Basis partition polynomials, overpartitions and the Rogers-Ramanujan identities (Q2346297)

Each partition \(\pi\) of a positive integer contains a largest square of nodes in its Ferrers graph called the Durfee square. If the Durfee square has side length \(d\), then the \(it\)h rank \(r_i\) of \(\pi (1 \leq i \leq d)\) is defined as the difference between the number of nodes in the \(it\)h row of the Ferrers graph of \(\pi\) and the number in the \(it\)h column. It is known that, for every rank vector \(\overset{\rightarrow}{r} = (r_1 , r_2 , \dots, r_d )\), there is a smallest integer that has a partition with rank vector \(\overset{\rightarrow}{r}\), and this partition is unique. It is called the basis partition of \(\overset{\rightarrow}{r}\). Let \(B(n)\) be the number of basis partitions of \(n\). It is also known that the generating series of the sequence \(\{B(n)\}_{n\geq 0}\) is equal to the series \(\sum^\infty_{ n=0}\frac{q^{n{}^2}(q;q)_n}{(q;q)_n}\), where \((A; q)_n := (1 - A)(1 -Aq)\cdots(1 - Aq)^{n -1}\). In this paper, the author focuses on the series \(G(a, x; q) = \sum^\infty_{n=0}\frac{ a^n q^{n{}^2}(x;q)_n}{(q;q)_n} \). He finds an identity for \(G(a, x; q)\) which both leads to the Rogers-Ramanujan identities and also provides a new representation for the generating function of \(\{B(n)\}_{n\geq 0}\). This identity involves polynomials called basic partition polynomials. They provide a new, more rapidly converging series expansion for the basis partition generating function. Finally, the author proves that the basis partitions of \(n\) are in one-to-one correspondence with the overpartitions of \(n\) in which there are no overlined 1's, the difference between all parts is at least 2, and greater than 2 between parts where the larger is overlined.