Page 792 - Handbook Of Integral Equations

The first four polynomials are
2
3
T 0 =1, T 1 = x, T 2 =2x – 1, T 3 =4x – 3x.
The recurrent formulas:
T n+1 (x)=2xT n (x) – T n–1 (x), n ≥ 2.
The functions T n (x) form an orthogonal system on the interval –1< x < +1, with

T n (x)T m (x)
+1 0 if n ≠ m,
√ dx = 1 2 π if n = m ≠ 0,
–1 1 – x 2 π if n = m =0.
The Chebyshev functions of the second kind,
U 0 (x) = arcsin x,
√
2
1 – x dT n (x)
U n (x) = sin(n arcsin x)= (n =1, 2, ... ),
n dx
just as the Chebyshev polynomials, also satisfy the differential equation (1).
The generating function is
∞
1 – sx n
= T n (x)s (|s| < 1).
1 – 2sx + s 2
n=0
Hermite polynomial
The Hermite polynomial H n = H n (x) satisfies the equation

y – 2xy +2ny =0
x
xx
and is defined by the formulas
d n
n
2
H n (x)=(–1) exp x 2 exp –x .
dx n
The first four polynomials are
3
2
H 0 =1, H 1 = x, H 2 =4x – 2, H 3 =8x – 12x.
The recurrent formulas:
H n+1 (x)=2xH n (x) – 2nH n–1 (x), n ≥ 2.
The functions H n (x) form an orthogonal system on the interval –∞ < x < ∞, with
∞

2 0 if n ≠ m,
exp –x H n (x)H m (x) dx = √ n
π 2 n!if n = m.
–∞
1 2
The Hermite functions ψ n (x) are introduced by the formula ψ n (x)=exp – x H n (x), where
2
n =0, 1, 2, ...
The generating function:
∞ n
2 s
exp –s +2sx = H n (x) .
n!
n=0
Jacobi polynomials
The Jacobi polynomials P n α,β = P n α,β (x) satisfy the equation
2

(1 – x )y + β – α – (α + β +2)x y + n(n + α + β +1)y =0
x
xx
and are defined by the formulas
n
(–1) n –α –β d n α+n β+n –n m n–m n–m m
α,β
P n = (1–x) (1+x) (1–x) (1+x) =2 C n+α C n+β (x–1) (x+1) ,
n
2 n! dx n
m=0
a
where C are binomial coefficients.
b
•
References for Supplement 10: H. Bateman and A. Erd´ elyi (1953, 1955), M. Abramowitz and I. A. Stegun (1964).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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