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

TL;DR

This work generalizes Watson's trigonometric sum by studying $C_n(\nu)=\sum_{l=1}^{n-1} \cos\frac{2\pi\nu l}{n}\csc\frac{\pi l}{n}$ and its special case $C_n$, situating it as a half-order Dowker sum. The authors develop integral and series representations, derive rich structural properties, and establish two complementary large-$n$ asymptotic regimes, including a leading term $-\frac{2n}{\pi}\ln\bigl(2\sin(\frac{\pi\nu}{n})\bigr)$. They provide sharp bounds, accurate approximations, and multiple digamma/gamma-related summation identities, revealing deep connections to Bernoulli polynomials of higher order and Fourier-type transforms. The results furnish practical tools for applications in number theory and physics where such cosecant-weighted sums arise, and they extend the understanding of Dowker-type sums by detailing both analytic and asymptotic structures of $C_n(\nu)$ and $C_n$. Overall, the paper delivers a comprehensive treatment of $C_n(\nu)$ with broad theoretical and applied relevance, linking cotangent derivatives, digamma functions, and discrete transforms in a cohesive framework.

Abstract

In this paper we study the finite trigonometric sum $\sum a_l\csc\big(πl/n\big)$, where $a_l$ are equal to $\cos(2πl ν/n)$ and where the summation index $l$ and the discrete parameter $ν$ 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 $\prod\big(\csc(πl/n)\big)^{\csc(πl/n)}$

Figures (6)

• Figure 1: The sum $C_{n}(\nu)$ as a function of $n$, where $n\in[2,300]$, for two different values of argument: $\nu=16$ (solid line) and $\nu=73$ (dash--dotted line). We deliberately take values of $\nu$, which lie outside the interval defined earlier, in order to observe the overall behaviour of $C_{n}(\nu)$.
• Figure 2: The sum $C_{300}(\nu)$ as a function of $\nu$, where $\nu\in[0,300]$. We clearly observe that it verifies the basic property $C_{n}(\nu)=C_{n}(n-\nu)$, see \ref{['39ufif']}, and that it may take both positive and negative values. It is also visible that the minimum of $\left|C_{n}(\nu)\right|$ occurs at $\nu=50=\frac{1}{6}n$ and $\nu=250=\frac{5}{6}n$. More generally, empirical studies show that at such points $C_{n}(\nu)$ is always very small, never equal to zero, but tends to zero when $n\to\infty$, $n$ being a multiple of 6 (see also Corollary \ref{['lpo3sai']} in Sect. \ref{['2093u0j']}).
• Figure 3: The difference between $C_n(\nu)$ and its approximation, provided by Theorem \ref{['lk7d3mf4']}, as a function of $n$ for $\nu=7$ (dashed line), $\nu=10$ (solid line) and $\nu=16$ (dash--dotted line). Note that the greater the argument $\nu$, the better the approximation [roughly the approximation error is proportional to $\nu^{-6}$]. One should also bear in mind that $C_{100}(7)\approx53$, $C_{100}(10)\approx31$ and $C_{100}(16)\approx2$, so that in all these cases the approximation is rather accurate (and it remains such at least for moderate values of $n$, e.g. for $n=10\,000$ and $\nu=7$ the approximation error is about $-2\times10^{-4}$, while $C_{10\,000}(7)\approx34\,541$).
• Figure 4: The approximation error for $C_{300}(\nu)$ as a function of $\nu$, where $\nu\in[10,270]$ (approximation given by Theorem \ref{['lk7d3mf4']}). For the sake of comparison: $C_{300}(10)\approx+299$, $C_{300}(20)=C_{300}(280)\approx+168$, $C_{300}(50)=C_{300}(250)\approx-3\times10^{-3}$, $C_{300}(100)=C_{300}(200)\approx-105$, $C_{300}(150)\approx-132$ (see Fig. \ref{['g6hytbhfw2']} for the graph of $C_{300}(\nu)$).
• Figure 5: The difference between $C_n(\nu)$ and its approximation, provided by the asymptotics from Theorem \ref{['lk7d3mf5']}, as a function of $n$ for $\nu=7$ (dashed line), $\nu=10$ (solid line) and $\nu=16$ (dash--dotted line). It is clearly visible that the approximation error quickly tends to zero and that it is very small (compare this graph to Fig. \ref{['937fybfe4']}).

Theorems & Definitions (21)

• Theorem 1: Integral representations for $\bm{C_n(\nu)}$
• proof
• Theorem 2: Finite series representations for $\bm{C_n(\nu)}$
• proof
• Lemma 1
• proof
• Theorem 3: Infinite series representations for $\bm{C_n(\nu)}$
• proof
• Theorem 4: Almost asymptotic expansion of $\bm{C_n(\nu)}$
• Corollary 1: Asymptotic representation of $\bm{C_n(\nu)}$ at large $\bm{n}$