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
The coefficients of the Fibonacci polynomials can be read off from a left-justified Pascal's triangle following the diagonals (shown in red). The sums of the coefficients are the Fibonacci numbers.
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.
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
The coefficients of the Fibonacci polynomials can be read off from a left-justified Pascal's triangle following the diagonals (shown in red). The sums of the coefficients are the Fibonacci numbers.
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.