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

DOI10.1007/S11139-022-00576-2arXiv2107.12543MaRDI QIDQ6373832

Publication date: 26 July 2021

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 = 2 p

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

.

Mathematics Subject Classification ID

Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis (42C05) Polynomials in number theory (11C08) Other special orthogonal polynomials and functions (33C47)

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