Lacunary partition functions
A function \(f(q)\) is lacunary if \(f(q)= \sum_{n\geq 0} a(n)q^n\) and \(a(n)\) is almost always \(0\). It is known by the work of J.-P. Serre, B. Gordon and S. Robins that there are approximately 60 pairs \((r,s)\) for which \[ \prod_{n=1}^\infty (1-q^n)^r(1-q^{2n})^s \] is lacunary. In this article, the author first establishes, using the theory of Bailey pairs, identities expressing multi-sum Rogers-Ramanujan type expressions in terms of Hecke type functions. An example is \[ \begin{split} &\sum_{n\geq 0}\sum_{j=-n}^n(-1)^{n+j}q^{((2k+3)n^2+(2k+1)n)/2-j(3j+1)/2}(1-q^{2n+1}) \\ &\qquad = \sum_{n_k\geq n_{k-1}\geq\cdots \geq n_1\geq 0} \frac{(q;q)_{n_k}(-1)^{n_k}q^{n_k(n_k+1)/2+n_{k-1}(n_{k-1}+1)+\cdots +n_1(n_1+1)}}{(q;q)_{n_k-n_{k-1}}\cdots (q;q)_{n_2-n_1}},\end{split}\tag{1} \] where as usual \[ (a;q)_n = (1-a)(1-aq)\cdots (1-aq^{n-1}). \] Using a result due to P. Bernays, the author deduces that the left hand side of (1) is lacunary. He then gives a partition-theoretic interpretation of the right hand side of (1) and explains why the difference of certain partition functions is almost always 0. Such results are analogous to Euler's result, which states that if \(E(n)\) and \(O(n)\) denote respectively the number of partitions of \(n\) into an even and odd number of distinct parts, then \(E(n)-O(n)=0\) for almost all \(n\).
scientific article; zbMATH DE number 1804047