How to show that $\sum_{n=1}^{\infty}\frac{\phi^{2n}}{n^2{2n \choose n}}=\frac{9}{50}\pi^2$

Question

Given:$$\sum_{n=1}^{\infty}\frac{\phi^{2n}}{n^2{2n \choose n}}=\frac{9}{50}\pi^2$$

Where $\phi=\frac{\sqrt{5}+1}{2}$

How can I we show that the above sum is correct? I have checked numerically, it seem correct, but i don't how to proves it

Answer 1

It can be shown $|x|<4\implies\sum_{n=1}^\infty\frac{x^n}{n^2\binom{2n}{n}}=2\arcsin^2\frac{\sqrt{x}}{2}$, so the choice $x=\phi^2$ gives $\sum_{n=1}^\infty\frac{\phi^{2n}}{n^2\binom{2n}{n}}=2\arcsin^2\frac{\phi}{2}=\frac{9\pi}{50}$ as desired. (The arcsine's evaluation is possible by chasing isosceles triangles in a regular pentagon with inscribed diagonals.)

Answer 2

Consider the well-known series expansion of the squared $\arcsin$ function. Namely

$$2\arcsin^2\left(\frac x2\right)=\sum_{n=1}^{\infty}\frac{x^{2n}}{n^2\binom{2n}n}$$

As you can see the RHS equals your given sum for the case that $x=\phi$. Plugging this in yields to

$$\begin{align} 2\arcsin^2\left(\frac \phi2\right)=\sum_{n=1}^{\infty}\frac{\phi^{2n}}{n^2\binom{2n}n}=\frac{9\pi^2}{50} \end{align}$$

which further reduces the problem to

$$\sin\left(\frac{3\pi}{10}\right)=\frac\phi2=\frac{1+\sqrt{5}}{4}$$

The latter one can be shown by only using the fact that $\cos(\pi/5)=1/4(1+\sqrt 5)$. It is sufficient to show that

$$\begin{align} \sin\left(\frac{3\pi}{10}\right)&=\cos\left(\frac{\pi}5\right)\\ \sin\left(\frac{3\pi}{10}\right)&=\cos\left(\frac{7\pi}{10}-\frac{\pi}2\right)\\ \sin\left(\frac{3\pi}{10}\right)&=\sin\left(\frac{7\pi}{10}\right)\\ \sin\left(\frac{3\pi}{10}\right)&=\sin\left(-\frac{3\pi}{10}+\pi\right)\\ \sin\left(\frac{3\pi}{10}\right)&=-\sin\left(-\frac{3\pi}{10}\right) \end{align}$$

and the last line turn out to be true since the sine function is odd.

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The assumed fact concerning the value of $\cos(\pi/5)$ can be easily verified by cosidering the fifths roots of unity. To be precise consider

$$\begin{align} x^4+x^3+x^2+x+1&=0\\ x^2+\frac1{x^2}+x+\frac1x+1&=0\\ \left(x+\frac1x\right)^2+\left(x+\frac1x\right)-1&=0 \end{align}$$

The latter can be solved by the substitution $z=1/x+x$ which boils the problem down to the solving of two quadratic equations. The real part of the first solution $($the one with positive real and imaginary part$)$ equals the cosine of $\pi/5$ and we are done.

Answer 3

Replace $\phi$ with $x$. That gives a function of $x$ with its Taylor series.
Now to find a differential equation that the Taylor series obeys.
Hint:

On one hand, $d^2f/dx^2=\sum (2n+2)(2n+1)a_{n+1}x^{2n}$. On the other hand $xdf/dx=\sum 2n a_n x^{2n}$.

Good luck with that. As a second hint, other people have supplied the answer.