Problem on existence of limit for special product of sines

Abstract

We consider the problem on the asymptotic behavior of ''``long'' products of sines, the arguments of which are generated by a given infinitely large sequence. We briefly discuss possible branches of the general problem. The main attention is focused on the case, when the considered sequence is a certain subsequence of the sequence of factorials of natural numbers. The scheme of proof of main results is based on the number-theoretic properties of sequences composed of factorials multiplied (or divided) by the Euler number. We employ little-known integral representations for the integer and fractional parts of elements of such sequences. We outline the ideas for developing the proposed approach, and promising aspects are indicated. The research is initiated by a recently arisen question in the theory of boundary value problems on the exact computation of the spectral radius for a parametric family of functional operators.