What are all the pairs of parameters $\alpha, \beta$ for which the symmetric stable distributions have a PDF that can be expressed in terms of analytic functions? For example, if $\alpha =2$ we get a Gaussian distribution, for $\alpha =1, \beta =0$ we get a Cauchy distribution, and for $\alpha =1/2, \beta=1$ we get a Levy distribution.
(1) Are these the only choices of parameters that give a PDF that's expressed in terms of analytic functions? If not, what are the others?
(2) For non-analytic PDFs, is it still possible to compute derivatives? I ask because I am interested in constructing differential equations from densities.
All of the one-dimensional stable distributions have smooth pdfs (even analytic in the symmetric case, if memory serves) but in very few cases (perhaps just those you have listed) does the pdf have a simpe closed form expression. The book One-dimensional stable distributions by V. Zolotarev contains a wealth of information about stable densities, including series expansions and asymptotics.