Classical Orthogonal Polynomials
General Structure
2
A family {Fn (x)}∞
n=0 of classical orthogonal polynomials in Lw (a, b)
is defined by
1 dn
1
w(x)(s(x))n
(1)
Fn (x) =
Kn w(x) dxn
where Kn is some normalization constant,
1. F1 (x) is a first-degree polynomial in x,
2. s(x) is a polynomial of degree less than or equal to two which has only
real roots,
3. and w(x) is a strictly positive, integrable function in the interval (a, b)
and satisfies w(a)s(a) = w(b)s(b) = 0.
Eq. 1 is called the generalized Rodrigues’ formula. These polynomials satisfy
1. each Fn (x) is a polynomial of degree n,
2. and
Rb
a
Fn (x)xk w(x)dx = 0 for all integers k < n.
A useful lemma:
5. An Fn − λn+1 (αn x + βn )w′ Fn+1 + γn λn−1 (αn x + βn )w′ Fn−1 +
′
′
Bn Fn+1
+ γn Dn Fn−1
= 0, where
• An = (αn x + βn )(2αn λn w + αn (ws)′′ + λn w′ (αn x + βn )) − α2n (ws)′
• Bn = αn (ws)′ + w(αn x + βn )(λn − λn+1 )
• Dn = −αn (ws)′ + w(αn x + βn )(λn − λn+1 )
Generating Function
A family {Fn }∞
n=0 of classical orthogonal polynomials can be generated
using a single function g of two variables.
g(x, t) =
∞
X
an Fn (x)tn
One can use recurrence relations to obtain a differential equation for g(x, t)
and then solve it to obtain g(x, t).
Hermite Polynomials
dm
dm
(wsn )
=
(wsn )
=0
m
dxm
dx
x=a
x=b
2
(2)
w(x) = e−x , s(x) = 1, a = −∞, b = +∞
Generalized Rodrigues Formula
for all integers m < n.
2
Kn = (−1)n , Hn (x) = (−1)n ex
Differential Equation
(m)
The coefficients kn
are defined by
(3)
Constants
(n)
s(x) = σ2 x + σ1 x + σ0 .
(4)
Classical orthogonal polynomials defined by Eq. 1 satisfy the differential
equation
dFn
d
w(x)s(x)
= λn w(x)Fn
(5)
dx
dx
(1)
λn = K1 k1 n + σ2 n(n − 1).
(6)
(n−1)
2. µ = ν = λ − 12 (λ > − 12 ): w(x) = 1 − x2
λ (x)
nomials Cn
2 −1/2
λ−1/2
; Gegenbauer poly-
; Chebyshev polynomials of the
hn =
(Fn (x))2 w(x)dx = (−1)
Kn
wsn dx
(7)
(n+1)
αn =
(n)
kn
(n)
, βn = αn
(n−1)
kn
kn+1
kn
−
(n+1)
(n)
!
, γn = −
hn
αn
. (8)
hn−1 αn−1
The following recurrence relations hold between Fn ’s.
1. Fn+1 = (αn x + βn )Fn + γn Fn−1
2. 2wsαn Fn′ + (αn (ws)′ + λn w(αn x + βn )) Fn − λn+1 wFn+1 +
γn λn−1 wFn−1 = 0
3. 2wsαn Fn′ + (αn (ws)′ + (λn − λn+1 )w(αn x + βn ))Fn + γn (λn−1 −
λn+1 )wFn−1 = 0
4. 2αn λn wFn + ((αn (ws)′ + (λn − λn−1 )w(αn x + βn ))Fn )′ − (λn+1 −
λn−1 )(wFn+1 )′ = 0
2
= −2ne−x Hn
(14)
(15)
Hn+1 = 2xHn − 2nHn−1
(16)
′
Hn
= 2nHn−1
(17)
2
g(x, t) = e2xt−t =
∞
X
1
n!
Hn (x)tn
(18)
n=0
Laguerre Polynomials
w(x) = xν e−x (ν > −1), s(x) = x, a = 0, b = +∞
Generalized Rodrigues Formula
Kn = n!, Lνn (x) =
1 −ν x dn
x e
xn+ν e−x
n
n!
dx
(19)
Constants
=
(−1)n
(−1)n−1 (n + ν)
Γ(n + ν + 1)
(n−1)
, kn
=
, hn =
n!
