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As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities, such as[3]
Closed form expressions, similar to Binet's formula are:[3]
where
are the solutions (in t) of
For Lucas Polynomials n > 0, we have
A relationship between the Fibonacci polynomials and the standard basis polynomials is given by[5]
For example,
Combinatorial interpretation
If F(n,k) is the coefficient of xk in Fn(x), namely
then F(n,k) is the number of ways an n−1 by 1 rectangle can be tiled with 2 by 1 dominoes and 1 by 1 squares so that exactly k squares are used.[1] Equivalently, F(n,k) is the number of ways of writing n−1 as an ordered sum involving only 1 and 2, so that 1 is used exactly k times. For example F(6,3)=4 and 5 can be written in 4 ways, 1+1+1+2, 1+1+2+1, 1+2+1+1, 2+1+1+1, as a sum involving only 1 and 2 with 1 used 3 times. By counting the number of times 1 and 2 are both used in such a sum, it is evident that
This gives a way of reading the coefficients from Pascal's triangle as shown on the right.
Hoggatt, V. E.; Long, Calvin T. (1974). "Divisibility properties of generalized Fibonacci Polynomials". Fibonacci Quarterly. 12: 113. MR0352034.
Ricci, Paolo Emilio (1995). "Generalized Lucas polynomials and Fibonacci polynomials". Rivista di Matematica della Università di Parma. V. Ser. 4: 137–146. MR1395332.