03/07/2021, 11:26 PM
(05/19/2011, 05:49 AM)Gottfried Wrote: I considered the iteration of the multiplication in the sense of the interpolation of the factorial. I adapted the idea of a truncation of a matrixoperator in the same way as Andy has described it with his slog-solution. The occuring powerseries are "automatically" that of the gamma/lngamma-function - it seems, that method provides a proper "uniqueness-criterion" implicitely (that's then also a backup for the possible justifications of Andy's method).
When I stuck at a certain problem I discussed this in the math.SE-forum, see the link here: math.stackexchange.com
I'll also put my results in my text on Uncompleting the Gamma (PDF) which I'll upload here if I'm finished as update of the provisorical chapter 4.
Gottfried
The incomplete gamma function is one of the most interesting ones !!
I had the idea of removing the singularities of the sexp in a similar way as you removed the poles and zero's of the gamma.
Not sure if that will work. What do you think ??
If G(z) is the uncomplete ( or incomplete ) gamma function.
G(z) =/= 0 implies that there is an entire logG(z) analogue to loggamma !
Also logG(exp(z)) must be interesting.
Notice that all derivatives of G(z) has positive derivatives.
Im referring to fake function theory again yes.
I wonder how close G(z) , fakegamma(z) and fakeG(z) are.
And how many zero's the fake will have. Only finite zero's ??
Forgive me, maybe I said all that before.
the pxp analogue of G(z) is also interesting.
continu sums and products relate too.
A kind of superfactorial is also worth considering.
For the superfunction of G(z) we could perhaps try using the recent auxilliary function strategy :
F(s) = G( F(s-1) - exp(-s) )
So clearly it connects to many things here at the tetration forum.
regards
tommy1729