Euler-type Recurrence Relations for Partition Functions with Congruence Conditions

We study partition functions $p_{\delta ,g}(n)$ counting partitions into parts congruent to $0$ or $\pm g(\mathrm{m}\mathrm{o}\mathrm{d}\delta )$. 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.