2026/06/30 by Bo Yang
#math.NA #cs.NA
An approach to generate three-term recurrence relations with arbitrary degree steps is proposed for orthogonal polynomials. Specifically, given any class of orthogonal polynomials \Qp(x)\p=0∞ defined by Favard's theorem, we employ the adjacent members Qp(x) and Qp-1(x) to compute Qp+s(x) of high degree and the one of low degree Qp-t(x), where (s,t) are parameters for degree step adjustment. The coefficients of both relations are analyzed, revealing novel properties that enable the derivation of three-term recurrence relations with respect to Qp+s(x), Qp(x) and Qp-t(x) by eliminating Qp-1(x). Furthermore, in addition to the standard recursive formula, which is characterized by degree increase, the formulas for degree decrease and end-to-middle directions are also formulated. Moreover, explicit recurrence relations with 2-degree steps are presented for Hermite, Gegenbauer and Legendre polynomials. The computation precision of the proposed recurrence relations is also compared with that of the standard ones.