Polynomial interpolation and sums of powers of integers |
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Authors: | José Luis Cereceda |
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Institution: | 1. Telefónica de Espa?a, Distrito Telefónica, Edificio Este 1, Madrid 28050, Spainjl.cereceda@movistar.es |
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Abstract: | In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of 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 k arbitrarily chosen (real or complex) values. Then, we focus on the case that fk(n) is given by the sum of powers of the first n positive 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. |
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Keywords: | Polynomial interpolation sums of powers of integers Eulerian numbers Stirling numbers of the second kind D-numbers recurrence relation |
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