I'm working with a series of powers whose general term contains this definite integral
$$F[a-n,b]:=\int_{-\pi}^{\pi}\frac{e^{-(a-n)(c+i\,u)}}{(c+i\,u)^{b+1}}du$$
with $a,c>0$, $b\in\mathbb{R}$, $i$ is the unit imaginary number and $n=1,2,3,...$.
To this end, and integratin by parts, one can arrive to the following recurrence relation ($t$ is the power base)
$$F[a-n,b]=\frac{A^n}{t}\,G(a,b)+\frac{a+1}{t}F[a+1-n,b]+\frac{b+1}{t}F[a+1-n,b+1]$$ where $A=(-e^c)^n$ and $G(a,b)=2\sin(a\pi+(b+1)\arctan(\pi/c))$ .
Note that $G(a+n,b)=(-1)^n\,G(a,b).$
Problem: My main aim is express $F[a-n, *]$ in terms of $F[a, *]$, that is, convert the first argument (that eventually may be negative) in positive. It seems clear that one must apply this recurrence relation $n$ times but I'm not able to find a closed formula.
Any help will be welcomed.
Since your recurrence relation is “linear” I would approach this problem using generating functions. I recommend Generatingfunctionology by Herbert Wilf.
Of course first solve the associated homogeneous recurrence relation before adding the $A$ business.