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Fibonacci polynomials

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In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials.


These Fibonacci polynomials are defined by a recurrence relation:[1]

The Lucas polynomials use the same recurrence with different starting values:[2]

They can be defined for negative indices by[3]

The Fibonacci polynomials form a sequence of orthogonal polynomials with and .


The first few Fibonacci polynomials are:

The first few Lucas polynomials are:


  • The degree of Fn is n − 1 and the degree of Ln is n.
  • The Fibonacci and Lucas numbers are recovered by evaluating the polynomials at x = 1; Pell numbers are recovered by evaluating Fn at x = 2.
  • The ordinary generating functions for the sequences are:[4]
  • The polynomials can be expressed in terms of Lucas sequences as
  • They can also be expressed in terms of Chebyshev polynomials and as
where is the imaginary unit.


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]


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[edit]

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.


  1. ^ a b Benjamin & Quinn p. 141
  2. ^ Benjamin & Quinn p. 142
  3. ^ a b c Springer
  4. ^ Weisstein, Eric W. "Fibonacci Polynomial". MathWorld.
  5. ^ A proof starts from page 5 in Algebra Solutions Packet (no author).

Further reading[edit]

  • Hoggatt, V. E.; Bicknell, Marjorie (1973). "Roots of Fibonacci polynomials". Fibonacci Quarterly. 11: 271–274. ISSN 0015-0517. MR 0332645.
  • Hoggatt, V. E.; Long, Calvin T. (1974). "Divisibility properties of generalized Fibonacci Polynomials". Fibonacci Quarterly. 12: 113. MR 0352034.
  • Ricci, Paolo Emilio (1995). "Generalized Lucas polynomials and Fibonacci polynomials". Rivista di Matematica della Università di Parma. V. Ser. 4: 137–146. MR 1395332.
  • Yuan, Yi; Zhang, Wenpeng (2002). "Some identities involving the Fibonacci Polynomials". Fibonacci Quarterly. 40 (4): 314. MR 1920571.
  • Cigler, Johann (2003). "q-Fibonacci polynomials". Fibonacci Quarterly (41): 31–40. MR 1962279.

External links[edit]