Page 337 - Determinants and Their Applications in Mathematical Physics

322 Appendix

Differential equation.

xL (x)+(1 − x)L (x)+ nL n (x)=0;

n n
Appell relation. If

1
,
n
x
φ n (x)= x L n
then

φ (x)= nφ n−1 (x).
n
φ n (x) is the Laguerre polynomial with its coefficients arranged in reverse
order.

Hermite Polynomial H n (x)
Definition.

N
r
(−1) (2x) n−2r 1
H n (x)= n! , N = 2 n .
r!(n − 2r)!
r=0
Rodrigues formula.
2 2 d
H n (x)=(−1) e D n e −x , D = ;
n x
dx
Generating function relation.
∞
2 H n (x)t n
e 2xt−t = ;
n!
n=0
Recurrence relation.
H n+1 (x) − 2xH n (x)+2nH n−1 (x)=0;
Differential equation.

H (x) − 2xH (x)+2nH n (x)=0;
n n
Appell relation.

H (x)=2nH n−1 (x).
n
Legendre Polynomials P n (x)

Definition.
1 (−1) (2n − 2r)! x n−2r 1
N
r
P n (x)= , N = 2 n .
2 n r!(n − r)! (n − 2r)!
r=0