Euler product about sine function is $\frac{\sin(x)}{x} = \prod_{n=1}^\infty \left ( 1- \left(\frac{x}{n\pi}\right)^2 \right)$
I wonder if there is known results about slight modification of above product.
Does there exists analytic expression about following infinite product?
$\prod_{n=1}^\infty \left( 1- \left( \frac{x}{n\pi + a}\right)^2\right)$
I can't find out what it is, even though it is a slight modification.
The generalizations of Euler's infinite product formula for the sine are applications of the Weierstrass factorization theorem: $$\prod_{n=1}^\infty \left( 1- \left( \frac{x}{n\pi + a}\right)^2\right)=\frac{\Gamma \left(\frac{a+\pi }{\pi }\right)^2}{\Gamma \left(\frac{a-x+\pi }{\pi }\right) \Gamma \left(\frac{a+x+\pi }{\pi }\right)}.$$ Check that this expression tends to $x^{-1}\sin x$ when $a\rightarrow 0$.