Partitions, quasimodular forms, and the Bloch–Okounkov theorem

Abstract

We give a very short proof of the Bloch–Okounkov theorem on the quasimodularity of certain functions defined by sums over partitions, and also show how to make their map $\mathfrak {s}\mathfrak {l}_2$-equivariant.

Appendix: Table of q-brackets up to weight 8

We give a list of $\langle f\rangle _q$ for all elements f of $\Lambda _*$ of even weight $\le 8$, using Ramanujan’s notations $P=E_2$, $Q=E_4$, $R=E_6$.

$\langle 1\rangle _q =1$	$\langle Q_2^4\rangle _q = \dfrac{-15P^4 + 180QP^2 - 320RP + 156Q^2}{331776}$
$\langle Q_2\rangle _q = \dfrac{-P}{24}$	$\langle Q_2^2Q_4\rangle _q = \dfrac{75P^4 - 144QP^2 - 128RP + 204Q^2}{3317760 }$
$\langle Q_2^2\rangle _q=\dfrac{-P^2+2Q}{576}$	$\langle Q_2Q_3^2\rangle _q = \dfrac{25P^4 - 57QP^2 + 2RP + 30Q^2}{622080}$
$\langle Q_4\rangle _q=\dfrac{5P^2+2Q}{5760}$	$\langle Q_2Q_6\rangle _q = \dfrac{-175P^4 - 168QP^2 + 160RP + 276Q^2}{69672960}$
$\langle Q_2^3\rangle _q =\dfrac{-3P^3 + 18QP - 16R}{13824}$	$\langle Q_3Q_5\rangle _q = \dfrac{-35P^4 - 21QP^2 + 26RP + 30Q^2}{4354560}$
$\langle Q_2Q_4\rangle _q=\dfrac{15P^3 - 6QP - 16R}{138240}$	$\langle Q_4^2\rangle _q = \dfrac{-2625P^4 - 1260QP^2 + 1600RP + 2628Q^2}{232243200}$
$\langle Q_3^2\rangle _q=\dfrac{5P^3 - 3QP - 2R}{25920}$	$\langle Q_8\rangle _q = \dfrac{175P^4 + 420QP^2 + 320RP + 228Q^2}{1393459200}$
$\langle Q_6\rangle _q=\dfrac{-35P^3 - 42QP - 16R}{2903040}$