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172   5. Further Determinant Theory

          5.1.2  The Generalized Geometric Series and Eulerian
                 Polynomials

          Notes on the generalized geometric series ψ n (x) and the Eulerian
          polynomials A n (x) are given in Appendix A.6.

                               A n (x)=(1 − x) n+1 ψ n (x).          (5.1.2)
          Theorem (Lawden).

                          1        1 − x


                        1/2!        1      1 − x


                        1/3!       1/2!      1    1 − x

                                                                     .

               A n
               n! x  =     ....................................
                       1/(n − 1)! 1/(n − 2)!  ···         1   1 − x

                        1/n!    1/(n − 1)!  ···         1/2!    1

                                                                   n
          The determinant is a Hessenbergian.
          Proof. It is proved in the section on differences (Appendix A.8) that

                                  m
                                              m

                         ∆ ψ 0 =    (−1) m−s      ψ s = xψ m .       (5.1.3)
                           m
                                              s
                                 s=0
          Put
                                   ψ s =(−1) s! φ s .                (5.1.4)
                                            s
          Then,
                  m−1

                              +(1 − x)φ m =0,  m =1, 2, 3,... .      (5.1.5)
                        φ s
                      (m − s)!
                  s=0
          In some detail,
           φ 0  +(1 − x)φ 1                                     =0,
           φ 0 /2! + φ 1     +(1 − x)φ 2                        =0,
                                                                     (5.1.6)
           φ 0 /3! + φ 1 /2  + φ 2            +(1 − x)φ 3       =0,
           ...........................................................
           φ 0 /n!+ φ 1 /(n − 1)! + φ 2 /(n − 2)! + ··· + φ n−1 +(1 − x)φ n =0.
          When these n equations in the (n + 1) variables φ r ,0 ≤ r ≤ n, are
          augmented by the relation
                                    (1 − x)φ 0 = x,                  (5.1.7)

          the determinant of the coefficients is triangular so that its value is
          (1 − x) n+1 . Solving the (n + 1) equations by Cramer’s formula (Sec-
          tion 2.3.5),
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