Determinants and Their Applications in Mathematical Physics

212 5. Further Determinant Theory

determinants with constant elements. It is a trivial exercise to find two

determinantsA=|aij|nandB=|bij|nsuch thataij=bijfor any pair

(i, j) and the elementsaijare not merely a rearrangement of the elements

bij, butA=B. It is an equally trivial exercise to find two determinants of

different orders which have the same value. If the elements are polynomials,

then the determinants are also polynomials and the exercises are more

difficult.

It is the purpose of this section to show that there exist families of

distinct matrices whose determinants are not distinct for the reason that

they represent identical polynomials, apart from a possible change in sign.

Such determinants may be described as equivalent.

5.6.2 Determinants with Binomial Elements

Letφm(x) denote an Appell polynomial (Appendix A.4):

φm(x)=

m

∑

r=0

(

m

r

)

αm−rx

r

. (5.6.1)

The inverse relation is

αm=

m

∑

r=0

(

m

r

)

φm−r(x)(−x)

r

. (5.6.2)

Define infinite matricesP(x),P
T
(x),A, and Φ(x) as follows:

P(x)=

[

(

i− 1

j− 1

)

x

i−j

]

,i,j≥ 1 , (5.6.3)

where the symbol←→ denotes that the order of the columns is to be

reversed.P
T
denotes the transpose ofP. BothAand Φ are defined in

Hankelian notation (Section 4.8):

A=[αm],m≥ 0 ,

Φ(x)=[φm(x)],m≥ 0. (5.6.4)

Now define block matricesMandM

∗

as follows:

M=

[

OP

T

(x)

P(x)Φ(x)

]

, (5.6.5)

M

∗

=

[

OP

T

(−x)

P(−x) A

]

. (5.6.6)

These matrices are shown in some detail below. They are triangular, sym-

metric, and infinite in all four directions. Denote the diagonals containing

the unit elements in both matrices by diag(1).

It is now required to define a number of determinants of submatrices of

eitherMorM
∗

. Many statements are abbreviated by omitting references

to submatrices and referring directly to subdeterminants.