Radial Limits of Partial Theta and Similar Series

TL;DR

This paper investigates the behavior of certain periodic series with unbounded exponents as they approach roots of unity, providing explicit limit calculations using elementary methods from $q$-theory.

Contribution

It introduces a method to compute radial limits of partial theta and similar series with periodic coefficients and unbounded exponents, expanding understanding of their boundary behavior.

Findings

Explicit limits at roots of unity for these series

Elementary proofs using $q$-integral techniques

Connection to quadratic polynomials in the exponent

Abstract

We study unilateral series in a single variable $q$ where its exponent is an unbounded increasing function, and the coefficients are periodic. Such series converge inside the unit disk. Quadratic polynomials in the exponent correspond to partial theta series. We compute limits of those series as the variable tends radially to a root of unity. The proofs use ideas from the $q$-integral and are elementary.