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

2021/07/27 by Alexei Zhedanov, Zhedanov, Alexei
Mathematics · #11Z05 #33C47 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2107.12543

openalex publication_date 2021/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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