We construct a non-polynomial generalization of the q-Askey scheme. Whereas the elements of the q-Askey scheme are given by q-hypergeometric series, the elements of the non-polynomial scheme are given by contour integrals, whose integrands are built from Ruijsenaars’ hyperbolic gamma function. Alternatively, the integrands can be expressed in terms of Faddeev’s quantum dilogarithm, Woronowicz’s quantum exponential, or Kurokawa’s double sine function. We present the basic properties of all the elements of the scheme, including their integral representations, joint eigenfunction properties, and polynomial limits.

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.

We show that the Virasoro fusion kernel is equal to Ruijsenaars' hypergeometric function up to normalization. More precisely, we prove that the Virasoro fusion kernel is a joint eigenfunction of four difference operators. We find a renormalized version of this kernel for which the four difference operators are mapped to four versions of the quantum relativistic hyperbolic Calogero-Moser Hamiltonian tied with the root system BC1. We consequently prove that the renormalized Virasoro fusion kernel and the corresponding quantum eigenfunction, the (renormalized) Ruijsenaars hypergeometric function, are equal.

We construct confluent conformal blocks of the second kind of the Virasoro algebra. We also construct the Stokes transformations which map such blocks in one Stokes sector to another. In the BPZ limit, we verify explicitly that the constructed blocks and the associated Stokes transformations reduce to solutions of the confluent BPZ equation and its Stokes matrices, respectively. Both the confluent conformal blocks and the Stokes transformations are constructed by taking suitable confluent limits of the crossing transformations of the four-point Virasoro conformal blocks.