What is the solution of the fractional differential equation $$ f^{(\alpha-1)}(t) = tf(t) $$

where $(\alpha)$ denotes the fractional derivative of order $\alpha$

EDIT: Background behind this question.

I am interested in this equation in relation with the alpha-stable version of the Stein's lemma. Recall, that if $X \in N(0,1)$ then $$ E(X g(X)) = E(g'(X)) $$ for every function $g$ for which the expectations in the left and right parts exist.

The simplest way to prove this is to use the property of normal density $f'(x) = -x f(x)$ and integration by parts.

Let $\phi$ be a characteristic function of the standard symmetric stable density $S(\alpha, 1, 0)$. If I apply the usual derivative to $\phi$ and take a Fourier transform I will get the equation above (modulo signs and coefficients). A closed form solution would give an analytic presentation of the stable density (highly unlikely).

2nd question: Is it possible to obtain a differential equation for the stable density to be used in the proof of the Stein's lemma?

Thanks, Vladimir