, δ, γ, c ∈ C ( j = 1, 2 · · · m) be such that m Proof. Using the definition of Eq (1.8) in (1.17), we obtain Changing the order of summation and integration, and after simplification of Eq (3.2), we get dt.
1+r in theorem 3.1, then following relation holds (3.5) Proof. Consider the definition of (1.17) for j = 1, and the right side of the Eq (3.6), we get Theorem 3.3. For the E 1 GMBF we have the following higher derivative formula for δ = 1, is given below Proof. Differentiation with respect to z in Eq (1.17), we get we can write the pochhammer symbols as (3.10) Now, using the Eq (3.10) in Eq (3.9), we have Again differentiation with respect to z in Eq (3.9), we have continue this technique up to n times, we obtain the desired result which state in the theorem 3.3. Proof. Replacing z by λz α j ···α j , b = −1 and δ = 1 in Eq (1.17), take its product z β 1 ···β j , and after taking differentiation with respect to z up to n times, we obtain our required result.

Integral transforms of E 1 GMBF
In this section, we establish some integral transforms (Euler, Mellin and Laplace transform) of E 1 GMBF in the form of theorems, and also discuss its sub cases.

Relation of the E 1 GMBF with the Laguerre polynomial and Whittaker function
In this section, the authors represent the E 1 GMBF in terms of Laguerre polynomial, and Whittaker function in the form of theorems.

Conclusions
In this research, we described extension of extended generalized multi-index Bessel function (E 1 GMBF) and developed some results with the Laguerre polynomial and Whittaker function, integral representation, derivatives and solved integral transforms (beta transform, Laplace transform, Mellin transforms). Moreover, we discussed the composition of the generalized fractional integral operator having Appell function as a kernel with the E 1 GMBF and obtained results in terms of Wright functions.