Mock modular forms and d-distinct partitions

Abstract

The generating function for partitions into (d+1)-distinct parts (d+1∈N) may be expressed as the q-hypergeometric seriesRd+1(1;q):=∑n⩾0qd+1(n)qn=∑n⩾0q(d+1)(n2)+n(q;q)n, where qd+1(n):=p(n|parts differ by at least d+1). Within combinatorial number theory, this function has long since been of historical importance: a famous identity of Euler and Sylvester implies that the distinct parts function q1/24R1(1;q) is an ordinary modular form, and the celebrated Rogers–Ramanujan identities imply that the 2-distinct parts function q−1/60R2(1;q) is an ordinary modular form. More recently Alder–Andrews, Zagier, and others have studied the general series Rd+1(1;q) for d∈N0. In particular, Zagier has proved for d>1 that these series are never ordinary modular forms; however, other than the cases d=0 (Euler) and d=1 (Rogers–Ramanujan), the precise modular properties of these combinatorial series remain unknown. Here, we prove for integers d⩾1, that the combinatorial q-series Rd+1(1;q), and relevant generalizations, are natural denominators of a new class of mixed mock modular forms. As such, we also obtain many new results as corollaries, including new expressions for Zwegersʼs lauded μ-function, and many of the well-known combinatorial q-series in the subject, confirming the central role that the q-hypergeometric series Rd+1(1;q) play within the theory. Once armed with this realization, we also obtain general theorems on the analytic behavior of Rd+1(1;q) and related series near the unit disk, a priori an impenetrable barrier littered with exponential singularities.