On a generalized three-parameter wright function of Le Roy type

Abstract

Recently S. Gerhold and R. Garra -F. Polito independently introduced a new function related to the special functions of the Mittag-Leffler family. This function is a generalization of the function studied byÉ. Le Roy in the period 1895-1905 in connection with the problem of analytic continuation of power series with a finite radius of convergence. In our note we obtain two integral representations of this special function, calculate its Laplace transform, determine an asymptotic expansion of this function on the negative semi-axis (in the case of an integer third parameter γ) and provide its continuation to the case of a negative first parameter α. An asymptotic result is illustrated by numerical calculations. Discussion on possible further studies and open questions are also presented.MSC 2010 : Primary 33E12; Secondary 30D10, 30F15, 35R11

Some results on the Complete Monotonicity of Mittag-Leffler Functions of Le Roy Type

“…All forms of the MLR function mentioned in the previous section satisfy the Laplace transform rule (11) where λ is a real or complex constant. The proof of this formula for various representation of the MLR function can be found, e.g., in [3,5] and also in Appendix C. After inverting Eq. (11) and making the change of variable st = z 1/α we arrive at (12) where L z denotes the Bromwich contour with Re(z) > 0.…”

“…The asymptotic behaviour of the MLR function for complex arguments as well as its integral representations in the complex domain are discussed in [5]. Moreover, in [22], it has been recently given the integral representation of the MLR function for α, β, γ, x > 0, and α + β ≥ x 0 where x 0 is the abscissa of the minimum of the Gamma function.…”

“…Inspired by the fact that the special cases of the MLR functions for n = 1 are CM we expect that under some conditions the MLR functions should be CM for n = 1 as well. Indeed, this is the essence of the unanswered question which closes the paper [5]:…”

“…The Mittag-Leffler (ML) functions of the Le Roy type (denoted here, for shortness, as MLR functions) are defined in [5] as (−x) r [Γ(β + αr)] n , x ∈ R, α > 0, β ∈ R, n = 1, 2, . .…”

“…For many years several efforts have been payed to study the complete monotonicity (CM) of the Mittag-Leffler functions which has been, however, proved just for some of the mentioned special cases but not for the more general case for which CM has been only conjectured [5].…”