Euler-type recurrences for $t$-color and $t$-regular partition functions

TL;DR

This paper derives Euler-like recursive formulas for t-colored and t-regular partition functions, including an infinite family of recurrences for the 3-colored partition function, using q-series identities.

Contribution

It introduces new recursive formulas for t-colored and t-regular partition functions, expanding the understanding of their combinatorial structures.

Findings

Derived recursive formulas for t=2 and t=3

Established an infinite family of recurrences for 3-colored partitions

Utilized q-series identities to prove the formulas

Abstract

We give Euler-like recursive formulas for the $t$-colored partition function when $t=2$ or $t=3,$ as well as for all $t$-regular partition functions. In particular, we derive an infinite family of ``triangular number" recurrences for the $3$-colored partition function. Our proofs are inspired by the recent work of Gomez, Ono, Saad, and Singh on the ordinary partition function and make extensive use of $q$-series identities for $(q;q)_{\infty}$ and $(q;q)_{\infty}^3.$