The Second Kummer Function with Matrix Parameters and Its Asymptotic Behaviour

Abstract

In the present article, we introduce the second Kummer function with matrix parameters and examine its asymptotic behaviour relying on the residue theorem. Further, we provide a closed form of a solution of a Weber matrix differential equation and give a representation using the second Kummer function.

Some Results on the Extended Hypergeometric Matrix Functions and Related Functions

“…e following integral representation for generalized Euler's beta matrix function in (13) holds well:…”

“…Proof. It is enough to use (13), interchange the order of integration, take transformations v � p/t(1 − t) and η � t in (13), and then use (12) in the left-hand side of the above equation, respectively. □ Theorem 4.…”

“…In particular, various results of gamma, Euler's beta, and hypergeometric matrix functions have been presented and investigated (see, e.g., [11][12][13][14][15][16][17]). Motivated by investigations of the extended gamma, beta, and Gauss hypergeometric matrix functions given in [14-16, 18, 19], we aim to derive certain properties of matrix generalizations of gamma, Euler's beta, Gauss, and confluent hypergeometric functions.…”

A Note on the Appell Hypergeometric Matrix Function F2

“…e proof of (23)-(26) is a direct consequence of definition (27). e relation (27) is obtained setting C 1 � B 1 in (18) and then using (25). Similarly, the relation (28) is derived setting C 2 � B 2 in (19) and then using (26).…”

“…On the other hand, many authors [18][19][20][21][22][23][24][25] generalized the hypergeometric series F(α, β, c; z) by extending parameters α, β, and c to square matrices A, B, and C in the complex space C d×d . Recently, the extension of the classical Appell hypergeometric functions F s , s � {1, 2, 3, 4}, to the Appell hypergeometric matrix functions has been a subject of intensive studies [26][27][28][29][30].…”