Rankin–Cohen bracket

In mathematics, the Rankin–Cohen bracket of two modular forms is another modular form, generalizing the product of two modular forms. Rankin (1956, 1957) gave some general conditions for polynomials in derivatives of modular forms to be modular forms, and Cohen (1975) found the explicit examples of such polynomials that give Rankin–Cohen brackets. They were named by Zagier (1994), who introduced Rankin–Cohen algebras as an abstract setting for Rankin–Cohen brackets.

If ${\displaystyle f(\tau )}$ and ${\displaystyle g(\tau )}$ are modular form of weight k and h respectively then their nth Rankin–Cohen bracket [f,g]_n is given by

It is a modular form of weight k + h + 2n. Note that the factor of ${\displaystyle (2\pi i)^{n}}$ is included so that the q-expansion coefficients of ${\displaystyle [f,g]_{n}}$ are rational if those of ${\displaystyle f}$ and ${\displaystyle g}$ are. ${\displaystyle d^{r}f/d\tau ^{r}}$ and ${\displaystyle d^{s}g/d\tau ^{s}}$ are the standard derivatives, as opposed to the derivative with respect to the square of the nome which is sometimes also used.

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