Determinants and Their Applications in Mathematical Physics

166 4. Particular Determinants

β 2 i, 2 i=λii,i≥ 1 ,

β 2 i+1, 2 j+1=λij, 0 ≤j≤i,

β 2 i+2, 2 j=λi+1,j−λij, 1 ≤j≤i+1, (4.13.6)

λij=

i

i+j

(

i+j

2 j

)

;

λii=

1

2

,i>0; λi 0 =1,i≥ 0. (4.13.7)

The functionsλijandfr(x) appear in Appendix A.10.

Theorem 4.60.

M=NKN

T

.

Proof. Let

G=[γij]n=NKN

T

. (4.13.8)

Then

G

T

=NK

T

N

T

=NKN

T

=G.

Hence,Gis symmetric, and sinceMis also symmetric, it is sufficient to

prove thatαij=γijforj≤i. There are four cases to consider:

i.i, jboth odd,

ii.iodd,jeven,

iii.ieven,jodd,

iv.i, jboth even.

To prove case (i), puti=2p+1 andj=2q+1 and refer to Appendix A.10,

where the definition ofgr(x) is given in (A.10.7), the relationships between

fr(x) andgr(x) are given in Lemmas (a) and (b) and identities among the

gr(x) are given in Theorem 4.61.

α 2 p+1, 2 q+1=u 2 q+2p+u 2 q− 2 p

=

N

∑

j=1

aj

{

f 2 q+2p(xj)+f 2 q− 2 p(xj)

}

=

N

∑

j=1

aj

{

gq+p(xj)+gq−p(xj)

}

=2

N

∑

j=1

ajgp(xj)gq(xj). (4.13.9)

It follows from (4.13.8) and the formula for the product of three matrices

(the exercise at the end of Section 3.3.5) with appropriate adjustments to