Page 336 - Determinants and Their Applications in Mathematical Physics

A.5 Orthogonal Polynomials 321

4. Prove that the vector Appell equation, namely
C = jC j−1 , j > 0,

j
is satisfied by the column vector

−1
−1
−1
j p +1 j +2
C j = φ j+1 φ j+2
0 φ j 1 2
−1 T

j + n − 1
··· φ j+n−1 , n ≥ 1.
n − 1
n
5. If

m
m

f nm = (−1) r φ r φ n−r , n ≥ m,
r
r=0
prove that
f =(n − m)f n−1,m .
nm
A.5 Orthogonal Polynomials

The following brief notes relate to the Laguerre, Hermite, and Legendre
polynomials which appear in the text.

Laguerre Polynomials L (α) (x) and L n (x)
n
Definition.
n (−1) x
r r

L (α) (x)=(n + α)! ,
r!(n − r)! (r + α)!
n
r=0

n n x r

L n (x)= L (0) (x)= (−1) r .
n r r!
r=0
Rodrigues formula.
e x d
L n (x)= D (e −x n D = .
x );
n
n! dx
Generating function relation.
∞

(1 − t) −1 −xt/(1−t) = L n (x)t ;
e
n
n=0
Recurrence relations.
(n +1)L n+1 (x) − (2n +1 − x)L n (x)=+nL n−1 (x)=0,

xL (x)= n[L n (x) − L n−1 (x)];
n