Page 339 - Determinants and Their Applications in Mathematical Physics

324 Appendix

It follows from (A.6.2) that
xψ = ψ m+1 , m ≥ 0. (A.6.4)
m
The formula
∆ ψ 0 = xψ m , m > 0,
m
is proved in the section on differences in Appendix A.8.
Other formulas for ψ m include the following:

m
(−1) m+r r! S m+1,r+1
ψ m = , m ≥ 0 (Comtet), (A.6.5)
(1 − x) r+1
r=0
x (−1) m+r
m
ψ m = r! S mr , m ≥ 0, (A.6.6)
1 − x (1 − x) r
r=1
where the S mr are Stirling numbers of the second kind (Appendix A.1).
1 ∂

ψ m = D r , D = (Zeitlin). (A.6.7)
1 − xe u ∂u
u=0
Let
1
t = φ 0 = .
1 − x
Then,
ψ 0 = −(1 − t),
ψ 1 = −t + t 2
= −t(1 − t),
2
ψ 2 = t − 3t +2t 3
= t(1 − t)(1 − 2t),
2
3
ψ 3 = −t +7t − 12t +6t 4
2
= −t(1 − t)(1 − 6t +6t ),
4
3
2
ψ 4 = t − 15t +50t − 60t +24t 5
2
3
= t(1 − t)(1 − 14t +36t − 24t ).
The function ψ m satisfies the linear recurrence relations

m
m

ψ m = x 1+ ψ r , m ≥ 0 (A.6.8)
r
r=0

x m−1 m

= 1+ ψ r , m ≥ 1 (A.6.9)
1 − x r
r=0
m
m
m m
x ψ m+r = (−1) m+r
r r ψ m+r
r=0 r=0
=∆ ψ m . (A.6.10)
m