(n − 1)!
n+1
(20)
1
2n + ν + 1
n+ν
, βn =
, γn = −
, λn = −n
n+1
n+1
n+1
(21)
Differential Equation
d
dx
kn+1
dHn
dx
a
Define αn , βn , and γn by
kn+1
2
e−x
Generating Function
αn = −
a
= 0, hn =
dHn
d2 Hn
− 2x
+ 2nHn = 0
dx2
dx
Recurrence Relations
(n)
Z b
√
(12)
d
dx
kn
(n)
n kn n!
(11)
Differential Equation
; Chebyshev polynomials of
Recurrence Relations
Define hn by
Z b
(10)
(13)
2
1. deg(s) = 0: s(x) = 1, w(x) = e−x , a = −∞, b = +∞; Hermite
polynomials Hn (x)
2. deg(s) = 1: s(x) = x, w(x) = xν e−x (ν > −1), a = 0, b = +∞; Laguerre polynomials Lνn (x)
3. deg(s) = 2: s(x) = 1 − x2 , w(x) = (1 + x)µ (1 − x)ν (µ, ν > −1), a =
−1, b = +1; Jacobi polynomials Pnµ,ν (x)
Jacobi polynomials are further classified by values of µ and ν:
1. µ = ν = 0: w(x) = 1; Legendre polynomials Pn (x)
1/2
αn = 2, βn = 0, γn = −2n, λn = −2n
= 2n , k n
Classification
Depending on the degree of s(x), we can classify classical orthogonal polynomials into three categories:
4. µ = ν = 12 : w(x) = 1 − x2
2nd kind, Un (x)
2
e−x
π2n n!
kn
2
1−x
Hn (−x) = (−1)n Hn (x)
The coefficients σi are defined by
3. µ = ν = − 12 : w(x) =
the 1st kind, Tn (x)
dn
dxn
Parity
(n)
(n−1) n−1
(1)
(0)
Fn (x) = kn xn + kn
x
+ · · · + kn x + kn .
where
(9)
n=0
xν+1 e−x
dLνn
dx
= −nxν e−x Lνn
d2 Lνn
dLν
+ (ν + 1 − x) n + nLνn = 0
2
dx
dx
Recurrence Relation
x
(n + 1)Lνn+1 = (2n + ν + 1 − x)Lνn − (n + ν)Lνn−1
(22)
(23)
(24)
Generating Function
g(x, t) =
xt
1
exp −
(1 − t)ν+1
1−t
=
∞
X
n=0
Lνn (x)tn
(25)
Classical Orthogonal Polynomials
Legendre Polynomials
Differential Equation
w(x) = 1, s(x) = 1 − x2 , a = −1, b = +1
1 − x2
Generalized Rodrigues Formula
n
(−1)n dn
1 − x2
n
n
2 n! dx
Kn = (−1)n 2n n!, Pn (x) =
(26)
d2 Cnλ
− (2λ + 1)x
dx2
λ
dCn
λ
=0
+ n(n + 2λ)Cn
dx
(44)
Recurrence Relation
λ
λ
λ
− (n + 2λ − 1)Cn−1
(n + 1)Cn+1
= 2(n + λ)xCn
(45)
Parity
Pn (−x) = (−1)n Pn (x)
(27)
Constants
Chebyshev Polynomials of the 1st Kind
−1/2
w(x) = 1 − x2
1
2n Γ(n + 2 )
(n)
kn =
,
n! Γ( 12 )
αn =
(n−1)
kn
= 0, hn =
2
2n + 1
(28)
2n + 1
n
, βn = 0, γn = −
, λn = −n(n + 1)
n+1
n+1
(29)
Differential Equation
d
dx
1−x
Rodrigues Formula
(2n)!
2n n!
n
n−1/2
2 n!
