Determinants and Their Applications in Mathematical Physics

5.8 Some Applications of Algebraic Computing 227

formulas in determinant theory contain products and quotients involving

several determinants of ordernor some function ofn.

Computers are invaluable in the initial stages of an investigation. They

can be used to study the behavior of determinants as their orders increase

and to assist in the search for patterns. Once a pattern has been observed,

it may be possible to formulate a conjecture which, when proved analyti-

cally, becomes a theorem. In some cases, it may be necessary to evaluate

determinants of order 10 or more before the nature of the conjecture be-

comes clear or before a previously formulated conjecture is realized to be

false.

In Section 5.6 on distinct matrices with nondistinct determinants, there

are two theorems which were originally published as conjectures but which

have since been proved by Fiedler. However, that section also contains a set

of simple isolated identities which still await unification and generalization.

The nature of these identities is comparatively simple and it should not be

difficult to make progress in this field with the aid of a computer.

The following pages contain several other conjectures which await proof

or refutation by analytic methods and further sets of simple isolated iden-

tities which await unification and generalization. Here again the use of a

computer should lead to further progress.

5.8.2 Hankel Determinants with Hessenberg Elements

Define a Hessenberg determinantHn(Section 4.6) as follows:

Hn=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

h 1 h 2 h 3 h 4 ··· hn− 1 hn

1 h 1 h 2 h 3 ··· ··· ···

1 h 1 h 2 ··· ··· ···

1 h 1 ··· ··· ···

··· ··· ··· ···

1 h 1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

,

H 0 =1. (5.8.1)

Conjecture 1.

∣ ∣ ∣ ∣ ∣ ∣ ∣

Hn+r Hn+r+1 ··· H 2 n+r− 1

Hn+r− 1 Hn+r ··· H 2 n+r− 2

··· ··· ··· ···

Hr+1 Hr+2 ··· Hn+r

∣ ∣ ∣ ∣ ∣ ∣ ∣ n

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

hn hn+1 ··· h 2 n+r− 1

hn− 1 hn ··· h 2 n+r− 2

··· ··· ··· ···

h 1 −r h 2 −r ··· hn

∣ ∣ ∣ ∣ ∣ ∣ ∣

n+r

.

h 0 =1,hm=0,m<0.

Both determinants are of Hankel form (Section 4.8) but have been ro-

tated through 90
◦
from their normal orientations. Restoration of normal

orientations introduces negative signs to determinants of orders 4mand

4 m+1,m≥1. Whenr= 0, the identity is unaltered by interchanging

Hsandhs,s=1, 2 , 3 .... The two determinants merely change sides. The