Fibonacci polynomials

In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalisation of the Fibonacci numbers.

Definition

These polynomials are defined by a recurrence relation:

:

Properties

The first few Fibonacci polynomials are:

:$F\_1(x)=1\; ,$:$F\_2(x)=x\; ,$:$F\_3(x)=x^2+1\; ,$:$F\_4(x)=x^3+2x\; ,$:$F\_5(x)=x^4+3x^2+1\; ,$:$F\_6(x)=x^5+4x^3+3x\; ,$

The Fibonacci numbers are recovered by evaluating the polynomials at "x" = 1. The degree of "F"_"n" is "n"-1. The ordinary generating function for the sequence is

:$sum\_\{m=0\}^infty\; F\_n(x)\; t^n\; =\; frac\{t\}\{1-xt-t^2\}\; .$

Lucas polynomials

The associated Lucas polynomials "L"_"n"("x") have a similar relationship to the Lucas numbers. They satisfy the same recurrence relationship, with different starting values:

The first few Lucas polynomials are:

:$L\_1(x)=x\; ,$:$L\_2(x)=x^2+2\; ,$:$L\_3(x)=x^3+3x\; ,$:$L\_4(x)=x^4+4x^2+2\; ,$:$L\_5(x)=x^5+5x^3+5x\; ,$:$L\_6(x)=x^6+6x^4+9x^2\; +\; 2\; ,$

The Lucas numbers are recovered by evaluating the polynomials at "x" = 1. The degree of "L"_"n" is "n". The ordinary generating function for the sequence is

:$sum\_\{m=0\}^infty\; L\_n(x)\; t^n\; =\; frac\{2-xt\}\{1-xt-t^2\}\; .$

References

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External links

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