This article is cited in 1 paper

Asymptotic behavior of “long” products of sines and the Pisot numbers

E. D. Alferova^ab, V. E. Podolskii^ab, V. B. Sherstyukov<sup>ab

a</sup> Lomonosov Moscow State University
^b Moscow Center for Fundamental and Applied Mathematics

Abstract: We study a problem concerning the exact calculation of the limit of a special sequence of trigonometric functions depending on a real parameter. The problem has recently arisen in the theory of functional-differential operators with affine transformations of the argument, and its complete solution is unknown. By a new elementary method, we find the exact value of the limit for the case in which the parameter is chosen from a certain family of quadratic irrationalities that are Pisot numbers. Estimates are given for the rate of convergence of the corresponding sequence to its limit. For individual parameter values not included in the main family, previously known estimates on the value of the desired limit are improved.

Keywords: product of sines, asymptotic formula, quadratic irrationality, Pisot number, silver ratio, golden ratio, Lucas sequence.

UDC: 517.521+511.2

Received: 15.09.2024

DOI: 10.4213/mzm14427

 English version: Mathematical Notes, 2025, 117:1, 14–27

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