Basis inversion in lambda-recursive families: triangular kernels and polynomial basis changes
Wanderson Matos
TL;DR
The paper develops a unified framework for polynomial families with triangular support of order m, introducing a direct kernel $\lambda_1$ and an inverse kernel $\lambda_3$ to realize basis inversions between $\{x^n\}$ and $\{f_n\}$. It proves existence and uniqueness of the inverse kernel under a discrete orthogonality condition and expresses $\lambda_3$ via determinants of a lower Hessenberg matrix, linking to a determinantal theory of inverses. It then introduces lambda-recursive sequences with principal factors and auxiliary components, deriving boundary and classwise recurrences for inverse-kernel coefficients and establishing a structured algebraic expansion matrix. The approach unifies classical families such as Chebyshev, Legendre, Hermite, Laguerre, Fibonacci, and Lucas under a single triangular-array machinery and suggests efficient, Clenshaw-type computational schemes and broad applications in basis changes and interpolation.
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.