Sums of Powers by Matrix Methods

This equation may be viewed as reducing the sum of n terms of the power sequence to a linear combination of the first r + 1 terms. Accordingly, there is an implicit assumption that n > v. Note that the matrix appearing as the middle factor on the right side of this equation is lower triangular. The zeros that should appear above the main diagonal have been omitted. The nonzero entries constitute a version of Pascal's triangle with alternating signs. The scalar equivalent of equation (2) has appeared previously ([4], eq. 57, p. 33) and can be derived by standard elementary manipulations of series expansions for exponential functions. The main virtue of the matrix form is esthetic: it reveals a nice connection between s^ and Pascal's triangle, and is easily remembered. The main idea we wish to present regarding the application of matrix analysis to difference equations may be summarized as follows. In general, an norder difference equation with constant coefficients is expressible as a firstorder vector equation. The solution of this first-order vector equation is given in terms of powers of the coefficient matrix. By reducing the coefficient matrix to its Jordan canonical form, the powers can be explicitly calculated, finally leading to a formula for a solution to the original difference equation. This approach was discussed previously [5] for the case in which the matrix is diagonalizable. In applying this method to the derivation of (2), the matrix is not diagonalizable. Another example with a nondiagonalizable matrix will also be presented, connected with reference [3]. In the interest of completeness, a few results about linear difference equations will be presented. These can also be found in any introductory text on the subject, for example [8].