Combinatorial sums and finite differences

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 k=0 ( n k ) ak, and let hn = ∑n k=0 ( n 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 k=0 ( n k ) (−1)ak, ∑ k ( n 2k ) ak, and ∑ k ( n 2k+1 ) ak, as well as sums involving unsigned and signed Stirling numbers of the first kind, ∑n k=0 [ n 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 = km, 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.