Page 335 - Determinants and Their Applications in Mathematical Physics

320 Appendix

The polynomial
m
m
ψ mn (x)= α n+r x r
r
r=0
satisfies the relations
ψ = mψ m−1,n+1 ,
mn
ψ mn − ψ m−1,n = xψ
mn
= mxψ m−1,n+1 .

Exercises

1. Prove that
m
m
φ m (x − h)= (−h) φ m−r (x)
r
r
r=0
=∆ φ 0 .
m
h
2. If

S m (x)= φ r φ s ,
r+s=m

T m (x)= φ r φ s φ t ,
r+s+t=m
prove that
S =(m +1)S m−1 ,
m
m
m +1
S m (x + h)= h S m−r (x),
r
r
r=0
T =(m +2)T m−1 ,
m
m
m +2

T m (x + h)= h T m−r (x).
r
r
r=0
3. Prove that
∞
1
−1
φ = (−1) c mn x ,
n
n
m
n=0
α m
where
c m0 =1,

m
1
c mn = , n ≥ 1.
α n m − i + j − 1 α m−i+j−1
m n
This determinant is of Hessenberg form, is symmetric about its sec-
ondary diagonal, and contains no more than (m + 1) nonzero diagonals
parallel to and including the principal diagonal.