On the q-factorization of power series

2025/01/30 by Schneider, Robert, Sills, Andrew V., Waldron, Hunter
#11P81 #30B10 #40A20 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2501.18744

Abstract

Any power series with unit constant term can be factored into an infinite product of the form ∏_{n≥ 1} (1-qⁿ)^-a_n. We give direct formulas for the exponents a_n in terms of the coefficients of the power series, and vice versa, as sums over partitions. As examples, we prove identities for certain partition enumeration functions. Finally, we note q-analogues of our enumeration formulas.

Related

• Infinite product of power series
• q-power symmetric functions and q-exponential formula
• An Alternative Generating Function for k-Regular Partitions
• Remarks on MacMahon's q-series
• Fractional Powers of the Generating Function for the Partition Function