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

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.