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

Q=











0

10

20

30

···











.

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,

Φ(x)=e

xQ

Φ(0).

Hence,

Φ(0) =e

−xQ

Φ(x)

that is,











α 0

α 1

α 2

α 3

···











=











1

−x 1

x

2

− 2 x 1

−x

3

3 x

2

− 3 x 1

....................





















φ 0 (x)

φ 1 (x)

φ 2 (x)

φ 3 (x)

···











,

which yields the relation which is inverse to the first line of (A.4.6), namely

αm=

m

∑

r=0

(

m

r

)

φr(x)(−x)

m−r

(A.4.9)

φm(x) is also given by the following formulas but with a lower limit for

min each case:

φm(x)=

m− 1

∑

r=0

∣

∣

∣

∣

αr αr+1

− 1 x

∣

∣

∣

∣

x

m−r− 1

,m≥ 1 ,

φm(x)=

m− 2

∑

r=0

∣ ∣ ∣ ∣ ∣ ∣

αr 2 αr+1 αr+2

− 1 x

− 1 x

∣ ∣ ∣ ∣ ∣ ∣

x

m−r− 2

,m≥ 2 , (A.4.10)

etc. The polynomialsφmand the constantsαmare related by the two-

parameter identity

p

∑

r=0

(−1)

r

(

p

r

)

φp+q−rx

r

=

q

∑

r=0

(

q

r

)

αp+rx

q−r

,p,q=0, 1 , 2 ,....

(A.4.11)