Euler-type Recurrence Relations for Partition Functions with Congruence Conditions
arXiv:2607.28245
The paper investigates partition functions that count partitions with parts congruent to 0 or ±g modulo δ, deriving Euler‑type recurrence relations via generalized Dedekind eta functions and Rankin‑Cohen brackets, and providing explicit recurrences (e.g., for δ=5) that yield Ramanujan‑type congruences and a Rademacher‑type formula.
Abstract
We study partition functions counting partitions into parts congruent to or . 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 , 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.