Work

Certain infinite products in terms of MacMahon type series

Year: 2025

Type: article

Abstract: Recently, Ono and the third author discovered that the reciprocals of the theta series [Formula: see text] and [Formula: see text] have infinitely many closed formulas in terms of MacMahon’s quasimodular forms [Formula: see text] and [Formula: see text]. 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 [Formula: see text] be the theta function corresponding to the odd quadratic character modulo 6. Then for any positive integer n, we have [Formula: see text] where [Formula: see text] and [Formula: see text].

Source: International Journal of Number Theory

DOI: 10.1142/s1793042126500314

Authors: Seokho Jin, Badri Vishal Pandey, Ajit Singh

Institutions: Chung-Ang University, University of Cologne, Indian Institute of Technology Dhanbad

Language: English

Topic: Advanced Mathematical Identities

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: European Commission, National Research Foundation, National Research Foundation of Korea, Ministry of Science and ICT, South Korea

Awards: G1874071222, G434337789