Recurrence equations involving orthogonal polynomials with related weight functions

Abstract: Every sequence of real polynomials $\{p_n\}_{n=0}^\infty=0$, orthogonal with respect to a positive weight function w(x) on the interval $(a,b)$, satisfies a three-term recurrence equation. We discuss the role played by the polynomials associated to $p_n$, especially as coefficient polynomials in the three-term recurrence equation involving polynomials $p_n$, $p_{n-1}$ and $p_{n-m}$, $m\in\{2,3,...,n-1\}$. Furthermore, we show how Christoffel's formula is used to obtain mixed three-term recurrence equations involving the polynomials $p_n$, $p_{n-1}$ and $g_{n-m,k}$, $m \in \{2,3,...,n-1\}$, where the sequence $\{g_{n,k}\}_{n=0}^\infty$, $k \in {\mathbb{N}}_0$, is orthogonal with respect to $c_k(x)w(x) > 0$ on (a,b) and $c_k$ is a polynomial of degree $k$ in $x$. We discuss the conditions on $k$, necessary and sufficient for these equations to be in such a form, that they can be applied in the study of the location of the zeros of the appropriate polynomials.

complex variables

Audience: researchers in the topic

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