
30.17. Orthogonal Polynomials 1559
Special cases:
L
α
0
(x) = 1, L
α
1
(x) = α + 1 − x, L
−n
n
(x) = (− 1)
n
x
n
n!
.
To calculate L
α
n
(x) for n ≥ 2, one can use the recurrence formulas
L
α
n+1
(x) =
1
n + 1
(2n + α + 1 −x)L
α
n
(x) − (n + α)L
α
n−1
(x)
.
Other recurrence formulas:
L
α
n
(x) = L
α
n−1
(x) + L
α−1
n
(x),
d
dx
L
α
n
(x) = −L
α+1
n−1
(x),
x
d
dx
L
α
n
(x) = nL
α
n
(x) −(n + α)L
α
n−1
(x).
The functions L
α
n
(x) form an orthogonal system on the interval 0 < x < ∞ with weight
x
α
e
−x
:
Z
∞
0
x
α
e
−x
L
α
n
(x)L
α
m
(x) dx =
(
0 if n 6= m,
Γ(α+n+1)
n!
if n = m.
The generating function is
(1 − s)
−α−1
exp
−
sx
1 − s
=
∞
X
n=0
L
α
n
(x)s
n
, |s| < 1.
30.17.2 Chebyshev Polynomials and Functions
◮ Chebyshev polynomials of the first kind . ...