Matrix-Valued Hermite and Laguerre polynomials via Quadratic Transformation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Pacharoni, Inés Torres, A. Victoria |
| author_facet | Pacharoni, Inés Torres, A. Victoria |
| contents | We present the first systematic extension of the classical Hermite-Laguerre quadratic correspondence to the matrix-valued setting. Starting from a Hermite-type weight matrix W(x) = exp(-x^2) Z(x) with W(x) = W(-x), the change of variables y = x^2 produces two Laguerre-type weights with parameters alpha = -1/2 and alpha = 1/2, and relates the corresponding sequences of matrix-valued orthogonal polynomials through an explicit decomposition into even and odd subsequences. We prove that this transformation preserves differential operators and Darboux transformations, thereby establishing a direct structural link between the Hermite and Laguerre sides and providing new constructive tools for the matrix Bochner problem. Concrete families - including a new 3x3 example and an arbitrary-size family built from block-nilpotent matrices - illustrate the theory and supply fresh sources of matrix-valued orthogonal polynomials endowed with non-trivial differential algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20287 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Matrix-Valued Hermite and Laguerre polynomials via Quadratic Transformation Pacharoni, Inés Torres, A. Victoria Classical Analysis and ODEs Functional Analysis 42C05, 33C45, 33C47, 34L10 We present the first systematic extension of the classical Hermite-Laguerre quadratic correspondence to the matrix-valued setting. Starting from a Hermite-type weight matrix W(x) = exp(-x^2) Z(x) with W(x) = W(-x), the change of variables y = x^2 produces two Laguerre-type weights with parameters alpha = -1/2 and alpha = 1/2, and relates the corresponding sequences of matrix-valued orthogonal polynomials through an explicit decomposition into even and odd subsequences. We prove that this transformation preserves differential operators and Darboux transformations, thereby establishing a direct structural link between the Hermite and Laguerre sides and providing new constructive tools for the matrix Bochner problem. Concrete families - including a new 3x3 example and an arbitrary-size family built from block-nilpotent matrices - illustrate the theory and supply fresh sources of matrix-valued orthogonal polynomials endowed with non-trivial differential algebras. |
| title | Matrix-Valued Hermite and Laguerre polynomials via Quadratic Transformation |
| topic | Classical Analysis and ODEs Functional Analysis 42C05, 33C45, 33C47, 34L10 |
| url | https://arxiv.org/abs/2508.20287 |