3F2(1) hypergeometric function and quadratic R-matrix algebra

Abstract

We construct a class of representations of the quadratic $R$-matrix algebra given by the reflection equation with the spectral parameter, $$ R{\,}(u-v)\,T^{(1)}(u)\,R{\,}(u+v)\,T^{(2)}(v)= T^{(2)}(v)\,R{\,}(u+v)\,T^{(1)}(u)\,R{\,}(u-v), $$ in terms of certain ordinary difference operators. These operators turn out to act as parameter shifting operators on the ${}_3F{}_2(1)$ hypergeometric function and its limit cases and on classical orthogonal polynomials. The relationship with the factorisation method will be discussed.