Finite Biorthogonal Polynomials Suggested by the Finite Orthogonal Polynomials Mnp,Qx

Abstract

Constructing a biorthogonal structure from scratch, that is, defining a biorthogonal pair is quite tough. Because here the orthogonality must be established between two different sets. There are four known univariate biorthogonal polynomial sets, suggested by Laguerre, Jacobi, Hermite and Szeg & odblac;-Hermite polynomials, in the literature. In this paper, we derive for the first time a pair of finite univariate biorthogonal polynomials suggested by the finite univariate orthogonal polynomials . The corresponding biorthogonality relation and some useful relations and properties, including differential equation and generating function, are presented. Further, a new family of finite biorthogonal functions is obtained using Fourier transform and Parseval identity. In addition, we compute the Laplace transform and fractional calculus operators for the new biorthogonal polynomial set .