The upper superexponential

[sheldonison]

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Instead we normalize it by \( \operatorname{usexp}(0)=a+1 \), which gives the formula:
\( \operatorname{usexp}_b(t)=a+\chi^{-1}\left(\ln(a)^x \chi(1)\right) \)

The interesting difference to the normal regular superexponential is that upper on is entire, while the normal one has a singularity at -2 and is no more real for \( x<-2 \).
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Does this upper super expoonential equation also hold for b=\( e^{1/e} \)?
Is this "chi" the same as the "Chi distribution" used in probability? Any links to a definition for
\( \chi \) and \( \chi^{-1} \)