Certain infinite products in terms of MacMahon type series

Recently, Ono and the third author discovered that the reciprocals of the theta series (q;q)_∞³ and (q²;q²)_∞(q;q²)_∞² have infinitely many closed formulas in terms of MacMahon's quasimodular forms Aₖ(q) and Cₖ(q). In this article, we use the well-known infinite product identities due to Jacobi, Watson, and Hirschhorn to derive further such closed formulas for reciprocals of other interesting infinite products. Moreover, with these formulas, we approximate these reciprocals to arbitrary order simply using MacMahon's functions and { MacMahon type} functions. For example, let Θ₆(q):=1/2∑ₙ χ₆(n) n qn²-1/24 be the theta function corresponding to the odd quadratic character modulo $6$. Then for any positive integer n, we have 1Θ₆(q)= q-3n²+n/2∑_k=r₁\\ k≡ n-0.2cm2r₂(-1)n-k/2Aₖ(q)C3n-k/2(q)+O(qⁿ⁺¹), where r₁:=3n-1-√12n+133+1 and r₂:=3n-1+√12n+133-1.