On Voigt-Type Functions Extended by Neumann Function in Kernels and Their Bounding Inequalities - Rakesh K. Parmar; Tibor K. Pogány; Uthara Sabu (Axioms, 2024)
The principal aim of this paper is to introduce the extended Voigt-type function <semantics>Vμ,ν(x,y)</semantics> and its counterpart extension <semantics>Wμ,ν(x,y)</semantics>, involving the Neumann function <semantics>Yν</semantics> in the kernel of the representing integral. The newly defined integral reduces to the classical Voigt functions <semantics>K(x,y)</semantics> and <semantics>L(x,y)</semantics>, 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 <semantics><msubsup>Vμ,ν(r)</msubsup>(x,y),<mspace width="0.166667em"></mspace><msubsup>Wμ,ν(r)</msubsup>(x,y)</semantics> and the r positive integer of the initial extensions <semantics>Vμ,ν(x,y),Wμ,ν(x,y)</semantics>. Several computable series expansions are obtained for the discussed Voigt-type functions in terms of Humbert confluent hypergeometric functions <semantics><msubsup>Ψ2(r)</msubsup></semantics>. 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 <semantics><msubsup>Qη−ν</msubsup></semantics> in the two-parameter and two-variable cases. Finally, functional bounding inequalities are given for <semantics>Vμ,ν(x,y)</semantics> and <semantics>Wμ,ν(x,y)</semantics>. Particularly interesting results are presented for the Neumann function <semantics>Yν</semantics> and for the Struve <semantics>Hν</semantics> function in the form of several functional bounds. The article ends with a thorough discussion and closing remarks.