Another question!

[tommy1729]

Integrate by parts, and for we get the spectacular identity that:

I do not even need to use integrate by parts to see a problem.
Its funny you say
because its more like an equality when we differentiate a given amount of times with respect to t.

You see : s is considered a constant with respect to t since s is not a function OF t NOR f.
There is big difference between a function , an operator , a variable and a constant.
ALthough that may sound belittling or trivial , your example shows this is an important concept !!

If you consider as a function F(s,f) then it is no surprise that taking the derivative with respect to f leaves s unchanged.

By the chain rule you then get the " wrong " / " correct "
if you take the derivative times.

This is similar to .

Hence by the very definition of the gamma function you also get
here which you already showed yourself with the - overkill - method integrate by parts.

This might not answer all your questions yet but I assume it helps.

It not completely formal either sorry.

It might affect your other posts about integral representations for fractional calculus , tetration and continuum sum.

Im still optimistic though and hope I did not discourage you to much.

regards

tommy1729