A note on congruences for the difference between even cranks and odd cranks

TL;DR

This paper provides elementary proofs and extensions of properties related to the crank parity function, including new Ramanujan-type congruences modulo 5 and 25, using identities involving the Rogers-Ramanujan continued fraction.

Contribution

It offers new elementary proofs and extends previous results on the crank parity function, establishing novel Ramanujan-type congruences.

Findings

Established Ramanujan-type congruences modulo 5 and 25 for sums involving C(n) and a(n)

Provided elementary proofs of key properties of the crank parity function

Extended previous results using Rogers-Ramanujan identities

Abstract

Recently, Amdeberhan and Merca proved some arithmetic properties of the crank parity function $C(n)$ defined as the difference between the number of partitions of $n$ with even cranks and those with odd cranks and the sequence $a(n)$ whose generating function is the reciprocal of that of $C(n)$. The function $C(n)$ was first studied by Choi, Kang, and Lovejoy. In this note, we give new elementary proofs of some of their main results and extend them. In particular, we establish Ramanujan-type congruences modulo $5$ and $25$ for certain finite sums involving $C(n)$ and $a(n)$. Our proofs employ the results of Cooper, Hirschhorn, and Lewis, and certain identities involving the Rogers-Ramanujan continued fraction $R(q)$ due to Chern and Tang.