Determinants and Their Applications in Mathematical Physics

A.5 Orthogonal Polynomials 321

4.Prove that the vector Appell equation, namely

C

′

j=jCj−^1 ,j>^0 ,

is satisfied by the column vector

Cj=

[

(

j

0

)− 1

φj

(

p+1

1

)− 1

φj+1

(

j+2

2

)− 1

φj+2

···

(

j+n− 1

n− 1

)− 1

φj+n− 1

]T

n

,n≥ 1.

5.If

fnm=

m

∑

r=0

(−1)

r

(

m

r

)

φrφn−r,n≥m,

prove that

f

′

nm

=(n−m)fn− 1 ,m.

A.5 Orthogonal Polynomials

The following brief notes relate to the Laguerre, Hermite, and Legendre

polynomials which appear in the text.

Laguerre PolynomialsL

(α)

n

(x)andLn(x)

Definition.

L

(α)

n (x)=(n+α)!

n

∑

r=0

(−1)

r

x

r

r!(n−r)! (r+α)!

,

Ln(x)=L

(0)

n (x)=

n

∑

r=0

(−1)

r

(

n

r

)

x

r

r!

.

Rodrigues formula.

Ln(x)=

e

x

n!

D

n

(e

−x

x

n

); D=

d

dx

.

Generating function relation.

(1−t)

− 1

e

−xt/(1−t)

=

∞

∑

n=0

Ln(x)t

n

;

Recurrence relations.

(n+1)Ln+1(x)−(2n+1−x)Ln(x)=+nLn− 1 (x)=0,

xL

′

n(x)=n[Ln(x)−Ln−^1 (x)];