Partitions: At the interface of \(q\)-series and modular forms
In this paper the author discusses a handful of problems related to the theory of partitions that require tools from both \(q\)-series and the theory of modular forms. The first two problems deal with the generating function for partitions into at most \(N\) parts, \[ \prod_{m = 1}^N \frac{1}{1-x^m} := \sum_{n \geq 0} p(n,N)x^n. \] The first of these is a conjecture of Rademacher, which we shall not reproduce here, about certain coefficients \(C_{hkj}(N)\) in the classical partial fraction decomposition of the function above. The second problem is to find nice formulas for \(p(n,N)\). In both cases the author makes a positive contribution. For Rademacher's conjecture, he describes a method for writing down explicit formulas for the \(C_{hkj}(N)\). Unfortunately, it does not appear possible to deduce Rademacher's conjecture from these formulas. For the problem of finding formulas for \(p(n,N)\), the author exploits a relationship between periodic sequences and the greatest integer function to simplify some of the results existing in the literature. For example, we have \[ p(n,5) = \{ (n+8)\left(n^3 + n + 22n^2 + 44n + 248 + 180 \left\lfloor \frac{n}{2} \right\rfloor \right)/ 2880 \}, \] where \(\lfloor \cdot \rfloor\) is the greatest integer function and \(\{ \cdot \}\) is the nearest integer function. For the next two problems the author provides no new results but suggests a number of interesting questions related to transformations and asymptotics of certain non-modular generating functions, like mock theta functions and the generating function for partitions into \(k\)th powers. The final problem, which seems to be the most wide open of the bunch, is to directly prove modular transformation properties of certain \(q\)-series. For example, we know that \[ \sum_{n\geq 0} \frac{q^{n^2}}{\prod_{k=1}^n (1-q^k)^2} = \prod_{n=1}^{\infty} \frac{1}{1-q^n}. \] Is it possible to deduce the implied modular transformations of the left-hand side without acknowledging the equality?
scientific article; zbMATH DE number 2022039