Rogers-Ramanujan subpartitions and their connections to other partitions
An ordinary partition of an integer \(n\) is a set of positive integers \[ a_1\geq a_2\geq \dots \geq a_r \] such that \[ a_1+a_2+\dots +a_r = n. \] A Rogers-Ramanujan partition of an integer \(n\) is a set of positive integers \(b_1>b_2>\dots >b_s\) such that \[ b_1+b_2+\dots +b_s=n \] and \(b_i-b_j>1\) for \(i<j\). A Rogers-Ramanujan subpartition of length \(s\) (if exists) of an ordinary partition \(a_1\geq a_2\geq \dots \geq a_r\) of \(n\) is the longest initial sequence of parts \(a_1,a_2,\dots , a_s\) such that \(a_s>a_{s+1}\) and \(a_1, a_2, \dots , a_s\) form a Rogers-Ramanujan partition of \(n-(a_{s+1}+\dots + a_r).\) In this paper, the author interpreted the identity \[ \begin{multlined} \prod_{k=1}^\infty\frac{1}{(1-q^k)}=\sum_{k=0}^\infty \frac{q^{k^2+k}}{\prod_{j=1}^k (1-q^j)\prod_{m=0}^\infty (1-q^{k+2+m})} \\ +\sum_{k=1}^\infty \frac{q^{k^2}}{\prod_{j=1}^{k-1}(1-q^j)\prod_{m=0}^\infty (1-q^{k+1+m})}\end{multlined}\tag{1} \] as an identity involving the Rogers-Ramanujan subpartitions. The author then considers the weighted version of the series on the right hand side of (1) and relates, for example, the series \[ \sum_{k=0}^\infty \frac{(2k+1)q^{k^2+k}}{\prod_{j=1}^k (1-q^j)\prod_{m=0}^\infty (1-q^{k+2+m})} +\sum_{k=1}^\infty \frac{(2k+1)q^{k^2}}{\prod_{j=1}^{k-1}(1-q^j)\prod_{m=0}^\infty (1-q^{k+1+m})} \] to 2-color Frobenius partitions.
scientific article; zbMATH DE number 5343448