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<p>Recently, Amdeberhan and Merca proved some arithmetic properties of the crank parity function <i>C</i>(<i>n</i>) defined as the difference between the number of partitions of <i>n</i> with even cranks and those with odd cranks, and the sequence <i>a</i>(<i>n</i>) whose generating function is the reciprocal of that for <i>C</i>(<i>n</i>). The function <i>C</i>(<i>n</i>) 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 <i>C</i>(<i>n</i>) and <i>a</i>(<i>n</i>). Our proofs employ the results of Cooper, Hirschhorn, and Lewis, and certain identities involving the Rogers–Ramanujan continued fraction <i>R</i>(<i>q</i>) due to Chern and Tang.</p>

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

  • 期刊论文

  • 发表日期：

    2025年9月21日

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  • Russelle Guadalupe

摘要

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 for 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.