
1564 SPECIAL FUNCTIONS AND THEIR PROPERTIES
Recurrence formulas:
C
(λ)
n+1
(x) =
2(n + λ)
n + 1
xC
(λ)
n
(x) −
n + 2λ −1
n + 1
C
(λ)
n−1
(x);
C
(λ)
n
(−x) = (−1)
n
C
(λ)
n
(x),
d
dx
C
(λ)
n
(x) = 2λC
(λ+1)
n−1
(x).
The generating function:
1
(1 − 2xs + s
2
)
λ
=
∞
X
n=0
C
(λ)
n
(x)s
n
.
The Gegenbauer polynomials are orthogonal on the interval −1 ≤ x ≤ 1 with weight
(1 − x
2
)
λ−1/2
:
Z
1
−1
(1 − x
2
)
λ−1/2
C
(λ)
n
(x)C
(λ)
m
(x) dx =
0 if n 6= m,
πΓ(2λ + n)
2
2λ−1
(λ + n)n! Γ
2
(λ)
if n = m.
30.18 Nonorthogonal Polynomials
30.18.1 Bernoulli Polynomials
◮ Definition. Basic properties.
The Bernoulli polynomials B
n
(x) are introduced by the formula
B
n
(x) =
n
X
k=0
C
k
n
B
k
x
n−k
(n = 0, 1, 2, . . . ),
where C
k
n
are the binomial coefficients ...