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# Lucas polynomials

The sequence of Lucas polynomials (or Cardan polynomials) is a sequence of polynomials which can be considered as a generalization of the Lucas numbers.

## Lucas polynomials

The sequence of Lucas polynomials
 Ln (x)
[1] is a sequence of polynomials defined by the recurrence relation
${\displaystyle L_{n}(x)={\begin{cases}2\,x^{0}=2,&{\mbox{if }}n=0,\\1\,x^{1}=x,&{\mbox{if }}n=1,\\L_{n-1}(x)\;x^{1}+L_{n-2}(x)\;x^{0},&{\mbox{if }}n\geq 2.\end{cases}}\,}$

## Formulae

${\displaystyle L_{n}(x)=\sum _{k=0}^{\lfloor {\frac {n}{2}}\rfloor }T_{(1,2)}{\bigg (}{\bigg \lceil }{\frac {n}{2}}{\bigg \rceil }+k,{\bigg \lfloor }{\frac {n}{2}}{\bigg \rfloor }-k{\bigg )}~x^{2k+n{\bmod {2}}}=\sum _{k=0}^{\lfloor {\frac {n}{2}}\rfloor }{\bigg \{}2{\binom {{\big \lceil }{\frac {n}{2}}{\big \rceil }+k-1}{{\big \lfloor }{\frac {n}{2}}{\big \rfloor }-k-1}}+{\binom {{\big \lceil }{\frac {n}{2}}{\big \rceil }+k-1}{{\big \lfloor }{\frac {n}{2}}{\big \rfloor }-k}}{\bigg \}}~x^{2k+n{\bmod {2}}}\,}$
where
 T(1, 2) (n, k) = 2 (  n  −  1 k  −  1  ) + (  n  −  1 k  )
is the term from column
 k
of row
 n
of the (1, 2)-Pascal triangle (Lucas triangle,) and
 (  n k  )
is a binomial coefficient.

## Lucas polynomials triangle

Lucas polynomials
 n
 Ln (x) =
 a0  x 0
 + a1  x 1
 + a2  x 2
 + a3  x 3
 + a4  x 4
 + a5  x 5
 + a6  x 6
 + a7  x 7
 + a8  x 8
 + a9  x 9
 + a10  x 10
 + a11  x 11
 + a12  x 12
 0
 L0 (x) =
 + 2

 1
 L1 (x) =

 + x

 2
 L2 (x) =
 + 2

 + x 2

 3
 L3 (x) =

 + 3 x

 + x 3

 4
 L4 (x) =
 + 2

 + 4 x 2

 + x 4

 5
 L5 (x) =

 + 5 x

 + 5 x 3

 + x 5

 6
 L6 (x) =
 + 2

 + 9 x 2

 + 6 x 4

 + x 6

 7
 L7 (x) =

 + 7 x

 + 14 x 3

 + 7 x 5

 + x 7

 8
 L8 (x) =
 + 2

 + 16 x 2

 + 20 x 4

 + 8 x 6

 + x 8

 9
 L9 (x) =

 + 9 x

 + 30 x 3

 + 27 x 5

 + 9 x 7

 + x 9

 10
 L10 (x) =
 + 2

 + 25 x 2

 + 50 x 4

 + 35 x 6

 + 10 x 8

 + x 10

 11
 L11 (x) =

 + 11 x

 + 55 x 3

 + 77 x 5

 + 44 x 7

 + 11 x 9

 + x 11

 12
 L12 (x) =
 + 2

 + 36 x 2

 + 105 x 4

 + 112 x 6

 + 54 x 8

 + 12 x 10

 + x 12

If you look at the Lucas polynomials triangle, you will see that the rising diagonals corresponding to even
 n
are the "(1, 2)-Pascal polynomials". And the column of degree
 1
have odd numbers as coefficients, the column of degree
 2
have square numbers as coefficients, the column of degree
 3
have square pyramidal numbers as coefficients, and so on... (Cf. rows of (1, 2)-Pascal triangle, i.e. Lucas triangle.)

## Lucas numbers

The Lucas numbers are recovered by evaluating the Lucas polynomials at
 x = 1
, i.e.
${\displaystyle L_{n}=L_{n}(1).\,}$
The degree of
 Ln
is
 n
.

## Generating function

The ordinary generating function of the Lucas polynomials is

${\displaystyle G_{\{L_{n}(x)\}}(t)=\sum _{n=0}^{\infty }L_{n}(x)~t^{n}={\frac {2-xt}{1-t(x+t)}}.\,}$

## Notes

1. Weisstein, Eric W., Lucas Polynomial, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/LucasPolynomial.html]