We study the recently introduced family of confluent Virasoro fusion kernels C-k(b,theta,sigma(s),nu). We study their eigenfunction properties and show that they can be viewed as non-polynomial generalizations of both the continuous dual q-Hahn and the big q-Jacobi polynomials. More precisely, we prove that: (i) C-k is a joint eigenfunction of four different difference operators for any positive integer k, (ii) C-k degenerates to the continuous dual q-Hahn polynomials when nu is suitably discretized, and (iii) C-k degenerates to the big q-Jacobi polynomials when sigma(s) is suitably discretized. These observations lead us to propose the existence of a non-polynomial generalization of the q-Askey scheme. The top member of this non-polynomial scheme is the Virasoro fusion kernel (or, equivalently, Ruijsenaars' hypergeometric function), and its first confluence is given by the C-k.