Exploring Eisenstein Series: Proving Recurrence Relations for G_k(z)

[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?
:uhh:

 

[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.
:grumpy:

 

[blue_raver22]

One approach to proving the recurrence relations for G_k(z) is to use the definition of Eisenstein series and the properties of modular forms. Specifically, you can use the fact that G_k(z) is a modular form of weight k for the full modular group, which means that it satisfies certain transformation properties under the action of the modular group. This can help you manipulate the expression for G_k(z) and show that it satisfies the recurrence relations. Additionally, you can also use the properties of the Dedekind eta function, which is closely related to the Eisenstein series, to help with the proof. Another helpful tool could be to use the Fourier expansion of G_k(z) and manipulate it to show the recurrence relations. Overall, it will likely require a combination of these techniques and possibly others to prove the recurrence relations for G_k(z).