A note on the rank parity function (Q965981)

The rank of a partition is defined as the largest part minus the number of parts. Let \(S(n)\) denote the number of partitions into distinct parts with even rank minus those with odd rank. In the paper under review the author proves the following identity for positive even integers \(n\): \[ S\left(\frac{n}{2}\right)=\sum_{\substack{ r\geq 0,\, |j|\leq r, \\ 24n+2=3(4r+1)^2-(6j+1)^2 }} (-1)^{r+j}-\sum_{\substack{ r\geq 0,\, |j|\leq r, \\ 24n+2=3(4r+3)^2-(6j+1)^2 }} (-1)^{r+j}. \] The author also establishes some results for the q-series related with the generating functions of \(S_E(n)\) and \(S_O(n)\), where \(S_E(n)\) (resp. \(S_O(n)\)) is the number of partitions of \(n\) into distinct parts with even rank minus the number of them with odd rank where the number of parts is even (resp. odd). The proofs of these results are achieved using two new Bailey pairs.