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. 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 |