How can the recurrence relations for Eisenstein series G_k(z) be proven?

[MathematicalPhysicist]

Can someone give me a hint how to show the recurrence relations for $$G_k(z)$$
,in wiki it's for the d_n's?

Other than proving it by induction I don't have clue what to do here.

Thanks.

 

[MathematicalPhysicist]

Obviously I need to use here some property of Eisenstein series, I just need to know which?

Anyone?

 

[Petek]

Were talking about the recurrence relation here, right?

I haven't worked out all the details, but here's how the proof is supposed to go:

The Weierstrass $\wp$ function (see here) satisfies the following differential equation:

$$[\wp'(z)]^2 = 4[\wp(z)]^3 - g_2\wp(z) - g_3$$

where $g_2$ and $g_3$ are the same as defined in the wiki article on Eisenstein series. Differentiate this equation and cancel $\wp'(z)$ to get the second order differential equation

$$\wp''(z) = 6[\wp(z)]^2 - \frac{1}{2}g_2$$

Now, as in the Eisenstein series article, we have

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

Differentiate twice and equate like powers of z.

 

[MathematicalPhysicist]

I am an idiot, the second ODE you gave me I proved before this task.

I shouldv'e known it would be that easy.

Thanks.