# Indefinite integral of $\frac{(z+2 n-3)\text{!!}}{(z-2 n+1)\text{!!}}$

Any chance i can find a definite integral for this complex function?

$$$$f_n(z)=\frac{2^{-\frac{1}{2} ( \sin (\pi z) ) ( \sin (2 \pi n))+2 n-2} \pi ^{\frac{1}{2} ( \sin (\pi z) ) ( \sin (2 \pi n) )} \Gamma \left(\frac{z-1}{2}+n\right)}{\Gamma \left(\frac{1}{2} (z-2 n+3)\right)} \quad n\in\mathbb{Z}$$$$

More specifically, this is the analytical continuation of $$\frac{(z+2 n-3)\text{!!}}{(z-2 n+1)\text{!!}}$$ to $$z\in\mathbb{C}$$ which i need to integrate and then find the value of it at $$z\in\mathbb{Z}$$.

Any sugestions are welcome :D

• For a given value of $n>0$, the integrand is just a polynomial. But which one ? Jul 4 at 3:19
• I am quite curious about the background of this problem. Jul 4 at 6:02