2021/10/25 by A. Jooste, Jooste, A. S., Daniel Duviol Tcheutia +3 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #33C05 #33C45 #33D15 #33F10 #Classical Analysis and ODEs (math.CA) #Electromagnetic Scattering and Analysis #FOS: Mathematics #Mathematical functions and polynomials #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2110.13305
openalex publication_date 2021/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider \pn\n=0∞, a sequence of polynomials orthogonal with respect to w(x)>0 on (a,b), and polynomials \gn,k\n=0∞,k ∈ ℕ0, orthogonal with respect to ck(x)w(x)>0 on (a,b), where ck(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 pn, pn-1 and gn-m,k,m∈\2,3,…, n-1\. In order for the zeros of pn and Gm-1gn-m,k to interlace (assuming pn and gn-m,k are co-prime), the coefficient of pn-1, namely Gm-1, should be of exact degree m-1, in which case restrictions on the parameter k are necessary. The zeros of Gm-1 can be considered to be inner bounds for the extreme zeros of the (classical or q-classical) orthogonal polynomial pn 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.