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+--- 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.
Anyway
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 Désired. Or Taylor series etc.
Regards
Tommy1729