Le jeu. 14 janv. 2021 à 01:45, Hilarie Orman <ho@alum.mit.edu> a écrit :
I am interested in some general ways of defining sequences s[i].
1. s[n+1] = f(s[n])*s[n-1] + f(s[n-1])*s[n] If f is the constant function "1", then this is the Fibonacci sequence.
This looks nontrivial and interesting. I don't know whether it has been studied, but I suppose so. However, you must certainly make more assumptions on f in order to be able to get interesting results beyond some very basic facts for special (e.g., 2-periodic) cases.
2. s[n+1] = g(s[n]){s[n], s[n-1], ..., s[0]} Here, g is a function that has a number as an input and produces a multivariate function that is applied to previous sequence values.
I think by the set you rather mean the family (s(n),...,s(0)) in particular, that there are n+1 values, even if they are all the same, and the order should be of importance ? (or not ?) But how can g() know, when it is given a single number s[n], how many arguments it should take ? I guess I understand that you want to generalize the idea of having for example s(n+1) = sum_{ k=0..n } g(k) s(k) or then also s(n+1) = sum_{ k=0..n } g(k) s(k)^alpha or then also s(n+1) = sum_{ k=0..n } g(k)( s(k) ) where g(k) is any function R -> R or then also s(n+1) = g(n)( s(0..n) ) where g(n) is any function R^(n+1) -> R . Yes, this could work. But certainly one would have to study very restricted classes of functions g in order to be able to say anything. 3. s[n+1] = g(s[n]){t[n], t[n-1], ... t[0]} where t is a sequence defined on
i=0...inf. As before, g is a multivariate function derived from the number s[n].
same as above : how does the number s[n] tell g how many arguments it must take? and again this is just an example of the earlier proposal where t = (t(n)) induces a sequence of functions g[t] = (g[t](n)). - Maximilian