Basis inversion in lambda-recursive families: triangular kernels and polynomial basis changes

TL;DR

This paper develops a unified algebraic framework for inverting polynomial basis expansions using triangular kernels, enabling explicit change-of-basis formulas and connecting classical polynomial families through determinant-based methods.

Contribution

It introduces a novel inverse kernel construction under orthogonality conditions, unifies basis change relations for various polynomial families, and provides determinant formulas for inversion.

Findings

Explicit inverse kernels for polynomial families derived

Unified framework for classical orthogonal polynomials and sequences

Determinantal expressions enable efficient basis change computations

Abstract

We study polynomial families {f_n(x)}_{n>=0} over a commutative ring R encoded by triangular arrays of order m, via expansions of the form f_n(x)=sum_{b=0}^{floor(n/m)} lambda_1(n,b) x^{n-mb}, where lambda_1 is the direct kernel supported on 0<=b<=floor(n/m). Under a simple discrete orthogonality condition, we prove the existence and uniqueness of an inverse kernel lambda_3 (triangular of the same order) giving the inversion formula x^n = sum_{b=0}^{floor(n/m)} lambda_3(n,b) f_{n-mb}(x). This reindexing principle yields explicit change-of-basis relations between two families, including the case of distinct step sizes m_1 and m_2, with connection coefficients obtained from a universal triangular sum once lambda_3 is known. On the algebraic side, lambda_1 defines a lower Hessenberg matrix M_(n,k) (the algebraic expansion matrix) whose determinant governs inversion, providing closed determinantal expressions for lambda_3(n,k). We introduce a class of lambda-recursive sequences of order m, specified by a principal factor (p_n) and auxiliary factors (h_(n,k)), for which det(M_(n,k)) satisfies a recurrence enabling direct computation of inverse-kernel and basis-change coefficients. Classical families (e.g., Chebyshev, Legendre, Hermite, Laguerre, Fibonacci, Lucas) fit naturally into this framework, unifying their connection coefficients via the same triangular-array computations and supporting structured Clenshaw-type schemes and related applications.