Consider the linear equation system:
$$\bf Mx = b$$
$${\bf M} = \left[\begin{array}{cccccccc} 1&0&0&0&0&0&0&0\\ -2&1&0&0&0&0&0&0\\ 0&-3&1&0&0&0&0&0\\ 0&0&-4&1&0&0&0&0\\ 0&0&0&-5&1&0&0&0\\ 0&0&0&0&-6&1&0&0\\ 0&0&0&0&0&-7&1&0\\ 0&0&0&0&0&0&-8&1 \end{array}\right]\hspace{0.5cm} {\bf b} = \left[\begin{array}{c} 1\\0\\0\\0\\0\\0\\0\\0\end{array}\right]$$
It is easy to see that these are equations corresponding to the recursive definition of the factorial function: $$n! = 1\cdot 2\cdots n, \forall n \in \{1,\cdots,8\}\\\text{Expressed recursively: }\\n! = n\cdot (n-1)! \Leftrightarrow\\ - n \cdot (n-1)!+n!=0$$
Now to the question: Can we use this fact to try and find interpolations / generalizations to arguments $\in \mathbb R$ to the factorial function the by some linear algebraic numerical method. If it is possible to do so, which particular one will yield the famous $\Gamma$ function?
$$t\to \Gamma(t), \forall t\in \mathbb R:\\\Gamma(t) = \int_0^\infty \tau^{t-1}e^{-\tau}d\tau$$Having the following connection to the factorial function :$$\Gamma(n) = (n-1)!\hspace{0.5cm} \forall n\in\mathbb N$$