Determinants and Their Applications in Mathematical Physics

324 Appendix

It follows from (A.6.2) that

xψ

′

m=ψm+1,m≥^0. (A.6.4)

The formula

∆

m

ψ 0 =xψm,m> 0 ,

is proved in the section on differences in Appendix A.8.

Other formulas forψminclude the following:

ψm=

m

∑

r=0

(−1)

m+r

r!Sm+1,r+1

(1−x)

r+1

,m≥0 (Comtet), (A.6.5)

ψm=

x

1 −x

m

∑

r=1

(−1)

m+r

r!Smr

(1−x)

r

,m≥ 0 , (A.6.6)

where theSmrare Stirling numbers of the second kind (Appendix A.1).

ψm=

[

D

r

(

1

1 −xe

u

)]

u=0

,D=

∂

∂u

(Zeitlin). (A.6.7)

Let

t=φ 0 =

1

1 −x

.

Then,

ψ 0 =−(1−t),

ψ 1 =−t+t

2

=−t(1−t),

ψ 2 =t− 3 t

2

+2t

3

=t(1−t)(1− 2 t),

ψ 3 =−t+7t

2

− 12 t

3

+6t

4

=−t(1−t)(1− 6 t+6t

2

),

ψ 4 =t− 15 t

2

+50t

3

− 60 t

4

+24t

5

=t(1−t)(1− 14 t+36t

2

− 24 t

3

).

The functionψmsatisfies the linear recurrence relations

ψm=x

[

1+

m

∑

r=0

(

m

r

)

ψr

]

,m≥ 0 (A.6.8)

=

x

1 −x

[

1+

m− 1

∑

r=0

(

m

r

)

ψr

]

,m≥1 (A.6.9)

x

m

∑

r=0

(

m

r

)

ψm+r=

m

∑

r=0

(−1)

m+r

(

m

r

)

ψm+r

=∆

m

ψm. (A.6.10)