Determinants and Their Applications in Mathematical Physics

206 5. Further Determinant Theory

a 3 =

|c 0 |

∣

∣

∣

∣

c 1 c 2

c 2 c 3

∣

∣

∣

∣

|c 1 |

∣

∣

∣

∣

c 0 c 1

c 1 c 2

∣

∣

∣

∣

, (5.5.19)

etc. Determinantal formulas fora 2 n− 1 ,a 2 n, and two other functions will be

given shortly.

Let

An=|ci+j− 2 |n,

Bn=|ci+j− 1 |n, (5.5.20)

withA 0 =B 0 = 1. Identities among these determinants and their cofactors

appear in Hankelians 1.

It follows from the recurrence relation (5.5.9) and the initial values ofPn

andQnthatP 2 n− 1 ,P 2 n,Q 2 n+1, andQ 2 nare polynomials of degreen.In

all four polynomials, the constant term is 1. Hence, we may write

P 2 n− 1 =

n

∑

r=0

p 2 n− 1 ,rx

r

,

Q 2 n+1=

n

∑

r=0

q 2 n+1,rx

r

,

P 2 n=

n

∑

r=0

p 2 n,rx

r

,

Q 2 n=

n

∑

r=0

q 2 n,rx

r

, (5.5.21)

where bothpmrandqmrsatisfy the recurrence relation

umr=um− 1 ,r+amum− 2 ,r− 1

and where

pm 0 =qm 0 =1, allm,

p 2 n− 1 ,r=p 2 n,r=0,r<0orr>n.

(5.5.22)

Theorem 5.11.

a.p 2 n− 1 ,r=

A

(n+1)

n+1,n+1−r

An

, 0 ≤r≤n,

b.p 2 n,r=

B

(n+1)

n+1,n+1−r

Bn

, 0 ≤r≤n,

c. a 2 n+1=−

AnBn+1

An+1Bn

,

d.a 2 n=−

An+1Bn− 1

AnBn

.