Associated Conformable Fractional Legendre Polynomials

Along with the work of Abul-Ez et al. [37], we introduce the associated conformable fractional Legendre polynomials (ACFLPs), from which the fractional differential equation of ACFLPs is established. Subsequently, some of interesting properties are derived such as generating function, hypergeometric representation, analytical formula, besides various of recurrence relations. Also, orthogonal properties of ACFLPs are developed in conformable context. We append our study by presenting the shifted ACFLPs and driving some of important properties such as Rodrigues' type representation formula of fractional order derivative and explicit formula. An interesting compact closed-form expression is derived from the definite integral using a convenient analytical formula for the shifted ACFLPs. This result is easily generalized for integrands involving products of an arbitrary number of shifted associated Legendre polynomials in conformable sense. (paper)