MacMahon's sums-of-divisors and allied -series
arXiv:2311.07496
Abstract
Here we investigate the -series \begin{align*} \mathcal{U}_a(q)&=\sum_{n=0}^{\infty} MO(a;n)q^n&:=\sum_{0< k_1<k_2<\cdots<k_a} \frac{q^{k_1+k_2+\cdots+k_a}}{(1-q^{k_1})^2(1-q^{k_2})^2\cdots(1-q^{k_a})^2},\\ \mathcal{U}_a^{\star}(q)&=\sum_{n=0}^{\infty}M(a;n)q^n&:=\sum_{1\leq k_1\leq k_2\leq\cdots\leq k_a} \frac{q^{k_1+k_2+\cdots+k_a}}{(1-q^{k_1})^2(1-q^{k_2})^2\cdots(1-q^{k_a})^2}. \end{align*} MacMahon introduced the in his seminal work on partitions and divisor functions. Recent works show that these series are sums of quasimodular forms with weights We make this explicit by describing them in terms of Eisenstein series. We use these formulas to obtain explicit and general congruences for the coefficients and Notably, we prove the conjecture of Amdeberhan-Andrews-Tauraso as the special case of the infinite family of congruences and we prove that We obtain further formulae using the limiting behavior of these series. For we obtain a ``hook length'' formulae for , and for , we find that
16 pages