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Combinatorial Sums and Finite Differences

By Michael Z. Spivey


We present a new approach to evaluating combinatorial sums by using finite differences. Let {ak} ∞ k=0 and {bk} ∞ k=0 be sequences with the property that ∆bk = ak for k ≥ 0. Let gn = �n � � n k=0 k ak, and let hn = �n � � n k=0 k bk. We derive expressions for gn in terms of hn and for hn in terms of gn. We then extend our approach to handle binomial sums of the form �n � � n k=0 k (−1) kak, � � � n � � � k 2k ak, and n k 2k+1 ak, as well as sums involving unsigned and signed Stirling numbers of the first kind, �n � � n k=0 k ak and �n k=0 s(n, k)ak. For each type of sum we illustrate our methods by deriving an expression for the power sum, with ak = k m, and the harmonic number sum, with ak = Hk = 1 + 1/2 + · · · + 1/k. Then we generalize our approach to a class of numbers satisfying a particular type of recurrence relation. This class includes the binomial coefficients and the unsigned Stirling numbers of the first kind

Topics: Binomial coefficient, binomial sum, combinatorial sum, finite difference, Stirling number 2
Year: 2009
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