Faulhaber’s Theorem for Arithmetic Progressions
نویسندگان
چکیده
Abstract. We show that the classical Faulhaber’s theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression a + b, a + 2b, . . . , a + nb is a polynomial in na+ n(n+ 1)b/2. The coefficients of these polynomials are given in terms of the Bernoulli polynomials. Following Knuth’s approach by using the central factorial numbers, we obtain formulas for r-fold sums of powers. Expressions for the alternating sums of powers are also presented by using the Bernoulli polynomials and Euler polynomials.
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