Determinants and Their Applications in Mathematical Physics

332 Appendix

Illustration.The function

f=Ax 0 x 2 x 4 x 6 +

Bx 0 x 2 x

2

3

x 5

x 1

+

Cx

2

0

x 1 x

5

3

+Dx

8

2

Ex

3

0 x^4 +Fx

4

1

(A.9.4)

is homogeneous of degree 4 in its variables and homogeneous of degree 12

in the suffixes of its variables. Hence,

6

∑

r=0

xr

∂f

∂xr

=4f,

6

∑

r=1

rxr

∂f

∂xr

=12f.

A.10 Formulas Related to the Function

(x+

√

1+x

2

)

2 n

Define functionsλnrandμnras follows. Ifnis a positive integer,

(x+

√

1+x

2

)

2 n

=

n

∑

r=0

λnrx

2 r

+

√

1+x

2

n

∑

r=1

μnrx

2 r− 1

, (A.10.1)

where

λnr=

n

n+r

(

n+r

2 r

)

2

2 r

, (A.10.2)

μnr=

rλnr

n

. (A.10.3)

Define the functionνias follows:

(1 +z)

− 1 / 2

=

∞

∑

i=0

νiz

i

. (A.10.4)

Then

νi=

(−1)

i

2

2 i

(

2 i

i

)

=P 2 i(0),

ν 0 =1, (A.10.5)

wherePn(x) is the Legendre polynomial.

Theorem A.7.

n

∑

j=1

λn− 1 ,j− 1 νi+j− 2 =

δin

2

2(n−1)

, 1 ≤i≤n.