The Mittag-Leffler-Legendre Polynomials and their Lie-Algebraic Relations

Abstract

In this paper, we introduce families of Mittag-Leffler-Legendre polynomials corresponding to two different forms of 2-variable Legendre polynomials. The Mittag-Leffler function plays a role of central importance in the theory of symbolic method. We take advantage from the monomiality and symbolic method to discuss Lie algebraic relations. The paper contains the method of symbolic evaluation to extend the studies of certain special functions including their properties. First, we obtain umbral and symbolic definitions, partial differential equations and umbral-symbolic operational representations for Legendre polynomials and then introduce Mittag-Leffler-Legendre polynomials by symbolic method. Then, we obtain the generating functions, series definitions and umbral-symbolic operational rules for these polynomials. We derive the multiplicative and derivative operators to study the quasi-monomiality property of these polynomials and also discuss the Lie algebraic relation. Further, we establish summation formulae and certain identities for these polynomials.