Fibonacci Polynomials - Definition

Definition

These Fibonacci polynomials are defined by a recurrence relation:

F_n(x)= \begin{cases} 0, & \mbox{if } n = 0\\ 1, & \mbox{if } n = 1\\ x F_{n - 1}(x) + F_{n - 2}(x),& \mbox{if } n \geq 2 \end{cases}

The first few Fibonacci polynomials are:

The Lucas polynomials use the same recurrence with different starting values: L_n(x) = \begin{cases} 2, & \mbox{if } n = 0 \\ x, & \mbox{if } n = 1 \\ x L_{n - 1}(x) + L_{n - 2}(x), & \mbox{if } n \geq 2. \end{cases}

The first few Lucas polynomials are:

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 degrees of Fn is n − 1 and the degree of Ln is n. The ordinary generating function for the sequences are:

The polynomials can be expressed in terms of Lucas sequences as

Read more about this topic:  Fibonacci Polynomials

Famous quotes containing the word definition:

    “Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
    —Elaine Heffner (20th century)

    “Was man made stupid to see his own stupidity?
    Is God by definition indifferent, beyond us all?
    Is the eternal truth man’s fighting soul
    Wherein the Beast ravens in its own avidity?”
    —Richard Eberhart (b. 1904)

    “No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
    —Florence Nightingale (1820–1910)