I'm quite new to the world of special functions.
Recently for research purposes, I've arrived to the following equation:
$$\Gamma(x) = 20(5x+1),$$
with $x \in \mathbb{R}$.
I've tried to exploit the fact that $\Gamma(x) = (x-1)!$ for integer $x$, and I found that $x=7$ is a solution.
Anyway, I've arrived to this solution just by plugging iteratively integer numbers into the equation. I'm wondering if there is a calculus approach which can be used to find also non-integer solutions, if any, for problem in the form
$$\Gamma(x) = p(x),$$
where $p(x)$ is a polynomial.
For the proposed problem, I've found numerically that also $x \simeq -0.24$ is a solution