Determinants and Their Applications in Mathematical Physics

(Chris Devlin) #1

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.
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