Determinants and Their Applications in Mathematical Physics

4.13 Hankelians 6 167

the upper limits that

γij=

i

∑

r=1

j

∑

s=1

βir 2

r+s− 1

kr+s− 2 βjs.

Hence,

γ 2 p+1, 2 q+1=2

2 p+1

∑

r=1

2 q+1

∑

s=1

β 2 p+1,r 2

r+s− 2

kr+s− 2 β 2 q+1,s. (4.13.10)

From the first line of (4.13.6), the summand is zero whenrandsare even.

Hence, replacerby 2r+ 1, replacesby 2s+ 1 and refer to (4.13.5) and

(4.13.6),

γ 2 p+1, 2 q+1=2

p

∑

r=0

q

∑

s=0

β 2 p+1, 2 r+1β 2 q+1, 2 s+1 2

2 r+2s

k 2 r+2s

=2

p

∑

r=0

q

∑

s=0

λprλqs

N

∑

j=1

aj(2xj)

2 r+2s

=2

N

∑

j=1

aj

p

∑

r=0

λpr(2xj)

2 r

q

∑

s=0

λqs(2xj)

2 s

=2

N

∑

j=1

ajgp(xj)gq(xj)

=α 2 p+1, 2 q+1, (4.13.11)

which completes the proof of case (i). Cases (ii)–(iv) are proved in a similar

manner.

Corollary.

|αij|n=|M|n=|N|

2

n|K|n

=|βij|

2

n

| 2

i+j− 1

ki+j− 2 |n

=

(

n

∏

i=1

βii

) 2

2

n

| 2

i+j− 2

ki+j− 2 |n. (4.13.12)

But,β 11 =1andβii=

1

2

, 2 ≤i≤n. Hence, referring to Property (e) in

Section 2.3.1,

|αij|n=2

n

2

− 2 n+2

|ki+j− 2 |n. (4.13.13)

Thus,Mcan be expressed as a Hankelian.