Elliptic orthogonal polynomials and OPRL
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Alves, Victor Martinez-Finkelshtein, Andrei |
| author_facet | Alves, Victor Martinez-Finkelshtein, Andrei |
| contents | We explore a class of meromorphic functions on elliptic curves, termed \emph{elliptic orthogonal a-polynomials} ($a$-EOPs), which extend the classical notion of orthogonal polynomials to compact Riemann surfaces of genus one. Building on Bertola's construction of orthogonal sections, we study these functions via non-Hermitian orthogonality on the torus, establish their recurrence properties, and derive an analogue of the Christoffel--Darboux formula. We demonstrate that, under real-valued orthogonality conditions, $a$-EOPs exhibit interlacing and simplicity of zeros similar to orthogonal polynomials on the real line (OPRL). Furthermore, we construct a general correspondence between families of OPRL and elliptic orthogonal functions, including a decomposition into multiple orthogonality relations, and identify new interlacing phenomena induced by rational deformations of the orthogonality weight. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19656 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Elliptic orthogonal polynomials and OPRL Alves, Victor Martinez-Finkelshtein, Andrei Classical Analysis and ODEs 42C05 (Primary) 14H52, 33E05 (Secundary) We explore a class of meromorphic functions on elliptic curves, termed \emph{elliptic orthogonal a-polynomials} ($a$-EOPs), which extend the classical notion of orthogonal polynomials to compact Riemann surfaces of genus one. Building on Bertola's construction of orthogonal sections, we study these functions via non-Hermitian orthogonality on the torus, establish their recurrence properties, and derive an analogue of the Christoffel--Darboux formula. We demonstrate that, under real-valued orthogonality conditions, $a$-EOPs exhibit interlacing and simplicity of zeros similar to orthogonal polynomials on the real line (OPRL). Furthermore, we construct a general correspondence between families of OPRL and elliptic orthogonal functions, including a decomposition into multiple orthogonality relations, and identify new interlacing phenomena induced by rational deformations of the orthogonality weight. |
| title | Elliptic orthogonal polynomials and OPRL |
| topic | Classical Analysis and ODEs 42C05 (Primary) 14H52, 33E05 (Secundary) |
| url | https://arxiv.org/abs/2507.19656 |