Could someone show me how to prove the following formula?
prove that when $a_n=\frac{ζ(2n)}{Π^{2n}}$ (n=1.2.3……)
$$(n+\frac{1}{2})a_n=\sum_{k=0}^{n-1}a_{n-k}{a_k}$$ (n≥2) and find out the value of $a_2,a_3,a_4,a_5,a_6$
zeta-function is soooo difficult and i almost give up
I need your help and thank you very much
The generating function for $\zeta(2n)/\pi^{2n}$ is
$$\frac{1}{2}-\frac{\pi x}{2}\cot(x)=\sum_{n=1}^\infty\frac{\zeta(2n)}{\pi^{2n}}x^{2n}.$$
Combining this with the fact that $\cot(x)$ satisfies the differential equation
$$f’(x)=-1-(f(x))^2$$
and using the Cauchy product gives the desired recurrence relation.