Difference operators of Sklyanin and van Diejen type

The Sklyanin algebra $S_{\eta }$ has a well-known family of infinite-dimensional representations $D(\mu )$, $\mu \in C^{\ast }$, in terms of difference operators with shift $\eta$ acting on even meromorphic functions. We show that for generic $\eta$ the coefficients of these operators have solely simple poles, with linear residue relations depending on their locations. More generally, we obtain explicit necessary and sufficient conditions on a difference operator for it to belong to $D(\mu )$. By definition, the even part of $D(\mu )$ is generated by twofold products of the Sklyanin generators. We prove that any sum of the latter products yields a difference operator of van Diejen type. We also obtain kernel identities for the Sklyanin generators. They give rise to order-reversing involutive automorphisms of $D(\mu )$, and are shown to entail previously known kernel identities for the van Diejen operators. Moreover, for special $\mu$ they yield novel finite-dimensional representations of $S_{\eta }$.