1/2 dn
1 − x2
1 − x2
Tn (x) = (−1)n
n
(2n)!
dx
Kn = (−1)n
Constants
1−x
2
2
2 d Pn
dPn dx
= −n(n + 1)Pn
(n)
kn
(30)
(n−1)
= 2n−1 , kn
(31)
(n + 1)Pn+1 = (2n + 1)xPn − nPn−1
d
dx
(32)
1/2 dTn 1 − x2
dx
1 − x2
Generating Function
1
1 − 2xt + t2
=
∞
X
(46)
(47)
(48)
(49)
Differential Equation
Recurrence Relation
g(x, t) = √
π
2
= 0, hn =
αn = 2, βn = 0, γn = −1, λn = −n2
dPn
− 2x
+ n(n + 1)Pn = 0
dx
dx2
, s(x) = 1 − x2 , a = −1, b = +1
d2 Tn
= −n2 1 − x2
−x
dx2
−1/2
Tn
dTn
+ n2 Tn = 0
dx
(50)
(51)
Recurrence Relation
Pn (x)tn
(33)
Tn+1 = 2xTn − Tn−1
(52)
n=0
Generating Function
Jacobi Polynomials
∞
ν
X
1 − t2
= T0 (x) + 2
Tn (x)tn
1 − 2xt + t2
g(x, t) =
w(x) = (1 + x) (1 − x) (µ, ν > −1), s(x) = 1 − x , a = −1, b = +1
µ
2
(53)
n=1
Rodrigues Formula
Kn = (−1)n 2n n!
(34)
(−1)n
(1 + x)−µ (1 − x)−ν
Pnµ,ν (x) = n
2 n!
n
dn
1 − x2
dxn
(35)
Constants
(n)
kn
Tn (cos θ) = cos nθ
=
n(ν − µ)
1 Γ(2n + µ + ν + 1)
(n−1)
=
, kn
kn
2n n! Γ(n + µ + ν + 1)
2n + µ + ν
(36)
2µ+ν+1 Γ(n + µ + 1)Γ(n + ν + 1)
n!(2n + µ + ν + 1)Γ(n + µ + ν + 1)
(37)
2 µ,ν
2 d Pn
dx2
+ (µ − ν − (µ + ν + 2)x)
dPnµ,ν
dx
+ n(n + µ + ν + 1)Pnµ,ν = 0
(55)
−1/2 dn
n+1/2
2n (n + 1)!
1 − x2
1 − x2
(2n + 1)!
dxn
(56)
Constants
(n)
kn
(n−1)
= 2n , kn
= 0, hn =
π
2
(57)
αn = 2, βn = 0, γn = −1, λn = −n(n + 2)
µ,ν
− 2(n + µ)(n + ν)(2n + µ + ν + 2)Pn−1
d
dx
1 − x2
(39)
1 − x2
Gegenbauer Polynomials
λ>−
1
2
(58)
Differential Equation
= (2n + µ + ν + 1) (2n + µ + ν)(2n + µ + ν + 2)x + ν 2 − µ2 Pnµ,ν
3/2 dUn dx
d2 Un
dx2
= −n(n + 2) 1 − x2
− 3x
1/2
dUn
+ n(n + 1)Un = 0
dx
Un
(59)
(60)
Recurrence Relation
, s(x) = 1 − x2 , a = −1, b = +1
Rodrigues Formula
Un+1 = 2xUn − Un−1
(61)
Generating Function
Γ(n + λ + 12 )Γ(2λ)
Kn = (−1)n 2n n!
Γ(n + 2λ)Γ(λ + 21 )
(−1)n Γ(n + 2λ)Γ(λ + 12 )
λ
Cn
(x) =
Γ(n + λ + 12 )Γ(2λ)
Constants
(2n + 1)!
2n (n + 1)!
(38)
2(n + 1)(n + µ + ν + 1)(2n + µ + ν)Pnµ,ν
λ−1/2 , s(x) = 1 − x2 , a = −1, b = +1
Kn = (−1)n
Recurrence Relation
w(x) = 1 − x2
1/2
Rodrigues Formula
Un (x) = (−1)n
Differential Equation
(54)
Chebyshev Polynomials of the 2nd Kind
w(x) = 1 − x2
hn =
1−x
A Useful Identity
2
(n)
kn =
n Γ(n + λ)
n!
√
hn =
1 − x2
Γ(λ)
1/2−λ dn
dxn
(40)
,
πΓ(n + 2λ)Γ(λ + 12 )
n!(n + λ)Γ(2λ)Γ(λ)
X
1
=
Un (x)tn
1 − 2xt + t2
(62)
sin(n + 1)θ
sin θ
(63)
n=0
1 − x2
n+λ−1/2
A Useful Identity
(41)
(n−1)
kn
=0
∞
g(x, t) =
(42)
(43)
Un (cos θ) =