Determinants and Their Applications in Mathematical Physics

322 Appendix

Differential equation.

xL

′′

n(x)+(1−x)L

′

n(x)+nLn(x)=0;

Appell relation.If

φn(x)=x

n

Ln

(

1

x

)

,

then

φ

′

n

(x)=nφn− 1 (x).

φn(x) is the Laguerre polynomial with its coefficients arranged in reverse

order.

Hermite PolynomialHn(x)

Definition.

Hn(x)=n!

N

∑

r=0

(−1)

r

(2x)

n− 2 r

r!(n− 2 r)!

,N=

[

1

2

n

]

.

Rodrigues formula.

Hn(x)=(−1)

n

e

x

2

D

n

(

e

−x

2 )

,D=

d

dx

;

Generating function relation.

e

2 xt−t

2

=

∞

∑

n=0

Hn(x)t

n

n!

;

Recurrence relation.

Hn+1(x)− 2 xHn(x)+2nHn− 1 (x)=0;

Differential equation.

H

′′

n

(x)− 2 xH

′

n

(x)+2nHn(x)=0;

Appell relation.

H

′

n

(x)=2nHn− 1 (x).

Legendre PolynomialsPn(x)

Definition.

Pn(x)=

1

2

n

N

∑

r=0

(−1)

r

(2n− 2 r)!x

n− 2 r

r!(n−r)! (n− 2 r)!

,N=

[

1

2

n

]

.