New developments for the Jacobi polynomials

Abstract In this work, first, a new and more general form of the Jacobi differential equation is developed, and the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -Jacobi polynomials are defined by means of the general solution of this equation and related generating functions and Rodrigues formula are obtained. Its orthogonality is also shown and its norm is derived. Subsequently, properties similar to those of the k-Jacobi polynomials are achieved by defining the k-Gegenbauer and k-Legendre differential equations and the k-Gegenbauer and k-Legendre polynomials corresponding to a special solution of them. These polynomials also have several new properties, including explicit formulas, generating functions, and recurrence relations. In addition, a certain class of bilateral and bilinear generating functions are derived and some examples are presented.