The Gamma function has certain properties that allow it being evaluated for certain complex arguments. Is there any known expression simplifying
$\Gamma(-n + i x) \Gamma(-n - i x)$
for integers $n \ge 0$ and real variables $x$? I checked Abramovitz and Stegun, but didn't find anything. Mathematica can simplify this for $n=0$ and $n=1$, but doesn't simplify for larger $n$.
Thanks in advance!
By the recursive relation $\Gamma(x+1)=x\Gamma(x)$ one has
$$\Gamma(z-n)=\Gamma(z)\prod_{k=1}^n\frac1{z-k}$$
and hence,
$$\Gamma(ix-n)\Gamma(-ix-n)=\Gamma(ix)\Gamma(-ix)\prod_{k=1}^n\frac1{k^2-x^2}$$
but by the reflection formula, $\Gamma(z)\Gamma(-z)=-\frac\pi{z\sin(\pi z)}$, giving us
$$\Gamma(ix-n)\Gamma(-ix-n)=-\frac\pi{ix\sin(\pi ix)}\prod_{k=1}^n\frac1{k^2-x^2}$$
or, without imaginary numbers,
$$\Gamma(ix-n)\Gamma(-ix-n)=\frac\pi{x\sinh(\pi x)}\prod_{k=1}^n\frac1{k^2-x^2}$$
where the product multiplies out into the form of
$$\prod_{k=1}^n(k^2-x^2)=\sum_{k=0}^na_kx^{2k}$$
where
$$a_k=(-1)^k\sum_{1\le i_1<i_2<\cdots<i_{n-k}\le n}(i_1i_2\cdots i_{n-k})^2$$
are given by Vieta's formulas.