another infinite product problem

Question

Is there a way to simplify the following $$ 1-\prod_{k=0}^\infty\frac{(k+a)^2-1}{(k+a)^2+b^2} $$ to a single infinite product? Assumptions: $a>0$, $b>0$.

Can the following idea prove useful? $$ \prod_{k=0}^\infty\frac{(k+a)^2-1}{(k+a)^2+b^2}=\prod_{k=0}^\infty\left(1-\frac{b^2+1}{(k+a)^2+b^2}\right). $$

Answer 1

• In general, expressible in terms of the $\Gamma$ function.
• For $a\in$ N, expressible in terms of the $($hyperbolic or trigonometric$)$ sine function$($s$)$.
• For $a=n+\dfrac12$, with $n\in$ N, expressible in terms of the $($hyperbolic or trigonometric$)$ cosine function$($s$)$.