An infinite product
An infinite product
Prove that
$$\prod_{n=1}^{\infty} \left ( 1 + \frac{x^2}{n^2+n-1} \right ) = \frac{1}{\cos \left ( \frac{\pi \sqrt{5}}{2} \right )} \frac{\cos \left ( \frac{\pi \sqrt{5-4x^2}}{2} \right )}{1-x^2}$$
$$\prod_{n=1}^{\infty} \left ( 1 + \frac{x^2}{n^2+n-1} \right ) = \frac{1}{\cos \left ( \frac{\pi \sqrt{5}}{2} \right )} \frac{\cos \left ( \frac{\pi \sqrt{5-4x^2}}{2} \right )}{1-x^2}$$
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
Re: An infinite product
Using the infinite product for cosines: $\displaystyle \cos (\pi x) = \prod\limits_{n=1}^{\infty} \left(1-\frac{x^2}{\left(n-\frac{1}{2}\right)^2}\right)$,
\begin{align*}\prod\limits_{n=1}^{\infty} \left(1 + \frac{x^2}{n^2+n-1}\right) &= \prod\limits_{n=2}^{\infty} \frac{\left(n-\frac{1}{2}\right)^2 - \frac{5}{4} + x^2}{\left(n-\frac{1}{2}\right)^2 - \frac{5}{4}} \\&= \frac{1}{(1-x^2)}\prod\limits_{n=1}^{\infty}\frac{\left(1 - \frac{\frac{5}{4} - x^2}{\left(n-\frac{1}{2}\right)^2}\right)}{1 - \frac{\frac{5}{4}}{\left(n-\frac{1}{2}\right)^2}} \\&= \frac{1}{1-x^2}\frac{\cos \left(\frac{\pi\sqrt{5 - 4x^2}}{2}\right)}{\cos \left(\frac{\pi\sqrt{5}}{2}\right)}\end{align*}
\begin{align*}\prod\limits_{n=1}^{\infty} \left(1 + \frac{x^2}{n^2+n-1}\right) &= \prod\limits_{n=2}^{\infty} \frac{\left(n-\frac{1}{2}\right)^2 - \frac{5}{4} + x^2}{\left(n-\frac{1}{2}\right)^2 - \frac{5}{4}} \\&= \frac{1}{(1-x^2)}\prod\limits_{n=1}^{\infty}\frac{\left(1 - \frac{\frac{5}{4} - x^2}{\left(n-\frac{1}{2}\right)^2}\right)}{1 - \frac{\frac{5}{4}}{\left(n-\frac{1}{2}\right)^2}} \\&= \frac{1}{1-x^2}\frac{\cos \left(\frac{\pi\sqrt{5 - 4x^2}}{2}\right)}{\cos \left(\frac{\pi\sqrt{5}}{2}\right)}\end{align*}
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