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The rational Heun operator and Wilson biorthogonal functions (1912.11571v1)
Published 25 Dec 2019 in math-ph, math.CA, and math.MP
Abstract: We consider the rational Heun operator defined as the most general second-order $q$-difference operator which sends any rational function of type $[(n-1)/n]$ to a rational function of type $[n/(n+1)]$. We shall take the poles to be located on the Askey-Wilson grid. It is shown that this operator is related to the one-dimensional degeneration of the Ruijsenaars-van Diejen Hamiltonians. The Wilson biorthogonal functions of type ${_{10}}\Phi_9$ are found to be solutions of a generalized eigenvalue problem involving rational Heun operators of the special "classical" kind.