Generalized eigenvalue problem and a new family of rational functions biorthogonal on elliptic grids

Abstract

It is well known that the eigenvalue problem for an arbitrary tridiagonal (Jacobi) matrix gives rise to generic orthogonal polynomials. We show that the generalized eigenvalue problem for two arbitrary Jacobi matrices gives rise to generic biorthogonal rational functions. The Favard theorem, three-term recurrence relations, Christoffel-Darboux formula  and spectral taransformations for these functions are obtained.

We construct a new family of such functions which are biorthogonal with discrete weight on the grid described by elliptic Jacobi functions. We find an explicit expression of these functions in terms of "modular" hypergeometric functions ${_{10}}W_9$ introduced by Frenkel and  Turaev. These functions can be considered  as an  "elliptic" generalization of the well-known Wilson rational biorthogonal fucntions.

It is shown that these functions possess all "classical" properties (duality, difference equation, self-similarity with respect to spectral transformations).
 
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