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5.1 Determinants Which Represent Particular Polynomials 171
α 0 α 1 α 2 α 3 ··· α n−1 α n
n x
n − 1 2x
b. ψ n (x)= 1 .
n − 2 3x
n!
.....................
1 nx
n+1
Both determinants are Hessenbergians (Section 4.6).
Proof of (a). Denote the determinant by H n+1 , expand it by the two
elements in the last row, and repeat this operation on the determinants of
lower order which appear. The result is
n
n
H n+1 (x)= H n+1−r (−x) +(−1) α n .
n
r
r
r=1
The H n+1 term can be absorbed into the sum, giving
n
n
(−1) α n = H n+1−r (−x) .
n
r
r
r=0
This is an Appell polynomial whose inverse relation is
n
n
H n+1 (x)= (−1) n−r α n−r x ,
r
r
r=0
which is equivalent to the stated result.
Proof of (b). Denote the determinant by H ∗ and note that some
n+1
of its elements are functions of n, so that the minor obtained by removing
its last row and column is not equal to H and hence there is no obvious
∗
n
recurrence relation linking H ∗ , H , H ∗ , etc.
∗
n+1 n n−1
The determinant H ∗ can be obtained by transforming H n+1 by a series
n+1
of row operations which reduce some of its elements to zero. Multiply R i
by (n+2−i), 2 ≤ i ≤ n+1, and compensate for the unwanted factor n!by
dividing the determinant by that factor. Now perform the row operations
i − 1
R = R i − xR i+1
n +1 − i
i
first with 2 ≤ i ≤ n, which introduces (n − 1) zero elements into C n+1 ,
then with 2 ≤ i ≤ n − 1, which introduces (n − 2) zero elements into C n ,
then with 2 ≤ i ≤ n − 2, etc., and, finally, with i = 2. The determinant
H ∗ appears.
n+1