Two families of orthogonal polynomials related to Jacobi polynomials (Q1178592)

The Jacobi polynomials \(P_ n^{(\alpha,\beta)}(x)\) satisfy a three term recurrence relation with recurrence coefficients that are simple rational functions of the degree \(n\), containing the two parameters \(\alpha\) and \(\beta\). When \(\alpha+\beta=0\) one must be careful in defining \(P_ 1(x)\). The classical way is to define \(P_ 1(x)=x+\alpha\), which leads to the standard Jacobi polynomials. However, the recurrence relation with initial values \(P_{-1}=0\) and \(P_ 0=1\) leads to \(P_ 1(x)=x\), and with this choice of \(P_ 1(x)\) one obtains the exceptional Jacobi polynomials studied in this paper. These polynomials are again orthogonal on \([-1,1]\) 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.