ANALYTIC ASPECTS OF PARTITION AND COLORED PARTITION FUNCTIONS USING q-SERIES

Abstract

A weighted partition generating function defined by a multi-factor Euler
product using analytic and computational methods. Explicit coefficient formulas are derived via
convolutions of colored partition functions and Euler-type linear recurrence relations involving
divisor sums are established. Asymptotic analysis shows that the coefficients exhibit exponential
growth governed by dominant singularities on the boundary of the unit disk. Numerical evaluation
based on truncated Euler products is investigated, and three-dimensional surface plots confirm
stable convergence within the unit disk while indicating increased sensitivity near the dominant
singularity. The combined results highlight the usefulness of integrating theoretical analysis with
numerical computation for partition-type generating functions.