A Burge tree of Virasoro-type polynomial identities

Abstract

Using a summation formula due to Burge, and a combinatorial identity between partition pairs, we obtain an infinite tree of q-polynomial identities for the Virasoro characters χ^{p, p&#39;}_{r, s}, dependent on two finite size parameters M and N, in the cases where: (i) p and p&#39; are coprime integers that satisfy 0 < p < p&#39;. (ii) If the pair (p&#39;, p) has a continued fraction (c_1, c_2, ... , c_{t-1}, c_t+2), where t >= 1, then the pair (s, r) has a continued fraction (c_1, c_2, ... , c_{u-1}, d), where 1 =< u =< t, and 1 =< d =< c_{u}. The limit M -> infinity, for fixed N, and the limit N -> infinity, for fixed M, lead to two independent boson-fermion-type q-polynomial identities: in one case, the bosonic side has a conventional dependence on the parameters that characterise the corresponding character. In the other, that dependence is not conventional. In each case, the fermionic side can also be cast in either of two different forms. Taking the remaining finite size parameter to infinity in either of the above identities, so that M -> infinity and N -> infinity, leads to the same q-series identity for the corresponding character.