Derivatives of any order of the confluent hypergeometric function F11(a,b,z) with respect to the parameter a or b

Abstract

The derivatives to any order of the confluent hypergeometric (Kummer) function F=F11(a,b,z) with respect to the parameter a or b are investigated and expressed in terms of generalizations of multivariable Kampé de Fériet functions. Various properties (reduction formulas, recurrence relations, particular cases, and series and integral representations) of the defined hypergeometric functions are given. Finally, an application to the two-body Coulomb problem is presented: the derivatives of F with respect to a are used to write the scattering wave function as a power series of the Sommerfeld parameter.

Design and implementation of spectrum sensing for cognitive radios with a frequency-hopping primary system

“…Ancarani and Gasaneo [31] have derived the following partial derivatives of the Gauss' hypergeometric function 2 F 1 (a, b; c; d)…”

“…and where ψ (0) is the digamma function, and 2 Θ (1) (·) is a Kampé de Fériet-like function [31] , defined below.…”

A Generic Simulation Approach for the Fast and Accurate Estimation of the Outage Probability of Single Hop and Multihop FSO Links Subject to Generalized Pointing Errors

“…where F 1×1×2 2×0×1 (·) is the Kampé de Fériet-like function [38]. Solving these equations results in a set of solutions.…”

Wronskian representations of hypergeometric integrals

“…Since covering arbitraryorder derivatives for general hypergeometric functions is beyond the scope of this work, we restrict ourselves to stating the first-order partial derivatives of the confluent hypergeometric function 1 F 1 and the hypergeometric function 2 F 1 in terms of KdF functions. For details on the construction of the following identities, the reader may refer to [2]- [4].…”