Express the PDF of StableDistribution in terms of FoxH for the case of alpha values grater than 1

Question

The PDF of StableDistribution in terms of FoxH for the case $\alpha\leq 1$ is available at https://reference.wolfram.com/language/ref/FoxH.html namely

StableDistributionPDF[x_, α_, β_, μ_, σ_] := 
 With[{Α = σ (1 + β^2 Tan[(π α)/2]^2)^(1/(2 α)), 
    Β = 2/(π α) Sign[x - μ] ArcTan[β Tan[(π α)/2]]}, 
  1/(α Α)*FoxH[{{{1 - 1/α, 1/α}}, {{(1 - Β)/2, (1 + Β)/
        2}}}, {{{0, 1}}, {{(1 - Β)/2, (1 + Β)/2}}}, 
    Abs[x - μ]/Α]]

How does this vary for the $\alpha> 1$ case?

I checked it numerically that the above formula really gives $S(x;\alpha,\beta,\mu,\sigma)$ for the case $0<\alpha<1$. For the case $1<\alpha<2$, however, the above formula gives completely wrong results.

I found two sources:

[1] Ralf Metzler, Joseph Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, Volume 339, Issue 1, December 2000, Pages 1-77 https://doi.org/10.1016/S0370-1573(00)00070-3 (relevant part: formulas (C.17) and (C.18) in Appendix C)

[2] A.M. Mathai, Ram Kishore Saxena, and Hans J. Haubold, The H-Function: Theory and Applications (Springer; 2010th edition) (relevant part: formulas (6.168) and (6.169))

The relevant formulas in the above sources seem not to be sufficient to get $S(x;\alpha,\beta,\mu,\sigma)$ in terms of a Fox H-function for the case $1<\alpha<2$.

Could you please help me to find $S(x;\alpha,\beta,\mu,\sigma)$ in terms of a Fox H function for the case $1<\alpha<2$?

Answer 1

Based on

P.N. Rathie, L.C.de S.M. Ozelim, C.E.G. Otiniano, Exact distribution of the product and the quotient of two stable Lévy random variables, Communications in Nonlinear Science and Numerical Simulation, Volume 36, 2016, Pages 204-218, ISSN 1007-5704, https://doi.org/10.1016/j.cnsns.2015.11.012.

the $\alpha$-stable probability density function $S(x;\alpha,\beta,\mu,\sigma)$ in terms of a Fox H-function for the case $1<\alpha<2$ is the following:

$S(x;\alpha,\beta,\mu,\sigma)=\frac{1}{\alpha A}H^{1,1}_{2,2} \left[ \frac{|x-\mu|}{A} \Bigg| \begin{matrix} (1-\frac {1}{\alpha},\frac {1}{\alpha}), & (\frac{1-B(\alpha-2)/\alpha}{2},\frac{1+B(\alpha-2)/\alpha}{2}) \\ (0,1), & (\frac{1-B(\alpha-2)/\alpha}{2},\frac{1+B(\alpha-2)/\alpha}{2}) \end{matrix}\right]$

where

$A=\sigma(1+\beta^2\tan^2(\pi\alpha/2))^\frac{1}{2\alpha}$

and

$B=\frac{2}{\pi(\alpha-2)}{\rm sign}(x-\mu)\arctan(\beta\tan(\pi(\alpha-2)/2)).$

The Wolfram Mathematica code for merging the $0<\alpha<1$ and $1<\alpha<2$ cases is:

Its test, e.g. :

with output