Resumen de Polynomial interpolation and sums of powers of integers
In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in nof degree k+ 1, Pk(n) and Qk(n), such that Pk(n) = Qk(n) = fk(n) for n= 1, 2,…?, k, where fk(1), fk(2),…?, fk(k) are karbitrarily chosen (real or complex) values. Then, we focus on the case that fk(n) is given by the sum of powers of the first npositive integers Sk(n) = 1k+ 2k+ ··· + nk, and show that Sk(n) admits the polynomial representations Sk(n) = Pk(n) and Sk(n) = Qk(n) for all n= 1, 2,…?, and k= 1, where the first representation involves the Eulerian numbers, and the second one the Stirling numbers of the second kind. Finally, we consider yet another polynomial formula for Sk(n) alternative to the well-known formula of Bernoulli.
