Page 347 - Determinants and Their Applications in Mathematical Physics

332 Appendix

Illustration. The function
2 2 5 8
Bx 0 x 2 x x 5 Cx x 1 x + Dx 2
0
3
3
f = Ax 0 x 2 x 4 x 6 + + 3 4 (A.9.4)
x 1 Ex x 4 + Fx
0 1
is homogeneous of degree 4 in its variables and homogeneous of degree 12
in the suffixes of its variables. Hence,
6
∂f
=4f,
x r
r=0 ∂x r
6
∂f

=12f.
rx r
r=1 ∂x r
A.10 Formulas Related to the Function
√
2 2n
(x + 1+ x )
Define functions λ nr and µ nr as follows. If n is a positive integer,
n n
+ + 2r−1
(x + 1+ x ) = λ nr x 2r + 1+ x 2 µ nr x , (A.10.1)
2 2n
r=0 r=1
where
n n + r
λ nr = 2 , (A.10.2)
2r
n + r 2r
µ nr = rλ nr . (A.10.3)
n
Define the function ν i as follows:
∞

−1/2
(1 + z) = ν i z . (A.10.4)
i
i=0
Then
(−1) i 2i
ν i =
2 2i i
= P 2i (0),
ν 0 =1, (A.10.5)
where P n (x) is the Legendre polynomial.

Theorem A.7.
n

λ n−1,j−1 ν i+j−2 = δ in , 1 ≤ i ≤ n.
2 2(n−1)
j=1