Just found out that it does not work for certain values of b because the gamma function is not defined for negative integers... Hmmm
I have this expression:
And I would very much like to express it in an alternative way without the product function (or a sum function for that matter).
Wolframalpha suggest this as an alternative form:
I have tried number examples, and they come up with the same results, so I guess they
Assuming it is correct, I really need a step-by-step derivation of why the two expressions are identical. As a note I can add that I have the following restrictions on the parameters:
I am aware that the gamma function works as an intrapolation of the faculty function, but not much more...
Here you can see it in wolframalpha: log[ product (1+b*(T-s)), s=0 to t] - Wolfram|Alpha
I hope someone has the time and skill to help me!
By the way, I am using the expression for an estimation. As I without problems can restrict b to 0.5<b<1 it is no problem that the gamme function is not defined for negative integers (and 0). Help as to how wolfram comes up with the beautiful expression is therefore still very much appreciated!