MacMahon’s sums-of-divisors and allied \(q\)-series.(English)Zbl 07916608
Summary: Here we investigate the \(q\)-series\[\begin{aligned}\mathcal{U}_a(q) &= \sum_{n = 0}^\infty MO(a; n) q^n \\&:= \sum_{0 < k_1 < k_2 < \cdots < k_a} \frac{q^{k_1 + k_2 + \cdots + k_a}}{(1 - q^{k_1})^2 (1 - q^{k_2})^2 \cdots (1 - q^{k_a})^2} \\\mathcal{U}_a^\star(q) &= \sum_{n = 0}^\infty M(a; n) q^n \\&:= \sum_{1 \leq k_1 \leq k_2 \leq \cdots \leq k_a} \frac{q^{k_1 + k_2 + \cdots + k_a}}{(1 - q^{k_1})^2 (1 - q^{k_2})^2 \cdots (1 - q^{k_a})^2}.\end{aligned}\]
MacMahon introduced the \(\mathcal{U}_a(q)\) in his seminal work on partitions and divisor functions. Recent works show that these series are sums of quasimodular forms with weights \(\leq 2a\). 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 \(M O(a; n)\) and \(M(a; n)\). Notably, we prove the conjecture of Amdeberhan-Andrews-Tauraso as the \(m = 0\) special case of the infinite family of congruences\[MO(11 m + 10; 11 n + 7) \equiv 0 \pmod{11},\]and we prove that\[MO(17 m + 16; 17 n + 15) \equiv 0 \pmod{17}.\]We obtain further formulae using the limiting behavior of these series. For \(n \leq a + \binom{a + 1}{2}\), we obtain a “hook length” formula for \(MO(a; n)\), and for \(n \leq 2 a\), we find that \(M(a; n) = \binom{a + n - 1}{n - a} + \binom{a + n - 2}{n - a - 1}\).
MacMahon introduced the \(\mathcal{U}_a(q)\) in his seminal work on partitions and divisor functions. Recent works show that these series are sums of quasimodular forms with weights \(\leq 2a\). 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 \(M O(a; n)\) and \(M(a; n)\). Notably, we prove the conjecture of Amdeberhan-Andrews-Tauraso as the \(m = 0\) special case of the infinite family of congruences\[MO(11 m + 10; 11 n + 7) \equiv 0 \pmod{11},\]and we prove that\[MO(17 m + 16; 17 n + 15) \equiv 0 \pmod{17}.\]We obtain further formulae using the limiting behavior of these series. For \(n \leq a + \binom{a + 1}{2}\), we obtain a “hook length” formula for \(MO(a; n)\), and for \(n \leq 2 a\), we find that \(M(a; n) = \binom{a + n - 1}{n - a} + \binom{a + n - 2}{n - a - 1}\).
MSC:
| 11F03 | Modular and automorphic functions |
| 11A25 | Arithmetic functions; related numbers; inversion formulas |
| 11F50 | Jacobi forms |
| 11M41 | Other Dirichlet series and zeta functions |
| 11F11 | Holomorphic modular forms of integral weight |
| 11F33 | Congruences for modular and \(p\)-adic modular forms |
Cite
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