The Fox H-function

FoxH[{{{1/3, 2/3}}, {{1/2, 1/2}}}, {{{0, 1}}, {{1/2, 1/2}}}, x]

has the same numerical value, 0.431029, at an arbitrary x value.

This specific Fox H-function appears in the $\alpha$-stable probability density function $S(x;\alpha=3/2,\beta=0,\mu=0,\sigma=1)$ when it is written in terms of a Fox H-function. For other applications, I need the mentioned $\alpha$-stable probability density function in terms of a Fox H-function and hence the mentioned specific Fox H-function.

More generaly, for an $0<\alpha<2$ value, i.e., in the case of $S(x;\alpha,\beta=0,\mu=0,\sigma=1)$ the relevant Fox H-function is

FoxH[{{{1-1/α, 1/α}}, {{1/2, 1/2}}}, {{{0, 1}}, {{1/2, 1/2}}}, x]

For $0<\alpha<1$ and $\alpha\neq1/2$, it works fine but when $1<\alpha<2$, the above Fox H-function has the same numerical value at an arbitrary x value.

Is this a bug in Wolfram Mathematica or is there a mathematical reason behind it?



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