Sums of non-integral powers

Sums of the Even Integral Powers ... Sums of the Even Integral Powers of the Cosecant and Secant

Special finite sums of the even powers of the cosecant and of the secant are studied, ∑ k csc (kπ/N) and ∑ k sec (kπ/N), with positive integers N ≥ 3,m and 1 ≤ k < N/2 . The main result of this article is that these power sums are even polynomials in N , of order 2m, whose coefficients are rational. The approach is based on new differential identities for the functions cscz and secz. The Mittag...

Sums of dth Powers in Non-commutative Rings

Sums of dth powers in central simple algebras and other non-commutative rings are investigated. MSC 2000: 11E76 (primary); 11E04, 12E05 (secondary)

Sums of powers via integration

Sum of powers 1 + · · · + n, with n, p ∈ N and n ≥ 1, can be expressed as a polynomial function of n of degree p + 1. Such representations are often called Faulhaber formulae. A simple recursive algorithm for computing coefficients of Faulhaber formulae is presented. The correctness of the algorithm is proved by giving a recurrence relation on Faulhaber formulae.

Exponential Sums modulo Prime Powers

where p is a prime power with m ≥ 2, χ is a multiplicative character (mod p), epm(·) is the additive character, epm(x) = e 2πix pm , and f, g are rational functions with integer coefficients. It is understood, that the sum is only over values of x for which g and f and both defined as functions on Z/(p), and g is nonzero (mod p). The sum is trivial if f and g are both constants, so we shall alw...

On sums which are powers