Page 227 - Determinants and Their Applications in Mathematical Physics

212 5. Further Determinant Theory

determinants with constant elements. It is a trivial exercise to find two
determinants A = |a ij | n and B = |b ij | n such that a ij = b ij for any pair
(i, j) and the elements a ij are not merely a rearrangement of the elements
b ij , but A = 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

Let φ m (x) denote an Appell polynomial (Appendix A.4):
m
m

φ m (x)= α m−r x . (5.6.1)
r
r
r=0
The inverse relation is
m
m
α m = φ m−r (x)(−x) . (5.6.2)
r
r
r=0
Define infinite matrices P(x), P (x), A, and Φ(x) as follows:
T

i − 1
P(x)= x i−j , i, j ≥ 1, (5.6.3)
j − 1
where the symbol ←↔ denotes that the order of the columns is to be
reversed. P T denotes the transpose of P. Both A and Φ are defined in
Hankelian notation (Section 4.8):
A =[α m ], m ≥ 0,
Φ(x)=[φ m (x)], m ≥ 0. (5.6.4)
∗
Now define block matrices M and M as follows:

O P (x)
T
M = , (5.6.5)
P(x) Φ(x)

O P (−x)
T
M = . (5.6.6)
∗
P(−x) A
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
either M or M . Many statements are abbreviated by omitting references
∗
to submatrices and referring directly to subdeterminants.