Padovan polynomials

In mathematics, Padovan polynomials are a generalization of Padovan sequence numbers. These polynomials are defined by:

P n ( x ) = { 1 , if  n = 1 0 , if  n = 2 x , if  n = 3 x P n − 2 ( x ) + P n − 3 ( x ) , if  n ≥ 4. {\displaystyle P_{n}(x)={\begin{cases}1,&{\mbox{if }}n=1\\0,&{\mbox{if }}n=2\\x,&{\mbox{if }}n=3\\xP_{n-2}(x)+P_{n-3}(x),&{\mbox{if }}n\geq 4.\end{cases}}}

The first few Padovan polynomials are:

P 1 ( x ) = 1 {\displaystyle P_{1}(x)=1\,}
P 2 ( x ) = 0 {\displaystyle P_{2}(x)=0\,}
P 3 ( x ) = x {\displaystyle P_{3}(x)=x\,}
P 4 ( x ) = 1 {\displaystyle P_{4}(x)=1\,}
P 5 ( x ) = x 2 {\displaystyle P_{5}(x)=x^{2}\,}
P 6 ( x ) = 2 x {\displaystyle P_{6}(x)=2x\,}
P 7 ( x ) = x 3 + 1 {\displaystyle P_{7}(x)=x^{3}+1\,}
P 8 ( x ) = 3 x 2 {\displaystyle P_{8}(x)=3x^{2}\,}
P 9 ( x ) = x 4 + 3 x {\displaystyle P_{9}(x)=x^{4}+3x\,}
P 10 ( x ) = 4 x 3 + 1 {\displaystyle P_{10}(x)=4x^{3}+1\,}
P 11 ( x ) = x 5 + 6 x 2 . {\displaystyle P_{11}(x)=x^{5}+6x^{2}.\,}

The Padovan numbers are recovered by evaluating the polynomials Pn−3(x) at x = 1.

Evaluating Pn−3(x) at x = 2 gives the nth Fibonacci number plus (−1)n. (sequence A008346 in the OEIS)

The ordinary generating function for the sequence is

∑ n = 1 ∞ P n ( x ) t n = t 1 − x t 2 − t 3 . {\displaystyle \sum _{n=1}^{\infty }P_{n}(x)t^{n}={\frac {t}{1-xt^{2}-t^{3}}}.}

See also

  • Polynomial sequences


References


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