  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.
