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This is something I discovered on my own a while back, but was unable to find much info about it online. I strongly believe that it is the closed form of the Fibonacci polynomials. I arrived at it by manipulating Binet's formula with the golden function.
I used the Wikipedia math generator to create an image for a school paper, and figured it would be a dumb idea to let the code to go to waste, so here it is:
Again, I have no sources to verify/mathematically prove this, so I'm just posting it here instead. Enjoy!
the summation on the left hand side for the generating functions are over m, while inside the sum they are over n. I think this is a mistake. I can't think how it could possibly be correct! —Preceding unsigned comment added by 126.96.36.199 (talk) 13:30, 21 August 2008 (UTC)
Formula in Definition
Has anybody noticed that the case for actually agrees with the case for ? I'll modify the formula to reflect this; if anyone has any qualms, feel free to address them here. --JB Adder | Talk 01:11, 23 September 2008 (UTC)
Please write legibly
- The degree of Fn is n-1.
- The degree of Fn is n − 1.
I found the first form above in this article. Notice that:
- A hyphen does not look like a minus sign; and
- Proper spacing should be used.
Relationship between Lucas sequences and Fibonacci/Lucas polynomials.
I think the relationship between Fibonacci/Lucas polynomials and Lucas sequences is more nuanced than is reflected in the article. Lucas sequences are a family of integer sequences with two parameters. The parameters are arbitrary but are assumed constant so each term in the sequence is regarded as an integer, not a function of the parameters. Fibonacci/Lucas polynomials on the other hand are two specific sequences of polynomials, i.e. functions rather than numbers. So, for example, it makes sense to talk about the zeros or the derivatives of Fibonacci polynomials but not about the zeros or the derivatives of Lucas sequences. Similarly you can talk about prime factors of elements of a Lucas sequence (as in Carmichael's theorem), but not about prime factors of Fibonacci/Lucas polynomials. Keeping this in mind, there are probably several places where the article should be rephrased to preserve this distinction.--RDBury (talk) 16:01, 9 December 2011 (UTC)
- I disagree. Lucas sequences may be viewed as sequences of polynomials in P and Q, and thus Fibonacci/Lucas polynomials may be viewed as particular cases of Lucas sequences. In particular, most identities for Fibonacci/Lucas polynomials are obtained from those of Lucas sequences by plugging in P = x and Q = -1 (similarly how identities for Fibonacci/Lucas numbers are obtained by plugging in P = 1, Q = -1 at Lucas sequence#Other relations). We can talk about derivatives of Lucas sequences as polynomials of P and Q, nothing is wrong with it. We can also talk about prime factors of the terms of a Lucas sequence as soon as P and Q are specified integers, nothing is wrong with it either. Maxal (talk) 05:00, 10 December 2011 (UTC)
The polynomials generated in the same way from the Lucas numbers are called Lucas polynomials?
According to the article the Lucas polynomials start with 2 and x, so unless I miss something it seems to me the description "The polynomials generated in the same way from the Lucas numbers are called Lucas polynomials" in the intro is misleading (it had me think they start with 2 and 1). 188.8.131.52 (talk) 14:50, 8 January 2012 (UTC)