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Zbl 0744.33004
Ismail, Mourad E.H.; Masson, David R.
Two families of orthogonal polynomials related to Jacobi polynomials. (English)
[J] Rocky Mt. J. Math. 21, No.1, 359-375 (1991). ISSN 0035-7596

The Jacobi polynomials $P\sb n\sp{(\alpha,\beta)}(x)$ 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 $P\sb 1(x)$. The classical way is to define $P\sb 1(x)=x+\alpha$, which leads to the standard Jacobi polynomials. However, the recurrence relation with initial values $P\sb{-1}=0$ and $P\sb 0=1$ leads to $P\sb 1(x)=x$, and with this choice of $P\sb 1(x)$ 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.\par 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.
[W.Van Assche (Heverlee)]

MSC 2000:: *33C45 Orthogonal polynomials and functions of hypergeometric type
42C05 General theory of orthogonal functions and polynomials
60J80 Branching processes

Keywords: Jacobi polynomials; recurrence relation; associated Jacobi polynomial; asymptotic results; generating function; weight function

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