On Voigt-Type Functions Extended by Neumann Function in Kernels and Their Bounding Inequalities

Abstract

The principal aim of this paper is to introduce the extended Voigt-type function Vμ,ν(x,y) and its counterpart extension Wμ,ν(x,y), involving the Neumann function Yν in the kernel of the representing integral. The newly defined integral reduces to the classical Voigt functions K(x,y) and L(x,y), and to their generalizations by Srivastava and Miller, by the unification of Klusch. Following an approach by Srivastava and Pogány, we also present the multiparameter and multivariable versions Vμ,ν(r)(x,y),Wμ,ν(r)(x,y) and the r positive integer of the initial extensions Vμ,ν(x,y),Wμ,ν(x,y). Several computable series expansions are obtained for the discussed Voigt-type functions in terms of Humbert confluent hypergeometric functions Ψ2(r). Furthermore, by transforming the input extended Voigt-type functions by the Grünwald–Letnikov fractional derivative, we establish representation formulae in terms of the associated Legendre functions of the second kind Qη−ν in the two-parameter and two-variable cases. Finally, functional bounding inequalities are given for Vμ,ν(x,y) and Wμ,ν(x,y). Particularly interesting results are presented for the Neumann function Yν and for the Struve Hν function in the form of several functional bounds. The article ends with a thorough discussion and closing remarks.

Keywords

• Voigt function
• generalized Voigt functions
• extended Voigt-type functions
• Neumann function
• Humbert confluent hypergeometric function of several variables
• Legendre function of the second kind