Page 331 - Determinants and Their Applications in Mathematical Physics

316 Appendix

The infinite triangular matrix in (A.4.8) can be expressed in the form
e xQ , where

0
 
 10 
20  .
 
Q = 
3 0
 
···
Identities among this and other triangular matrices have been developed
by Vein. The triangular matrix in (8) with its columns arranged in reverse
order appears in Section 5.6.2.
Denote the column vector on the left of (A.4.8) by Φ(x). Then,
xQ
Φ(x)= e Φ(0).
Hence,
−xQ
Φ(0) = e Φ(x)
that is,
     
α 0 1 φ 0 (x)
 −x 1
 α 1    φ 1 (x) 
 α 2  =  x 2 −2x 1   φ 2 (x)  ,


 


   3 2   
α 3 −x 3x −3x 1 φ 3 (x)
··· .................... ···
which yields the relation which is inverse to the first line of (A.4.6), namely
m
m

α m = φ r (x)(−x) m−r (A.4.9)
r
r=0
φ m (x) is also given by the following formulas but with a lower limit for
m in each case:
m−1
m−r−1

φ m (x)= α r α r+1 x , m ≥ 1,
−1 x

r=0

m−2 α r 2α r+1 α r+2

φ m (x)= −1 x x m−r−2 , m ≥ 2, (A.4.10)

r=0 −1 x
etc. The polynomials φ m and the constants α m are related by the two-
parameter identity

q
p
p q
(−1) r φ p+q−r x = α p+r x q−r , p, q =0, 1, 2,... .
r
r r
r=0 r=0
(A.4.11)