Hi everyone

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:

0<b<1, T>t>=0

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!