We consider all genus zero and genus one correlation functions for the Virasoro vacuum descendants of a vertex operator algebra. These are described in terms of explicit generating functions that can be combinatorially expressed in terms of graph theory related to derangements in the genus zero case and to partial permutations in the genus one case.
We consider the partition function of a general vertex operator algebra V on a genus two Riemann surface formed by sewing together two tori. We consider the non-trivial degeneration limit where one torus is pinched down to a Riemann sphere and show that the genus one partition function on the degenerate torus is recovered up to an explicit universal V -independent multiplicative factor raised to the power of the central charge.
ABSTRACT. For several classical polynomials un(x) satisfying a second order linear differential equation Dn{x), there is a generating function u(x,t) = 2^L0un{x)tn.We provide expansions v(x,t) = X)°°_0 vn{x)tn where vn(x) is a second solution of Dn (x).
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