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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)
[1] is a sequence of polynomials defined by the recurrence relation

Formulae

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.
The degree of
Ln
is
n
.

Generating function

The ordinary generating function of the Lucas polynomials is

See also



Notes

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