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Rankin-Cohen brackets on quasimodular forms. (English) Zbl 1206.11052
In this paper Rankin-Cohen brackets on quasimodular forms are defined. Specifically, let $$f(z)$$ and $$g(z)$$ be quasimodular forms of positive integer weights $$k$$ and $$\ell$$ and depths at most $$s \in \{0,\dots,\lfloor\frac{k}{2}\rfloor \}$$ and $$t \in \{0,\dots,\lfloor\frac{\ell}{2}\rfloor \}$$, respectively, on a given subgroup $$\Gamma$$ of finite index in $$\text{SL}_2(\mathbb{Z})$$. For $$n \geq 0$$ the $$n$$th Rankin-Cohen bracket $$\Phi_{n;k,s;\ell,t}(f,g)$$ is defined by $\Phi_{n;k,s;\ell,t}(f,g) := \sum_{r=0}^n (-1)^r \binom{k-s+n-1}{n-r}\binom{\ell-t+n-1}{r}D^rfD^{n-r}g,$ where $$D := \frac{1}{2\pi i} \frac{d}{dz}$$. It is shown that $$\Phi_{n;k,s;\ell,t}(f,g)$$ is a quasimodular form on $$\Gamma$$ having weight $$k+\ell+2n$$ and depth at most $$s+t$$. In some cases, a more precise description of $$\Phi_{n;k,s;\ell,t}(f,g)$$ is obtained. These Rankin-Cohen brackets generalize those for modular forms. As applications, elegant derivations of certain formulas of Neibur and van der Pol and certain differential equations of Ramanujan and Chazy are presented.

##### MSC:
 11F11 Holomorphic modular forms of integral weight 11F22 Relationship to Lie algebras and finite simple groups 16W25 Derivations, actions of Lie algebras