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arXiv · 2107.12543

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

Alexei Zhedanov
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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 $q$.

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Alexei Zhedanov. 2021-07-27. Ramanujan's trigonometric sums and para-orthogonal polynomials on the unit circle. https://arxiv.org/abs/2107.12543

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