Certain infinite products in terms of MacMahon type series

초록

Recently, Ono and the third author discovered that the reciprocals of the theta series (q; q)(infinity)(3) and (q(2); q(2)) infinity(q; q(2))(infinity)(2) have infinitely many closed formulas in terms of MacMahon's quasimodular forms A(k)(q) and C-k(q). In this paper, 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 Theta(6)(q) := 1/2 Sigma(n is an element of Z chi 6)(n)nq(n2-1/24) be the theta function corresponding to the odd quadratic character modulo 6. Then for any positive integer n, we have 1/Theta 6(q) = q(-3n2+n/2) Sigma(k=r) (r2)1k equivalent to n(mod2)(-1)n-k 2Ak(q)C3n-k/2(q) + O(q(n+1)), where r(1) := left perpendicular3n-1-root 12n+13/3right perpendicular + 1 and r(2) := inverted right perpendicular3n-1+root 12n+13-3inverted left perpendicular - 1.

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