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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2512.16057 |
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| _version_ | 1866918254550712320 |
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| author | Matos, Wanderson |
| author_facet | Matos, Wanderson |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16057 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Basis inversion in lambda-recursive families: triangular kernels and polynomial basis changes Matos, Wanderson Combinatorics Primary 05A, Secondary 15A 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. |
| title | Basis inversion in lambda-recursive families: triangular kernels and polynomial basis changes |
| topic | Combinatorics Primary 05A, Secondary 15A |
| url | https://arxiv.org/abs/2512.16057 |