On a generalization of Watson's trigonometric sum (on Dowker's sum of order one half)

TL;DR

This paper generalizes Watson's trigonometric sum involving cosecant and cosine functions, providing integral, series, asymptotic representations, bounds, and approximations, with applications in mathematics, physics, and engineering.

Contribution

It introduces new integral and series representations, bounds, and approximations for the generalized Dowker sum, extending previous work on Watson's sum and exploring its properties.

Findings

Derived integral and series representations of the sum

Established asymptotic expansions and bounds

Proposed simple accurate approximation formulas

Abstract

In this paper we study the finite trigonometric sum $\sum a_l\csc\big(\pi l/n\big)$, where $a_l$ are equal to $\cos(2\pi l \nu/n)$ and where the summation index $l$ and the discrete parameter $\nu$ both run through $1$ to $n-1$. This sum is a generalization of Watson's trigonometric sum, which has been extensively studied in a series of previous papers, and also may be regarded as the so-called Dowker sum of order one half. It occurs in various problems in mathematics, physics and engineering, and plays an important role in some number-theoretic problems. In the paper, we obtain several integral and series representations for the above-mentioned sum, investigate its properties, derive various, including asymptotic, expansions for it, and deduce very accurate upper and lower bounds for it (both bounds are asymptotically vanishing). In addition, we obtain two relatively simple approximate formulae containing only several terms, which are also very accurate and can be particularly appreciated in applications. Finally, we also derive several advanced summation formulae for the digamma functions, which relate the gamma and the digamma functions, the investigated sum, as well as the product of a sequence of cosecants $\prod\big(\csc(\pi l/n)\big)^{\csc(\pi l/n)}$