Page 791 - Handbook Of Integral Equations

Laguerre polynomials
The Laguerre polynomials L n = L n (x) satisfy the equation

xy +(1 – x)y + ny =0
x
xx
and are defined by the formulas
2
d n n –x n n 2 n–1 n (n – 1) 2 n–2
x
L n (x)= e x e (–1) x – n x + x + ··· .
dx n 2!
The first four polynomials have the form
3
2
2
L 0 =1, L 1 = –x +1, L 2 = x – 4x +2, L 3 = –x +9x – 18x +6.
To calculate L n for n ≥ 2, one can use the recurrent formulas
2
L n+1 (x)=(2n +1 – x)L n (x) – n L n–1 (x).

The functions L n (x) form an orthogonal system on the interval 0 < x < ∞, with
∞ 0 if n ≠ m,

–x
e L n (x)L m (x) dx = 2
0 (n!) if n = m.
The associated Laguerre polynomials of degree n – k and order k are given by

d k
k
L (x)= L n (x).
n k
dx
These satisfy the differential equation

xy +(k +1 – x)y +(n – k)y =0,
xx x
where n =1, 2, ... and k =0, 1, 2, ...
The generating function is

∞
1 sx s n
exp – = L n (x) .
1 – s 1 – s n!
n=0
Chebyshev polynomials
The Chebyshev polynomials T n = T n (x) satisfy the equation

2
2
(1 – x )y – xy + n y = 0 (1)

xx x
and are defined by the formulas
n
(–2) n! √ d n 2 n– 1
T n (x) = cos(n arccos x)= 1 – x 2 (1 – x ) 2
(2n)! dx n
[n/2]
n m (n – m – 1)! n–2m
= (–1) (2x) (n = 0,1,2, ... ),
2 m!(n – 2m)!
m=0
where [A] stands for the integer part of a number A.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 776