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arXiv · 2607.28245

Euler-type Recurrence Relations for Partition Functions with Congruence Conditions

Wissam Raji·Hasan Saad
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Abstract

We study partition functions $p_{\delta,g}(n)$ counting partitions into parts congruent to $0$ or $\pm g \pmod\delta$. Using generalized Dedekind eta functions and Rankin-Cohen brackets, we derive infinite families of Euler-type recurrences involving divisor sums and Fourier coefficients of cusp forms. We also obtain an explicit recurrence for $\delta=5$, which, as a corollary, gives a Ramanujan-type congruence. As a corollary of our method of proof, we obtain a Rademacher-type formula involving Kloosterman sums and Bessel functions.

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Wissam Raji, Hasan Saad. 2026-07-30. Euler-type Recurrence Relations for Partition Functions with Congruence Conditions. https://arxiv.org/abs/2607.28245

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