Continued Fractions for Square Series Generating Functions

TL;DR

This paper introduces new series expansions for square series generating functions using Jacobi-type continued fractions, providing exact convergents, infinite q-series representations, and applications to special sequences and theta functions.

Contribution

It develops novel expansions of square series functions via J-fractions, including exact convergents and infinite q-series, extending previous generating function transformations.

Findings

Derived new exact expansions for convergents of continued fractions.

Established infinite q-series representations involving q-Pochhammer symbols.

Applied results to generate new series for special sequences and theta functions.

Abstract

We consider new series expansions for variants of the so-termed ordinary geometric square series generating functions originally defined in the recent article titled "Square Series Generating Function Transformations" (arXiv: 1609.02803). Whereas the original square series transformations article adapts known generating function transformations to construct integral representations for these square series functions enumerating the square powers of $q^{n^2}$ for some fixed non-zero $q$ with $|q| < 1$, we study the expansions of these special series through power series generated by Jacobi-type continued fractions, or J-fractions. We prove new exact expansions of the $h^{th}$ convergents to these continued fraction series and show that the limiting case of these convergent generating functions exists. We also prove new infinite $q$-series representations of special square series expansions involving square-power terms of the series parameter $q$, the $q$-Pochhammer symbol, and double sums over the $q$-binomial coefficients. Applications of the new results we prove within the article include new $q$-series representations for the ordinary generating functions of the special sequences, $r_p(n)$, and $\sigma_1(n)$, as well as parallels to the examples of the new integral representations for theta functions, series expansions of infinite products and partition function generating functions, and related unilateral special function series cited in the first square series transformations article.