Exactly solvable inhomogeneous periodic quantum spin chain from q-ultraspherical polynomials at roots of unity

We present a closed inhomogeneous quantum XX spin chain on the periodic integer lattice $$\mathbb {Z}_m$$ which is diagonalized by means of Slater determinants built from Rogers’ q-ultraspherical polynomials with $$q^m=1$$ . The hermiticity of our periodic quantum spin Hamiltonian encodes a discrete orthogonality relation for these q-ultraspherical polynomials that is of a type studied previously by Spiridonov and Zhedanov.