A related discussion on interpolation: factorial and gamma-function

[Gottfried]

However as I now see the Hadamard and Luschny definitions hava a major drawback, they dont satisfy
\( (x+1)!=(x+1)x! \)
they merely interpolate n!. So there is no (mentioned) alternative definition of the gamma function that satisfies the above equation.

For tetration we too demand that
\( b[4](x+1)=b^{b[4]x} \)
and that it is not just an interpolation for b[4]n.
However here there seem to be several competing definitions.

upps? I thought(and was sure by reading the article) that for the positive integers there is identity with the "x! = x*(x-1)!" ? Did I miss something?
--- Ah , well, you mean for fractional x? Well - I'll consult the article again.

Thanks for the hint...

Gottfried