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

The Jacobi polynomials ${P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)$ 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 ${P}_{1}\left(x\right)$. The classical way is to define ${P}_{1}\left(x\right)=x+\alpha$, which leads to the standard Jacobi polynomials. However, the recurrence relation with initial values ${P}_{-1}=0$ and ${P}_{0}=1$ leads to ${P}_{1}\left(x\right)=x$, and with this choice of ${P}_{1}\left(x\right)$ one obtains the exceptional Jacobi polynomials studied in this paper. These polynomials are again orthogonal on $\left[-1,1\right]$ 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.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 42C05 General theory of orthogonal functions and polynomials 60J80 Branching processes