The aim of this paper is to introduce matrix polynomials L(M,?,?) n (x) and establish some properties viz, hypergeometric representation, generating relations, integral representations, recurrence summation formulas, series relation, fractional derivative operators.

We present an algebraic Bethe ansatz for certain submanifolds of the bilayer vertex models proposed by Shiroishi and Wadati as coupled asymmetric six-vertex free-fermion models. A peculiar feature of our formulation is the presence of a diagonal monodromy matrix element that does not generate unwanted terms. The model contains two free-parameters entering into the Bethe ansatz equations as a pu...