A bispectral q-hypergeometric basis for a class of quantum integrable models

For the class of quantum integrable models generated from the $q-$Onsager algebra, a basis of bispectral multivariable $q-$orthogonal polynomials is exhibited. In a first part, it is shown that the multivariable Askey-Wilson polynomials with $N$ variables and $N+3$ parameters introduced by Gasper and Rahman [1] generate a family of infinite dimensional modules for the $q-$Onsager algebra, whose fundamental generators are realized in terms of the multivariable $q-$difference and difference operators proposed by Iliev [2]. Raising and lowering operators extending those of Sahi [3] are also constructed. In a second part, finite dimensional modules are constructed and studied for a certain class of parameters and if the $N$ variables belong to a discrete support. In this case, the bispectral property finds a natural interpretation within the framework of tridiagonal pairs. In a third part, eigenfunctions of the $q-$Dolan-Grady hierarchy are considered in the polynomial basis. In particular, invariant subspaces are identified for certain conditions generalizing Nepomechie's relations. In a fourth part, the analysis is extended to the special case $q=1$. This framework provides a $q-$hypergeometric formulation of quantum integrable models such as the open XXZ spin chain with generic integrable boundary conditions ($q\neq 1$).