Lucas polynomials

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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)

is a sequence of polynomials defined by the recurrence relation

$L_{n}(x)={\begin{cases}2\,x^{0}=2,&{\mbox{if }}n=0,\\1\,x^{1}=x,&{\mbox{if }}n=1,\\L_{n-1}(x)\;x^{1}+L_{n-2}(x)\;x^{0},&{\mbox{if }}n\geq 2.\end{cases}}\,$

Formulae

$L_{n}(x)=\sum _{k=0}^{\lfloor {\frac {n}{2}}\rfloor }T_{(1,2)}{\bigg (}{\bigg \lceil }{\frac {n}{2}}{\bigg \rceil }+k,{\bigg \lfloor }{\frac {n}{2}}{\bigg \rfloor }-k{\bigg )}~x^{2k+n{\bmod {2}}}=\sum _{k=0}^{\lfloor {\frac {n}{2}}\rfloor }{\bigg \{}2{\binom {{\big \lceil }{\frac {n}{2}}{\big \rceil }+k-1}{{\big \lfloor }{\frac {n}{2}}{\big \rfloor }-k-1}}+{\binom {{\big \lceil }{\frac {n}{2}}{\big \rceil }+k-1}{{\big \lfloor }{\frac {n}{2}}{\big \rfloor }-k}}{\bigg \}}~x^{2k+n{\bmod {2}}}\,$

where 

T(1, 2) (n, k) = 2 (  n  −  1k  −  1  )  + (  n  −  1k  )

is the term from column 

k

of row 

n

of the (1, 2)-Pascal triangle (Lucas triangle,) and 

(  nk  )

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.

$L_{n}=L_{n}(1).\,$

The degree of 

Ln

is 

n

.

Generating function

The ordinary generating function of the Lucas polynomials is

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

See also

• (1, 2)-Pascal triangle (Lucas triangle)

• (1, 1)-Pascal polynomials (compared to Fibonacci polynomials)
• (1, 2)-Pascal polynomials (compared to Lucas polynomials)

• Polynomial sequences (number sequences described by a polynomial formula, which are not to be confused with sequences of polynomials)