  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 $BC_1$. We consequently
prove that the renormalized Virasoro fusion kernel and the corresponding
quantum eigenfunction, the (renormalized) Ruijsenaars hypergeometric function,
are equal.
