Results Concerning the Analysis of Multi-Index Whittaker Function

A variety of functions, their extensions, and variants have been extensively investigated, mainly due to their potential applications in diverse research areas. In this paper, we aim to introduce a new extension of Whittaker function in terms of multi-index confluent hypergeometric function of first kind. We discuss multifarious properties of newly defined multi-index Whittaker function such as integral representation, integral transform (i.e., Mellin transform andHankel transform), and derivative formula. 'e results presented here, being very general, are pointed out to reduce to yield some known or new formulas and identities for relatively functions.


Introduction
Generalized and multivariable forms of the special functions of mathematical physics have witnessed a significant evolution during recent years. In particular, the special functions of more than one variable provided new means of analysis for the solution of large classes of partial differential equations often encountered in physical problems. Most of the special functions of mathematical physics and their generalization have been suggested by physical problems. In mathematics, the Whittaker function is a solution of Whittaker equation, which is a modified form of confluent hypergeometric function of first kind, and it has various applications in multifarious area such as mathematical physics and many research areas, which are studied by various mathematicians (see [1][2][3][4]). Recently, many authors give an extension and generalization of several special functions such as beta function, gamma function, hypergeometric function, confluent hypergeometric function, and Whittaker function (see [2,3,[5][6][7][8][9][10][11][12]). Ghayasuddin et al. [7] defined a new type of confluent hypergeometric function by using extended beta function in terms of multi-index Mittag-Leffler function. Inspired by the abovementioned work, in this paper, we introduced a new extension of Whittaker function in terms of multi-index Mittag-Leffler function by using an extended confluent hypergeometric function and studied their various properties such as integral transform, integral representation, and derivative formula of it. We remember below the following basic definition and extension of special function. e classical beta function B(u, v) is defined as (see [13]) where By using the series expansion of (1 − ωt) − u and exp(ωt) in (3) and (4), respectively, the hypergeometric and confluent hypergeometric functions are written in terms of beta function as In 1997, Aslam Chaudhary et al. [5] give an extension of beta function defined as where Remark 1. If ρ � 0, then extended beta function (6) is reduced to classical beta function (1).
In 2004, Chaudhary et al. [6] introduced the extended hypergeometric and confluent hypergeometric functions in terms of extended beta function (6) as follows: eir integral representation is 2 Journal of Mathematics Shadab et al. [12] introduced an extension of beta function using generalized Mittag-Leffler function as follows: where E α (.) is the classical Mittag-Leffler (see [15,16]) function defined by where ω ∈ C, α ∈ R + 0 .
Shadab et al. [12] expressed the extended hypergeometric and confluent hypergeometric functions in terms of extended beta function (11) as follows: and their integral representation is Ghayasuddin et al. [7] introduced an extension of beta function using multi-index Mittag-Leffler function as follows: For s � 2, if we set 1/a 1 � α, 1/a 2 � 0, and b 1 � b 2 � 1 in (15), then we obtain the extended beta function defined by Shadab et al. [12].
In 2020, Ghayasuddin et al. [7] expressed the extended hypergeometric and confluent hypergeometric functions in terms of extended beta function (15) as follows: and their integral representation is e extension of Kummer's relation to the generalized extended confluent hypergeometric function of the first kind is as follows: For α � β � η � ] � 1 and ρ � 0, (20) is reduced to Kummer's formula of first kind for the classical confluent hypergeometric function (see [14]). e Whittaker function M κ,ζ (ω) in terms of confluent hypergeometric function of first kind (see [4,18]) is defined as In 2013, Nagar et al. [3] generalized the Whittaker function by using extended confluent hypergeometric function Φ ρ which is defined as

Multi-Index Whittaker Function
In this section, we give a new generalization of Whittaker function of the first kind by applying the multi-index confluent hypergeometric function ( If we take ρ � 0 and a i � · · · � a s � b i � · · · � b s � 1, (23) reduced to the classical Whittaker function (21).

Journal of Mathematics
Now, writing the right-hand side of the above representation by using (23), we get the desired result.
e following formula holds true: Proof. Using integral representation of the multi-index Whittaker function, we obtain dtdω.

(37)
Now, interchanging the order of integration and using the definition of gamma function, we obtain dtdω. (38)
Proof. Using (23) and (18), expanding multi-index Whittaker function in terms of generating extended beta function and changing the order of integration and summation, we obtain On using the known result (see p. 182 (9)of [21]), where R(ζ + m) > − 1, r � (p 2 + a 2 ) 1/2 and P m ζ (z) is the Legendre function (see [20]).By using (43) in (42) and after some simplification, we get the desired result.

Derivative of Multi-Index Whittaker Function
Theorem 5. e following differential formula holds true: Proof.

Conclusion
In the present paper, we introduce a multi-index Whittaker function in terms of extended confluent hypergeometric function. We have provided some important properties of Whittaker function such as integral representation, integral transform, and derivative formula. We have known that most of the special function of mathematical physics such as modified Bessel function and Laguerre and Hermite polynomial can be written in terms of Whittaker function. erefore, extensions and generalization of the Whittaker function are playing important roles in applied mathematics and mathematical physics.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.