Partition identities with mixed mock modular forms
We seek them here, we seek them there, partition identities are everywhere. This fascinating paper makes the case that many more partition identities are just mock theta functions in disguise. The authors begin by claiming that that there are countless partition identities related to infinite products which are usually modular forms. They go on to give a new example related to a mock modular form. Theorem. Let \(C_k(n)\) denote the number of partitions of \(n\) of the following form \(f_11+f_22+\cdots+f_nn=n\) where (1) \(0\leq f_i\leq 2k+1\), (2) if \(f_i<2k\) then \(\lfloor f_i/2\rfloor+\lfloor f_{i+1}/2\rfloor\leq k-1\) and (3) if \(f_i=2k\) or \(2k+1\), then \(i\) is odd, say \(2j+1\), each of \(f_1, f_3, \ldots, f_{2j+1}\) is either \(2k\) or \(2k+1\) and each of \(f_2, f_4, \ldots, f_{2j+2}\) is zero, then \[ \sum_{n=0}^\infty C_k(n)q^n = \frac{1+\sum_{j=1}^\infty (q^{(2k+3)j^2+j} s(j,q^2) - q^{(2k+3)j^2-j} s(j-1,q^2))} {\prod_{n=1}^\infty(1-q^n)}, \] where \[ s(j,q) = \sum_{n=-j}^j (-1)^j q^{-j^2}. \] Here, \(C_k(n)\) is the number of partitions of \(n\) in which the first parts consist entirely of consecutive odd numbers starting with 1 appearing \(2k\) or \(2k+1\) times and the larger parts are required to appear fewer than \(2k\) times and are subject to the conditions. Let \(F_k(x;q)= \sum_{m,n\geq 0} c_k(m,n)x^mq^n\), where \(c_k(m,n)\) is the number of partitions enumerated by \(C_k(n)\) that have exactly \(m\) parts. Then \(F(x;q)\) satisfies a first order nonlinear difference equation relating \(F(x;q)\) and \(F(xq^2;q)\) which can be proved by a simple counting argument. Iterating this equation \(N\) times, letting \(N\rightarrow\infty\) and putting \(x=1\) gives the identity for \(\sum_{n=0}^\infty C_k(n)q^n\). This is Fine's somewhat neglected elementary method for solving such difference equations. Along the way to the identity, we are treated to a masterclass of manipulations from the Andrews toolbox of partition theorems and Bailey pairs. For \(k=0\) and 1, the behemoth simplifies considerably: \[ \sum_{n=0}^\infty C_1(n)q^n = f_0(q^2) \prod_{n=1}^\infty(1+q^n), \] where \(f_0(q) = 1 + \sum_{n=1}^\infty q^{n^2}/(1+q)(1+q^2)\cdots(1+q^n)\) is one of Ramanujan's fifth order mock theta functions, and from \(k=0\), \[ 1+\sum_{n=1}^\infty(q^{3n^2+n}s(n,q^2) - q^{3n^2-n} s(n-1, q^2)) = (-q;-q)_\infty (q;q)_\infty. \] As a final challenge, the authors ask for purely bijective proofs of the last two identities.
scientific article; zbMATH DE number 6479856
scientific article