Properties of the multi-index special function W^(α,ν)(z)

2023/01/02 by Droghei, Riccardo
#26A33 #33E12 #34A08 #34K37 #35R11 #FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2301.04640

Abstract

In this paper, we investigate some properties related to a multi-index special function W^(α,ν) that arose from an eigenvalue problem for a multi-order fractional hyper-Bessel operator, involving Caputo fractional derivatives. We show that for particular values of the parameters involved in this special function W^(α,ν), this leads to the hyper-Bessel function of Delerue. The Laplace transform of the W^(α,ν) is discussed obtaining, in particular cases, the well-known functional relation between hyper-Bessel function and multi-index Mittag-Leffler function, or, quite simply, between classical Wright and Mittag-Leffler functions. Moreover, it is shown that the multi-index special function satisfies the recurrence relation involving fractional derivatives. In a particular case, we derive, to the best of our knowledge, a new differential recurrence relation for the Mittag-Leffler function. We also provide derivatives of the 3-parameters function W_α,β,ν with respect to parameters, leading to infinite power series with coefficients being quotients of digamma and gamma functions.

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