Elliptic Hypergeometric Laurent Biorthogonal Polynomials with a Dense Point Spectrum on the Unit Circle

Using the technique of the elliptic Frobenius determinant, we construct new elliptic solutions of the $QD$-algorithm. These solutions can be interpreted as elliptic solutions of the discrete-time Toda chain as well. As a by-product, we obtain new explicit orthogonal and biorthogonal polynomials in terms of the elliptic hypergeometric function ${_3}E_2(z)$. Their recurrence coefficients are expressed in terms of the elliptic functions. In the degenerate case we obtain the Krall-Jacobi polynomials and their biorthogonal analogs.

💡 Research Summary

The paper presents a comprehensive construction of new elliptic solutions to the QD‑algorithm and the discrete‑time Toda chain by exploiting the elliptic Frobenius determinant. Starting from the classical theory of Laurent biorthogonal polynomials (LBP), the authors recall that moments (c_n) generate Toeplitz determinants (\Delta_n) and shifted determinants (\Delta_n^{(j)}). Under the non‑degeneracy conditions (\Delta_n\neq0) and (\Delta_n^{(1)}\neq0), the LBP satisfy a three‑term recurrence \