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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
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