Fibonacci polynomials
Main article: Lucas sequence

From Wikipedia the free encyclopedia

Top View

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.

Definition

[edit]

These Fibonacci polynomials are defined by a recurrence relation:

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

They can be defined for negative indices by

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

Examples

[edit]

The first few Fibonacci polynomials are:

The first few Lucas polynomials are:

Properties

[edit]

  • 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:

  • 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.

Identities

[edit]

As particular cases of Lucas sequences, Fibonacci polynomials satisfy a number of identities, such as

Closed form expressions, similar to Binet's formula are:

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

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 x 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. 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.

References

[edit]
  1. ^ Benjamin & Quinn p. 141
  2. ^ Benjamin & Quinn p. 142
  3. ^ 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.