Three-term Recurrence Relation with Arbitrary Degree Step for Orthogonal Polynomials

Abstract

An approach to generate three-term recurrence relations with arbitrary degree step is proposed. Specifically, given any class of orthogonal polynomials $\{Q_{p}(x)\}_{p=0}^{\infty}$ defined by Favard's theorem, we employ the adjacent members $Q_{p}(x)$ and $Q_{p-1}(x)$ to compute the one of high degree $Q_{p+s}(x)$ and that of low degree $Q_{p-t}(x)$, where $(s,t)\in \mathbb{N^{+}}$. Therefore, it is able to derive a three-term recurrence relation with respect to $Q_{p+s}(x)$, $Q_{p}(x)$ and $Q_{p-t}(x)$ by taking $Q_{p-1}(x)$ as the bridge. Moreover, as the extensions of standard recursive formula which is characterized with degree increase, the ones for degree decrease and end-to-middle directions are formulated as well. The explicit recurrence relations with two-degree step are offered for Hermite, Gegenbauer and Legendre polynomials. Precision comparison is also performed between the standard three-term recurrence relation and the proposed ones.