Tetration Forum
F " (x) = F(x) F(x-1) F(x-2) ... and the alike. - Printable Version

+- Tetration Forum (https://math.eretrandre.org/tetrationforum)
+-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)
+--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)
+--- Thread: F " (x) = F(x) F(x-1) F(x-2) ... and the alike. (/showthread.php?tid=1180)

F " (x) = F(x) F(x-1) F(x-2) ... and the alike. - tommy1729 - 09/19/2017

I want reintroduce a kind of equations again.

I used to call them " chaos equations " originally , but as a kid I was unaware of mathematical chaos as a formal thing in math.

I mention them from time to time , here , on sci.math or elsewhere.


Consider C^oo functions that are strictly nondecreasing for x > 1.

The functions satisfy

F^(n)(x) = F(x-a) F(x-a-1) F(x-a-2) ...

Where ^(n) means differentiate n times.

n is a positive real ( in particular I consider integer mainly ) and a is a real number.

In particular the real-analytic solutions are intresting.

## I assume F needs to be analytic to be very intresting ##

I call the solutions Chaos(n,a,x).

Chaos(0,1,x) can be expressed by the gamma function.

Chaos(1,0,x) can be expressed by sexp(x).

But What is a solution to chaos(2,0,x) ?

F " (x) = F(x) F(x-1) F(x-2) ... ??

Closed forms are unlikely , even with sexp , slog , LambertW etc. But asymptotics and understanding is D├ęsired. Or Taylor series etc.