Encountered this function in one of my research problems
$$\frac{\Gamma \left(1-\dfrac{i c}{a}-\gamma \right) \Gamma \left(1+\dfrac{i c}{a}+\dfrac N 2-\gamma \right)}{\Gamma \left(1+\dfrac{i c}{a}-\gamma \right) \Gamma \left(1-\dfrac{i c}{a}+\dfrac N 2-\gamma \right)}-\frac{\left(a \gamma +b-\dfrac{i c}{2}\right) \left(\dfrac{a \gamma +b-\dfrac{i c}{2}-1}{a \gamma +b+\dfrac{i c}{2}-1}\right)^N}{a \gamma +b+\dfrac{i c}{2}}=0\quad\text{where } \{a,b,c\}\in \mathbb{C} \text{ and } N\in \mathbb{Z^+}$$
I am wondering if it is possible to solve for $\gamma$ analytically.
I solved it numerically for various parameters and the answers are interesting ($\gamma^\prime s$ lie on some sort of nice curve). For example for $$N\to 20,a\to 1.53\, -0.02 i,b\to 0.098\, -0.3 i,c\to -0.3+2 i$$
solutions look like
and for another set of random parameters
$$N\to 30,a\to 1.5\, +2 i,b\to -0.5+0.3 i,c\to -3-2 i$$
Solution looks like
Playing with different parameters, I always got most of the solutions on a seemingly nice curve. So, I am wondering if there is a way to find the solution analytically.
