2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/223615Using 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.Classical Analysis and ODEsExactly Solvable and Integrable Systems33E05, 33E30, 33C47text