Page 129 - Determinants and Their Applications in Mathematical Physics

114 4. Particular Determinants

Proof of (F). Denote the sum by S and apply the Hankelian relation
φ r+s−3 = a r,s−1 = a r−1,s .

n n n n

S = (s − 1)A sj a r,s−1 A + (r − 1)A ir a r−1,s A sj
ir
s=1 r=1 r=1 s=1
n n

= (s − 1)A δ s−1,i + (r − 1)A δ r−1,j .
sj
ir
s=1 r=1
The proof of (F) follows. Equation (E) is proved in a similar manner.

Exercises

Prove the following:

1. A ij,pq =0.
p+q=m+2
2n−2

2. φ m A ip,jq =(n − 1)A ij .
m=0 q+q=m+2
2n−2
2
3. mφ m A ip,jq =(n − n − i − j +2)A ij .
m=1 p+q=m+2
2n−2

4. φ m A ijp,hkq = nA ij,hk .
m=0 p+q=m+2
2n−2
2
5. mφ m A ijp,hkq =(n − n − i − j − h − k − 4)A ij,hk .
m=1 p+q=m+2
2n−2

6. mφ m−1 A ijp,hkq
m=1 p+q=m+2
= iA i+1,j;hk + jA i,j+1;hk + hA ij;h+1,k + kA ij;h,k+1 .
2n−2

7. φ p+r−1 φ q+r−1 A pq = φ 2r , 0 ≤ r ≤ n − 1.
m=0 p+q=m+2
2n−2

8. m φ p+r−1 φ q+r−1 A pq =2rφ 2r , 0 ≤ r ≤ n − 1.
m=1 p+q=m+2
9. Prove that

n−1 n

rA r+1,j φ m+r−2 A im = iA i+1,j
r=1 m=1
by applying the sum formula for Hankelians and, hence, prove (F 1 )
directly. Use a similar method to prove (E 1 ) directly.