5.6 Distinct Matrices with Nondistinct Determinants 213Define a TuranianTnr(Section 4.9.2) as follows:Tnr=∣ ∣ ∣ ∣ ∣ ∣ ∣
φr− 2 n+2 ... φr−n+2.
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φr−n+1 ... φr∣ ∣ ∣ ∣ ∣ ∣ ∣ n,r≥ 2 n− 2 , (5.6.7)which is a subdeterminant ofM.
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... 1 ...
... 14 x ...... 13 x 6 x2
...... 12 x 3 x2
4 x3
...... 1 xx2
x3
x4
...... 1 φ 0 φ 1 φ 2 φ 3 φ 4 ...... 1 xφ 1 φ 2 φ 3 φ 4 φ 5 ...... 12 xx2
φ 2 φ 3 φ 4 φ 5 φ 6 ...... 13 x 3 x2
x3
φ 3 φ 4 φ 5 φ 6 φ 7 ...... 14 x 6 x2
4 x3
x4
φ 4 φ 5 φ 6 φ 7 φ 8 ....
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The infinite matrixM
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... 1 ...
... 1 − 4 x ...... 1 − 3 x 6 x2
...... 1 − 2 x 3 x2
− 4 x3
...... 1 −xx2
−x3
x4
...... 1 α 0 α 1 α 2 α 3 α 4 ...... 1 −xα 1 α 2 α 3 α 4 α 5 ...... 1 − 2 xx2
α 2 α 3 α 4 α 5 α 6 ...... 1 − 3 x 3 x2
−x3
α 3 α 4 α 5 α 6 α 7 ...... 1 − 4 x 6 x2
− 4 x3
x4
α 4 α 5 α 6 α 7 α 8 ...
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The infinite matrixM∗The elementαroccurs (r+ 1) times inM
∗. Consider all the subdeter-
minants ofM
∗
which contain the elementαrin the bottom right-hand
corner and whose ordernis sufficiently large for them to contain the el-
ementα 0 but sufficiently small for them not to have either unit or zero
elements along their secondary diagonals. Denote these determinants by
B
nr
s
,s=1, 2 , 3 ,.... Some of them are symmetric and unique whereas oth-
ers occur in pairs, one of which is the transpose of the other. They are