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

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 \((q;q)_{\infty}^3.\)