The Jacobi polynomials satisfy a three term recurrence relation with recurrence coefficients that are simple rational functions of the degree , containing the two parameters and . When one must be careful in defining . The classical way is to define , which leads to the standard Jacobi polynomials. However, the recurrence relation with initial values and leads to , and with this choice of one obtains the exceptional Jacobi polynomials studied in this paper. These polynomials are again orthogonal on and the authors explicitly compute the weight function.
A second family of orthogonal polynomials studied in this paper is a class of associated Jacobi polynomials arising in birth and death processes without absorption at zero. Explicit formulas are given for these associated Jacobi polynomials and also asymptotic results and a generating function. The asymptotic behaviour then leads to an explicit formula for the weight function.