We study polynomial families fₙ (x) ₍>=₀ over a commutative ring R encoded by triangular arrays of order m, via expansions of the form fₙ (x) =sum₁=₀^floor (n/m) lambda₁ (n, b) x^n-mb, where lambda₁ 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 lambda₃ (triangular of the same order) giving the inversion formula xⁿ = sum₁=₀^floor (n/m) lambda₃ (n, b) f₍-₌₁ (x). This reindexing principle yields explicit change-of-basis relations between two families, including the case of distinct step sizes m₁ and m₂, with connection coefficients obtained from a universal triangular sum once lambda₃ is known. On the algebraic side, lambda₁ defines a lower Hessenberg matrix M_ (n, k) (the algebraic expansion matrix) whose determinant governs inversion, providing closed determinantal expressions for lambda₃ (n, k). We introduce a class of lambda-recursive sequences of order m, specified by a principal factor (pₙ) 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.
Wanderson Matos (Thu,) studied this question.