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