Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 0 down vote favorite

I google a lot but I'm not able to find any answer. The problem is the following:

For recurrent sequence using the characteristic polynomial it is possible to have a generic formula that gives the $n$-th term in terms of exponential. So for example the Fibonacci sequence $$F_n = F_{n-1} + F_{n-2}; F_0 = 1, F_1 = 1$$ has a Binet formula like

$$F_n = \frac{\phi_n - (-\phi^n)}{\sqrt{5}}$$

Considering a recurrent relation like

$$F_{n,k} = F_{n-1,k} + F_{n-2,k} + \cdots + F_{n-k,k}$$

where all terms from $1$ to $k$ are integers and considering that the characteristic polynomial has a solution, is it possible to have a kind of Binet formula for such a sequence?

Any help would be appreciate

share|improve this question
Yes. It's the same principle but the answer might not be pretty or even expressible in elementary terms. For $k=3$, see wolframalpha.com/input/…. – lhf Jan 9 '12 at 15:06
1  
en.wikipedia.org/wiki/… might help you. – Pedro Jan 9 '12 at 15:09
1  
Have you seen this? The problem of course is that the roots of the characteristic polynomial do not really have tidy expressions... – J. M. Jan 9 '12 at 15:09
Yet another reference: en.wikipedia.org/wiki/… – Peter Taylor Jan 9 '12 at 15:13
What's $F(n,k)$? Did you mean just $F_n$? – Hans Lundmark Jan 9 '12 at 16:03

Know someone who can answer? Share a link to this question via email, Google+, Twitter, or Facebook.

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.