Determinants and Their Applications in Mathematical Physics

164 4. Particular Determinants

=

∞

∑

r=0

(p)re

(r+p)t

r!

,

where

(p)r=p(p+ 1)(p+2)···(p+r−1) (4.12.52)

and denote the corresponding determinant byE

(p)

n :

E

(p)

n =

∣

∣

ψ

(p)

m

∣

∣

n

, 0 ≤m≤ 2 n− 2 ,

where

ψ

(p)

m

=D

m

f

=

∞

∑

r=0

(p)r(r+p)

m

e

(r+p)t

r!

. (4.12.53)

Theorem 4.59.

E

(p)

n =

e

n(2p+n−1)t/ 2

(1−e

t

)

n(p+n−1)

n− 1

∏

r=1

r!(p)r.

Proof. Put

gr=

αre

t

(1−e

t

)

2

,αrconstant,

and note that, from (4.12.48),

g 1 =D

2

(logf)

=

pe

t

(1−et)^2

,

so thatα 1 =pand

loggr= logαr+t−2 log(1−e

t

),

D

2

(loggr)=

2 e

t

(1−e

t

)

2

. (4.12.54)

Substituting these functions into the differential–difference equation, it is

found that

αn=nα 1 +2

n− 1

∑

r=1

(n−r)

=n(p+n−1). (4.12.55)

Hence,

gn=

n(p+n−1)e

t

(1−e

t

)

2

,

gn−r=

(n−r)(p+n−r−1)e

t

(1−e

t

)

2

. (4.12.56)