[math-fun] Sequence frameworks
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. 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. g could be, for example, a linear function with k terms using coefficients derived from s[n]. 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]. Is there some standard terminology for these methods that I could use to find prior work? The keywords just cause Google to latch onto well-known recursive sequences. Thanks, Hilarie
You might take a look at the book by A Gill, Linear Sequential Circuits, McGraw Hill, 1965. The subject called Systems Theory, very popular in the 1960s, studies problems like those There are many books (Zadeh & Desoer is a classic) The titles usually say Linear Systems Theory, but that covers a lot of ground. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Thu, Jan 14, 2021 at 12:45 AM Hilarie Orman <ho@alum.mit.edu> wrote:
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.
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. g could be, for example, a linear function with k terms using coefficients derived from s[n].
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].
Is there some standard terminology for these methods that I could use to find prior work? The keywords just cause Google to latch onto well-known recursive sequences.
Thanks,
Hilarie
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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
participants (3)
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Hilarie Orman -
M F Hasler -
Neil Sloane