I was thinking about the Gamma function, which for an integer positive argument is nothing but the factorial function.

Using the integral representation, namely

$$\Gamma[x] = \int_0^{+\infty}\ t^{x-1}e^{-t}\ \text{d} t$$

one can extend the "factorial" also to the whole set or Real numbers, except for the negative integers which are all poles.

Observing the plot of the function, and precisely with respect to the negative $X$ axis, I noticed that there are different values of $x$ in which $\Gamma[x]$ returns the same value:

enter image description here

In my example, there are different values of $x$ for which $\Gamma[x] = 7$.

Now the main question: assuming there are two values, $x$ and $x + h$ (for $h \neq 0$) such that

$$\Gamma[x] = \Gamma[x + h]$$

how could I solve such an equation?

And more generally: how could I find all the different values $x_{\alpha}$ for which $\Gamma[x_{\alpha}] = n$ for a fixed real $n$ ($n\neq \{-1; -2; -3; \ldots \}$)


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