Sums Of Powers Of Integers And Generalized Stirling Numbers Of The Second Kind

By applying the Newton-Gregory expansion to the polynomial associated with the sum of powers of integers S-k(n) = 1(k) + 2(k) + ... + n(k), we derive a couple of infinite families of explicit formulas for S-k(n). One of the families involves the r-Stirling numbers of the second kind {(k) (j)}(r), j = 0, 1, ... , k, while the other involves their duals {(k) (j)}(-r), with both families of formulas being indexed by the non-negative integer r. As a by-product, we obtain three additional formulas for S-k(n) involving the numbers {(k) (j)}(n+m), {(k) (j)}(n-m), and {(k) (j)}(k-j), where m is any given non-negative integer. Furthermore, we provide several formulas for the Bernoulli polynomials in terms of the generalized Stirling numbers of the second kind, the harmonic numbers, and the so-called harmonic polynomials.