6 - Hypergeometric supertrigonometric and superhyperbolic functions via Laguerre polynomials

Abstract

In this chapter, we introduce the concept, properties, and theorems of the Laguerre polynomials. We consider the supertrigonometric and superhyperbolic functions via Laguerre polynomials, Szegö function of first type, Rainville function, and Szegö function of second type. Moreover, we investigate in detail the Laplace transforms for the new special functions, such as the hypergeometric supersine, hypergeometric supercosine, hypergeometric superhyperbolic supersine, and hypergeometric superhyperbolic supercosine. We also present the Brafman theorem, Hille theorem, Feldheim theorem, Weisner theorem, and the other theorems for new special functions.