Page 221 - Determinants and Their Applications in Mathematical Physics

206 5. Further Determinant Theory

c

1 c 2
|c 0 |
c 2 c 3

a 3 = , (5.5.19)
c
0 c 1

|c 1 |
c 1 c 2

etc. Determinantal formulas for a 2n−1 , a 2n , and two other functions will be
given shortly.
Let
A n = |c i+j−2 | n ,
B n = |c i+j−1 | n , (5.5.20)
with A 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 of P n
and Q n that P 2n−1 , P 2n , Q 2n+1 , and Q 2n are polynomials of degree n.In
all four polynomials, the constant term is 1. Hence, we may write

n

P 2n−1 = p 2n−1,r x ,
r
r=0
n

Q 2n+1 = q 2n+1,r x ,
r
r=0
n

P 2n = p 2n,r x ,
r
r=0
n

Q 2n = q 2n,r x , (5.5.21)
r
r=0
where both p mr and q mr satisfy the recurrence relation
u mr = u m−1,r + a m u m−2,r−1
and where
p m0 = q m0 =1, all m,
(5.5.22)
p 2n−1,r = p 2n,r =0, r < 0or r> n.
Theorem 5.11.
(n+1)
A
a. p 2n−1,r = n+1,n+1−r , 0 ≤ r ≤ n,
(n+1) A n
B
b. p 2n,r = n+1,n+1−r , 0 ≤ r ≤ n,
B n
A n B n+1
c. a 2n+1 = − ,
A n+1 B n
A n+1 B n−1
d. a 2n = − A n B n .