Continued Fractions for partition generating functions

TL;DR

This paper derives continued fraction representations for partition generating functions using classical techniques, with potential extensions to vector partitions such as binary and n-ary partitions.

Contribution

It introduces new continued fraction formulas for partition generating functions based on Euler's and Ramanujan's methods, extending their applicability.

Findings

Derived continued fractions for partition generating functions.

Demonstrated potential for extending to vector partitions.

Connected classical techniques to modern partition theory.

Abstract

We derive continued fractions for partition generating functions, utilizing both Euler's techniques and Ramanujan's techniques. Although our results are for integer partitions there is scope to extend this work to vector partitions, including for binary and n-ary partitions.