Determinants and Their Applications in Mathematical Physics

118 4. Particular Determinants

5.If

An=|φm|n, 0 ≤m≤ 2 n− 2 ,

Fn=|φm|n, 1 ≤m≤ 2 n− 1 ,

Gn=|φm|n, 2 ≤m≤ 2 n,

whereφm is an Appell polynomial, apply Exercise 3a in which the

cofactors are scaled to prove that

D(A

(n)

ij

)=−

(

iA

(n)

i+1,j

+jA

(n)

i,j+1

)

in which the cofactors are unscaled. Hence, prove that

a.D

r

(Fn)=(−1)

n+r

r!A

(n+1)

r+1,n+1,^0 ≤r≤n;

b.D

n

(Fn)=n!An;

c.Fnis a polynomial of degreen;

d.D

r

(Gn)=(−1)

r

r!

∑

p+q=r+2

A

(n+1)

pq ,^0 ≤r≤^2 n;

e.D

2 n

(Gn)=(2n)!An;

f.Gnis a polynomial of degree 2n.

6.LetBndenote the determinant of order (n+ 1) obtained by bordering

An(0) by the row

R=

[

1 −xx

2

−x

3

···(−x)

n− 1

• 
  

]

n+1

at the bottom and the columnR

T

on the right. Prove that

Bn=−

(^2) ∑n− 2
r=0
(−x)
r

∑

p+q=r+2

A

(n)

pq(0).

Hence, by applying a formula in the previous exercise and then the

Maclaurin expansion formula, prove that

Bn=−Gn− 1.

7.Prove that

D

r

(Aij)=

(−1)

r

r!

(i−1)!(j−1)!

r

∑

s=0

(i+r−s−1)!(j+s−1)!

s!(r−s)!

Ai+r−s,j+s.

8.Apply the double-sum relation (A 1 ) in Section 4.8.7 to prove thatGn

satisfies the differential equation

2 n− 1

∑

m=0

(−1)

m

φmD

m+1

(Gn)

m!

=0.