Determinants and Their Applications in Mathematical Physics

114 4. Particular Determinants

Proof of (F).Denote the sum bySand apply the Hankelian relation

φr+s− 3 =ar,s− 1 =ar− 1 ,s.

S=

∑n

s=1

(s−1)A

sj

∑n

r=1

ar,s− 1 A

ir

+

∑n

r=1

(r−1)A

ir

∑n

s=1

ar− 1 ,sA

sj

=

n

∑

s=1

(s−1)A

sj

δs− 1 ,i+

n

∑

r=1

(r−1)A

ir

δr− 1 ,j.

The proof of (F) follows. Equation (E) is proved in a similar manner.

Exercises

Prove the following:

1.

∑

p+q=m+2

A

ij,pq

=0.

2.

2 n− 2

∑

m=0

φm

∑

q+q=m+2

Aip,jq=(n−1)Aij.

3.

2 n− 2

∑

m=1

mφm

∑

p+q=m+2

Aip,jq=(n

2

−n−i−j+2)Aij.

4.

2 n− 2

∑

m=0

φm

∑

p+q=m+2

A

ijp,hkq

=nA

ij,hk

.

5.

2 n− 2

∑

m=1

mφm

∑

p+q=m+2

A

ijp,hkq

=(n

2

−n−i−j−h−k−4)A

ij,hk

.

6.

2 n− 2

∑

m=1

mφm− 1

∑

p+q=m+2

A

ijp,hkq

=iA

i+1,j;hk

+jA

i,j+1;hk

+hA

ij;h+1,k

+kA

ij;h,k+1

.

7.

2 n− 2

∑

m=0

∑

p+q=m+2

φp+r− 1 φq+r− 1 A

pq

=φ 2 r, 0 ≤r≤n−1.

8.

2 n− 2

∑

m=1

m

∑

p+q=m+2

φp+r− 1 φq+r− 1 A

pq

=2rφ 2 r, 0 ≤r≤n−1.

9.Prove that

n− 1

∑

r=1

rA

r+1,j

n

∑

m=1

φm+r− 2 A

im

=iA

i+1,j

by applying the sum formula for Hankelians and, hence, prove (F 1 )

directly. Use a similar method to prove (E 1 ) directly.