Finite biorthogonal polynomials suggested by the finite orthogonal polynomials $M_{n}^{(p,q)}(x)$

Abstract

In this paper, we derive a pair of finite univariate biorthogonal polynomials suggested by the finite univariate orthogonal polynomials $M_{n}^{(p,q)}(x)$. The corresponding biorthogonality relation is given. Some useful relations and properties, concluding 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 polynomials $M_{n}(p,q,\upsilon;x)$.