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LAGUERRE'S EQUATION
But
hence
@ n h
lim
z!0 @z n …1 z†1 exp x i ˆ dn
1 z dx n
L n …x† ˆe x d n
dx n …xn e x †:
… †;
xn e x
The orthogonal Laguerre functions
The Laguerre polynomials, L n …x†, do not by themselves form an orthogonal set.
But the functions e x=2 L n …x† are orthogonal in the interval (0, 1). For any two
Laguerre polynomials L m …x† and L n …x† we have, from Laguerre's equation,
xL 00
m ‡…1 x†L 0 m ‡ mL m ˆ 0;
xL 00
n ‡…1 x†L 0 n ‡ mL n ˆ 0:
Multiplying these equations by L n …x† and L m …x† respectively and subtracting, we
®nd
x‰L n Lm 00 L m Ln 00 Š‡…1 x†‰L n Lm 0 L m LnŠˆ…n 0 m†L m L n
or
d
dx ‰L nLm 0 L m LnŠ‡ 0 1 x
x
‰L nLm 0 L m L nŠˆ…n 0 m†L mL n
:
x
Then multiplying by the integrating factor
Z
exp ‰…1 x†=xŠdx ˆ exp…ln x x† ˆxe x ;
we have
d
dx fxex ‰L n Lm 0 L m LnŠg 0 ˆ …n m†e x L m L n :
Integrating from 0 to 1 gives
Thus if m 6ˆ n
…n m†
Z 1
0
e x L m …x†L n …x†dx ˆ xe x ‰L n L 0 m L m L 0 nŠj 1 0 ˆ 0:
Z 1
0
e x L m …x†L n …x†dx ˆ 0 …m 6ˆ n†; …7:62†
which proves the required result.
Alternatively, we can use Rodrigues' formula (7.61). If m is a positive integer,
Z 1
0
e x x m L m …x†dx ˆ
Z 1
0
x m d n
dx n …xn e x †dx ˆ…1† m m!
Z 1
0
d nm
dx nm …xn e x †dx;
…7:63†
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