Recurrence equations involving different orthogonal polynomial sequences and applications

Abstract

Consider $\{p_n{\}}_{n=0}^{\infty }$, a sequence of polynomials orthogonal with respect to $w(x)>0$ on $(a,b)$, and polynomials $\{g_{n,k}{\}}_{n=0}^{\infty },k\in {\mathbb{N}}_0$, orthogonal with respect to $c_k(x)w(x)>0$ on $(a,b)$, where $c_k(x)$ is a polynomial of degree $k$ in $x$. We show how Christoffel's formula can be used to obtain mixed three-term recurrence equations involving the polynomials $p_n$, $p_{n-1}$ and $g_{n-m,k},m\in \{2,3,\dots ,n-1\}$. In order for the zeros of $p_n$ and $G_{m-1}{g}_{n-m,k}$ to interlace (assuming $p_n$ and $g_{n-m,k}$ are co-prime), the coefficient of $p_{n-1}$, namely $G_{m-1}$, should be of exact degree $m-1$, in which case restrictions on the parameter $k$ are necessary. The zeros of $G_{m-1}$ can be considered to be inner bounds for the extreme zeros of the (classical or $q$-classical) orthogonal polynomial $p_n$ and we give examples to illustrate the accuracy of these bounds. Because of the complexity the mixed three-term recurrence equations in each case, algorithmic tools, mainly Zeilberger's algorithm and its $q$-analogue, are used to obtain them.

18 pages