I want to propose a question that may not have a solution
Here I had asked a question: if it was possible to define the integral representation of the polygamma function $\psi^{(\nu)}(x)$ for $\nu>0$ but $\nu\not\in\mathbb{N}$.
(Among the comments it was pointed out to me that the solution doesn't exist yet as Wolfram is also looking for it.)
I thought about it for a bit and came up with this generalization, I calculated the following integral some time ago: $$\int_{0}^{\infty}e^{-x^{s}}\ln(x)^{n}\mathrm{d}x=\frac{1}{s^{n+1}}\Gamma\left(\frac{1}{s}\right)B_{n}\left(\psi^{(0)}\left(\frac{1}{s}\right),...,\psi^{(n-1)}\left(\frac{1}{s}\right)\right)$$
Where $B_n(x_1,...,x_n)$ is the n-th complete Bell polynomial
Modifying it appropriately we get: $$\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}e^{-t}t^{s-1}\ln(t)^{n}dt=B_{n}\left(\psi^{(0)}\left(s\right),...,\psi^{(n-1)}\left(s\right)\right)$$
Now, the idea would be this: to generalize the Bell polynomial in such a way as to make it act not on a sequence, but on a function, i.e. transform $$B_n(x_1,...,x_n)\mapsto B_{\nu}\left(f(\nu)\right)$$ In this case: $$\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}e^{-t}t^{s-1}\ln(t)^{\nu}dt=B_{\nu}\left(\psi^{(\nu-1)}(s)\right)$$
Can anyone think of any possible generalizations for the complete Bell polynomial? (Any idea is welcome, also an alternative resolution with the hypergeometric functions or some generic function)
Thus we would be working on polygamma functions with $\nu$ not integer