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I am interested in knowing how to calculate infinite products like (or reading any reference about it):

$$\prod_{j=1}^{\infty}\left( 1-\left( \frac{x}{a+j\pi} \right) ^2 \right)$$

Inserting it into a Mathematica worksheet (Wolfram research), it returns the following beautiful formula:

$$\frac{\pi^2\Gamma(\frac{\pi+a}{\pi})^2}{\Gamma(\frac{a-x}{\pi})\Gamma(\frac{a+x}{\pi})}$$

where $\Gamma(x)$ is the Euler's Gamma function, and $x$ and $a$ are positive real numbers.

Thanks in advance,

Gustavo

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closed as unclear what you're asking by user44191, Sean Lawton, Stefan Waldmann, Jan-Christoph Schlage-Puchta, David Handelman Mar 16 at 23:53

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Improve your $\LaTeX$ formulas. $\endgroup$ – user64494 Mar 13 at 19:00
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    $\begingroup$ Weierstrass products $\endgroup$ – reuns Mar 13 at 19:08
  • $\begingroup$ According to Maple, you left out a factor $a^2 - x^2$ in the denominator. $\endgroup$ – Robert Israel Mar 13 at 23:11
  • $\begingroup$ In general there is no method to give closed formulae for infinite sums or products. The reason is simply that most expressions involving limits do not have any easier expression. So in a way every closed formula is a lucky accident. $\endgroup$ – Jan-Christoph Schlage-Puchta Mar 15 at 18:49
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The Weierstrass product identity $$ \frac{1}{\Gamma(z)} = e^{\gamma z} z \prod_{n=1}^\infty \left(1 + \frac{z}{n}\right) e^{-z/n} $$ implies $$ \frac{\Gamma(s)^2}{\Gamma(s-z) \Gamma(s+z)} = \frac{s^2-z^2}{s^2} \prod_{n=1}^\infty \left(1 - \frac{z^2}{(n+s)^2}\right)$$ (valid wherever you don't run into a division by $0$ or a pole of $\Gamma$). You're essentially looking at the case $s = a/\pi$, $z = x/\pi$.

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