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

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