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Two families of orthogonal polynomials related to Jacobi polynomials. (English) Zbl 0744.33004

The Jacobi polynomials P n (α,β) (x) satisfy a three term recurrence relation with recurrence coefficients that are simple rational functions of the degree n, containing the two parameters α and β. When α+β=0 one must be careful in defining P 1 (x). The classical way is to define P 1 (x)=x+α, 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.

33C45Orthogonal polynomials and functions of hypergeometric type
42C05General theory of orthogonal functions and polynomials
60J80Branching processes