A note on congruences for the difference between even cranks and odd cranks
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914050136342528 |
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| author | Guadalupe, Russelle |
| author_facet | Guadalupe, Russelle |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_16267 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on congruences for the difference between even cranks and odd cranks Guadalupe, Russelle Number Theory Combinatorics 11P83, 05A17, 11P81 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. |
| title | A note on congruences for the difference between even cranks and odd cranks |
| topic | Number Theory Combinatorics 11P83, 05A17, 11P81 |
| url | https://arxiv.org/abs/2506.16267 |