Ramanujan's trigonometric sums and para-orthogonal polynomials on the unit circle

TL;DR

This paper explores the connection between Ramanujan's trigonometric sums and para-orthogonal polynomials on the unit circle, providing explicit formulas for special cases and generalizing to Kronecker polynomials.

Contribution

It introduces explicit expressions for para-orthogonal polynomials related to Ramanujan sums and generalizes the approach using Kronecker polynomials.

Findings

Explicit formulas for special q values like p, 2p, p^k

Extension of the method to Kronecker polynomials

Representation of moments as sums of Ramanujan sums

Abstract

Ramanujan's trigonometric sum $c_q(n)$ can be interpreted as a set of trigonometric moments of a finite measure concentrated at primitive $q$-th roots of unity with equal masses. This gives rise to sets of corresponding polynomials orthogonal on the unit circle. We present explicit expressions of these polynomials for special values of $q$, e.g. when $q=p$ or $q=2p$ or $q=p^k$, where $p$ is a prime number. We generalize this procedure taking the Kronecker polynomial instead of cyclotomic one. In this case the moments are expressed as finite sums of $c_q(n)$ with different $q$.