Page 190 - Determinants and Their Applications in Mathematical Physics

5.1 Determinants Which Represent Particular Polynomials 175

But

j − 1 j − 1 (j−i+1)
a + a i−1,j = + u

ij i − 1 i − 2

j − 1 j − 1 (j−i+2)
− + v
i − 2 i − 3

j (j−i+1) j (j−i+2)
= u − v
i − 1 i − 2
= a i,j+1 . (5.1.13)
Hence,
C + C = C j+1 , (5.1.14)

∗
j j
n

A = C 1 C 2 ··· C C j+1 ··· C n−1 C n ,

n j
n
j=1
(n+1)
A = − C 1 C 2 ··· C j C j+1 ··· C n−1 C n+1 . (5.1.15)

n+1,n
n
Hence,
n
(n+1)

A + A =
n n+1,n C 1 C 2 ··· (C − C j+1 ) C j+1 ··· C n
j
n
j=1
n

∗
= −
C 1 C 2 ··· C ··· C n
j
j=1
=0
by Theorem 3.1 on cyclic dislocations and generalizations in Section 3.1,
which proves (a).
Expanding A n+1 by the two elements in its last row,
(n+1)

A n+1 =(u − nv )A n − vA
n+1,n

=(u − nv )A n + vA
n
u nv

= v A + − A n ,
v v
n

yA n+1 y y nv

= A + −
v n+1 v n n y v A n
yA 1 y 2
= n +
v n v n A n

= D yA n
v n

= D 2 yA n−1
v n−1

yA n−r+1
= D r , 0 ≤ r ≤ n
v n−r+1