Generalized Frobenius Partitions Modulo Powers of $2$
Abstract
Let cϕk(n) denote the number of k-colored generalized Frobenius partitions of n. We prove that, for every m≥2 and every k≡2pmod2m, ∑n≥0cϕk(n)qn≡(φ(q) (q2;q2)∞)/(q;q)∞2∑n≥0cϕk/2(n)q2npmod2m, where φ(q) is the classical theta function. For m=2 this recovers a congruence of Chan, Wang, and Yang. Applied with k=18, we determine cϕ18(2n+1) modulo $16$ completely. In particular, we also prove ∑n≥0cϕ18(6n+1)qn ≡4∑r∈ℤqr(3r−1)/2pmod16, which proves the congruences cϕ18(30n+19)≡ cϕ18(30n+25)≡0pmod16 recently conjectured by Das, Nath, and Sarma (2026). It also yields further congruences modulo $16$ and a simple modulo-$8$ characterization that recovers and extends a recent congruence of those authors. As a second application we set k=10 and determine cϕ10(2n+1) modulo $8$.
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BibTeX
@misc{saikia2026generalized,
title = {{Generalized Frobenius Partitions Modulo Powers of $2$}},
author = {Manjil P. Saikia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11813},
eprint = {2609.11813},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11813},
note = {CC0},
}
RIS
TY - UNPB
TI - Generalized Frobenius Partitions Modulo Powers of $2$
AU - Manjil P. Saikia
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11813
UR - https://arxiv.org/abs/2609.11813
AB - Let cϕ_k(n) denote the number of k-colored generalized Frobenius partitions of n. We prove that, for every m≥2 and every k≡2pmod2^m, ∑_n≥0cϕ_k(n)q^n≡(φ(q) (q^2;q^2)_∞)/(q;q)_∞^2∑_n≥0cϕ_k/2(n)q^2npmod2^m, where φ(q) is the classical theta function. For m=2 this recovers a congruence of Chan, Wang, and Yang. Applied with k=18, we determine cϕ_18(2n+1) modulo $16$ completely. In particular, we also prove ∑_n≥0cϕ_18(6n+1)q^n ≡4∑_r∈ℤq^r(3r−1)/2pmod16, which proves the congruences cϕ_18(30n+19)≡ cϕ_18(30n+25)≡0pmod16 recently conjectured by Das, Nath, and Sarma (2026). It also yields further congruences modulo $16$ and a simple modulo-$8$ characterization that recovers and extends a recent congruence of those authors. As a second application we set k=10 and determine cϕ_10(2n+1) modulo $8$.
ER -