Page 144 - Determinants and Their Applications in Mathematical Physics

4.10 Henkelians 3 129

n j−1
V nj (−x)
.
h + n + j − 1
S n (x, h)= K n (h)V nn
j=1
Hence,

n j−1
h + n − 1 S n (x, h) n+1 V nj (−x)
=(−1)
h S n (0,h) h + n + j − 1
j=1

n j−1
1 (h + n + j − 2)!x
=
(n − j)!(h + j − 1)! (j − 1)!
j=1
n
1 h + n − 1
h+n−1 h+n+j−2
= D (x ),
(h + n − 1)! h + j − 1
j=1
 
2
n
(h + n − 1)! S n (x, h) h+n−1 n−1 h + n − 1
h+j−1
= D  x x 
h!(n − 1)! S n (0,h) h + j − 1
j=1

h+n−1
h + n − 1
h+n−1
n−1
= D x x r
r
r=h

= D h+n−1 x n−1 (1 + x) h+n−1 − p h+n−2 (x) ,
where p r (x) is a polynomial of degree r. Formula (a) follows. To prove (b),
put x = −1 − t. The details are elementary.
Further formulas of the Rodrigues type appear in Section 4.11.4.
4.10.3 Bordered Yamazaki–Hori Determinants — 1
Let
A = |a ij | n = |θ m | n ,
B = |b ij | n = |φ m | n , 0 ≤ m ≤ 2n − 1, (4.10.22)
denote two Hankelians, where
1 2 2(i+j−1) 2 2(i+j−1)
a ij = p x + q y − 1 ,
i + j − 1
1 2 2m+2 2 2m+2
θ m = p x + q y − 1 ,
m +1
1 2 i+j−1 2 i+j−1
b ij = p X + q Y ,
i + j − 1
1 2 m+1 2 m+1
φ m = p X + q Y ,
m +1
2
2
p + q =1,
2
X = x − 1,
2
Y = y − 1. (4.10.23)