On the number of parts in all partitions enumerated by the Rogers-Ramanujan identities

Abstract

The celebrated Rogers-Ramanujan identities equate the number of integer partitions of $n$ ($n\in {\mathbb{N}}_0$) with parts congruent to $\pm 1(\mathrm{mod}5)$ (respectively $\pm 2(\mathrm{mod}5)$) and the number of partitions of $n$ with super-distinct parts (respectively super-distinct parts greater than $1$). In this paper, we establish companion identities to the Rogers-Ramanujan identities on the number of parts in all partitions of $n$ of the aforementioned types, in the spirit of earlier work by Andrews and Beck on a partition identity of Euler.

arXiv comment16 pages