Asymptotic analogs of the Rogers-Ramanujan identities
Let p(n,S) be the number of partitions of n with parts belonging to the set S; let q(n,S) be the number of partitions of n with parts distinct and belonging to the set S; let \(q_ d(n)\) be the number of partitions of n with parts differing by at least d. Asymptotic formulas for p(n,S), q(n,S), and \(q_ d(n)\) are derived. Using these formulas necessary and/or sufficient conditions are obtained on sets S and S' for the various asymptotic relations p(n,S)\(\sim q(n,S')\), q(n,S)\(\sim q(n,S')\), and \(q_ d(n)\sim q(n,S)\). 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.
scientific article; zbMATH DE number 4008476
scientific article