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 F " (x) = F(x) F(x-1) F(x-2) ... and the alike. tommy1729 Ultimate Fellow     Posts: 1,358 Threads: 330 Joined: Feb 2009 09/19/2017, 09:06 PM 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 « Next Oldest | Next Newest »

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