Tangent power sums and their applications

TL;DR

This paper investigates tangent power sums, proving their integrality and polynomial nature in relation to parameters, and explores their connections with digit sums in specific bases, providing formulas and asymptotic behaviors.

Contribution

It establishes the integrality and polynomial structure of tangent power sums, introduces formulas, and links these sums to digit sums in base 2m.

Findings

Tangent power sums are integers for all m, p.

These sums form polynomials in m of degree 2p for fixed p.

Connections are made between tangent sums and digit sums in base 2m.

Abstract

For integer $m, p,$ we study tangent power sum $\sum^m_{k=1}\tan^{2p}\frac{\pi k}{2m+1}.$ We prove that, for every $m, p,$ it is integer, and, for a fixed p, it is a polynomial in $m$ of degree $2p.$ We give recurrent, asymptotical and explicit formulas for these polynomials and indicate their connections with Newman's digit sums in base $2m.$