2024/07/05 by Seokho Jin, Jin, Seokho, Badri Vishal Pandey +3 · 1 citation
Mathematics · #05A17 #11P81 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2407.04798
openalex publication_date 2024/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
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 \it MacMahon type functions. For example, let Θ6(q):=(1)/(2)∑n∈ℤ χ6(n) n q(n2-1)/(24) be the theta function corresponding to the odd quadratic character modulo 6. Then for any positive integer n, we have \frac1Θ6(q)= q-(3n2+n)/(2)∑_\substackk=r1
k≡ n\hspace-0.2cm\pmod2r2(-1)(n-k)/(2)Ak(q)C(3n-k)/(2)(q)+O(qn+1), where r1:=\lfloor(3n-1-√(12n+13))/(3)\rfloor+1 and r2:=\lceil(3n-1+√(12n+13))/(3)\rceil-1.