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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 [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
of row
of the
(1, 2)-Pascal triangle (
Lucas triangle,) and
is a
binomial coefficient.
Lucas polynomials triangle
If you look at the
Lucas polynomials triangle, you will see that the rising diagonals corresponding to even
are the "
(1, 2)-Pascal polynomials". And the column of degree
have
odd numbers as coefficients, the column of degree
have
square numbers as coefficients, the column of degree
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
, i.e.
The degree of
is
.
Generating function
The ordinary generating function of the Lucas polynomials is
See also
Notes