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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
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