*(English)*Zbl 0744.33004

The Jacobi polynomials ${P}_{n}^{(\alpha ,\beta )}\left(x\right)$ 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 $[-1,1]$ 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.

##### MSC:

33C45 | Orthogonal polynomials and functions of hypergeometric type |

42C05 | General theory of orthogonal functions and polynomials |

60J80 | Branching processes |