de of Table C-1, based on the CRC Hand- book of Tables for Probability and Statistis1, depicts such formulae for pow- ers up to 10. These are usually derived directly from the fundamental theorem of sum calculus, or via the Bernoulli polynomials as Formulae for the case where the increment is not one cannot apparently be found explicitly in the literature: [4]-[10]. Conceptually, these formulae are simple to obtain; however, the algebraic manipulations required tend to be overwhelming. In this note we present the first 10 formulae, as obtained on a computer by formal string manipulations. 2. Approach. Let where q=(m - 1)/d is an integer, d > 0. We desire a formal closed-form expression for Sm(1,d,m). Clearly s(1,d,m(( md(la -+ Copyright 2002 The McGraw-Hill Companies, Inc. Click Here for Terms of Use. Appendix C Table C-1 Inductive Formulae for General Sum Operations s°(1,1,m)="m - + 2) + + + - + 9m - + + - + + 6m - 3) s10(1, 1,m)(m 1)(2m 1)(3m8 + 12m7 + 8m6 - 18m5 - 10m4 + 24m3 + 2m2 - 15m + 5) The individual terms in the second expression are indeed the entries of the left side of Table 1. The only remaining task is obtaining a formal expres- sion for the first summation by collecting appropriate terms; this is a rather long and tedious task, particularly for high values of N. The algebraic manipulations have been carried out by a computer pro- gram. CPU time on a dedicated DEC PDP 11/70 was 2 hours; the code con- sisted of about 400 statements. The results are depicted on the right-hand side of Table C-1. We now have closed-form expressions for summations such as (1 + jVA)2 or (1 + jui)3. 3. Related Facts. Fact 1. Besides brute force computation, the results of Table C-1 may be proved by induction on q, for a fixed d, and N. The following facts can also be proved. Fact 2. For all N, d, Appendix C (1,d,m)m1mm1 7d 8d + + 11 9d 1) + 1 1 2 + 1 1m + - which is exact for n = 1. Fact 3. For sufficiently small d, f(x) Riemann integrable, m = 1 mod d, there exists a 8 such that In particular, if f(x) = xn Appendix C Th