Combinatorial sums and finite differences

Abstract

We present a new approach to evaluating combinatorial sums by using finite differences. Let { a k } k = 0 ∞ and { b k } k = 0 ∞ be sequences with the property that Δ b k = a k for k ⩾ 0 . Let g n = ∑ k = 0 n n k a k , and let h n = ∑ k = 0 n n k b k . We derive expressions for g n in terms of h n and for h n in terms of g n . We then extend our approach to handle binomial sums of the form ∑ k = 0 n n k ( - 1 ) k a k , ∑ k n 2 k a k , and ∑ k n 2 k + 1 a k , as well as sums involving unsigned and signed Stirling numbers of the first kind, ∑ k = 0 n n k a k and ∑ k = 0 n s ( n , k ) a k . For each type of sum we illustrate our methods by deriving an expression for the power sum, with a k = k m , and the harmonic number sum, with a k = H k = 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.