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

We give Euler-like recursive formulas for the t -colored partition function when $$t=2$$ t = 2 or $$t=3,$$ 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)∞ ( q ; q ) ∞ and (q;q)∞³. ( q ; q ) ∞ 3 .