Eisenstein Series - Recurrence Relation

Recurrence Relation

Any holomorphic modular form for the modular group can be written as a polynomial in and . Specifically, the higher order 's can be written in terms of and through a recurrence relation. Let . Then the satisfy the relation

for all . Here, is the binomial coefficient and and .

The occur in the series expansion for the Weierstrass's elliptic functions:

$\wp(z) =\frac{1}{z^2} + z^2 \sum_{k=0}^\infty \frac {d_k z^{2k}}{k!} =\frac{1}{z^2} + \sum_{k=1}^\infty (2k+1) G_{2k+2} z^{2k}$

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