Partition identities from third and sixth order mock theta functions (Q449199)

The authors interpret some third and sixth order mock theta functions in terms of \(n\)-color partitions and \(n\)-color overpartitions. For example, the sixth order mock theta function NEWLINE\[NEWLINE \sum_{n \geq 1} \frac{(-q;q)_{2n-2}q^n}{(q;q^2)_n} NEWLINE\]NEWLINE is the generating function for \(n\)-color overpartitions where \((i)\) the smallest part is non-overlined and of the form \(k_k\), and \((ii)\) the ``weighted difference'' between two consecutive parts is \(0\) or \(-2\).NEWLINENEWLINEThen they consider a few identities involving these mock theta functions, such as NEWLINE\[NEWLINE \sum_{n \geq 0} \frac{(-1)^nq^{(n+1)^2}(q;q^2)_n}{(-q;q)_{2n+1}} + 2\sum_{n \geq 1} \frac{(-q;q)_{2n-2}q^n}{(q;q^2)_n} = 3q\frac{(q^6;q^6)_{\infty}^3}{(q;q)_{\infty}(q^2;q^2)_{\infty}}. NEWLINE\]NEWLINE They interpret the identities combinatorially and prove some congruence properties of the modular forms on the right-hand sides. For example, if \(c(n)\) is defined by NEWLINE\[NEWLINE \sum_{n \geq 1} c(n)q^n = q\frac{(q^6;q^6)_{\infty}^3}{(q;q)_{\infty}(q^2;q^2)_{\infty}}, NEWLINE\]NEWLINE then NEWLINE\[NEWLINE c(3n) \equiv 0 \pmod{3}. NEWLINE\]NEWLINE They also explore crank-type statistics for such congruences.