2025/07/25 by Alves, Victor, Martinez-Finkelshtein, Andrei
#33E05 (Secundary) #42C05 (Primary) 14H52 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.19656
We explore a class of meromorphic functions on elliptic curves, termed 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.