Asymptotic analogs of the Rogers-Ramanujan identities

AbstractLet p(nS) be the number of partitions of n with parts belonging to the set S; let q(nS) be the number of partitions of n with parts distinct and belonging to the set S; let qd(n) be the number of partitions of n with parts differing by at least d. Asymptotic formulas for p(nS), q(nS), and qd(n) are derived. Using these formulas necessary and/or sufficient conditions are obtained on sets S and S′ for the various asymptotic relations p(nS) ∼ q(nS′), q(nS) ∼ q(nS′), and qd(n) ∼ q(nS). The last case leads to a nonexistence theorem analogous to those of Lehmer for equality. The other comparisons lead to infinite families of cases of asymptotic equality without strict equality. These new formulas can be interpreted as asymptotic analogs of classical Rogers-Ramanujan identities