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Approximations by special values of multiple cosine and sine functions

Su Hu, Min-Soo Kim

TL;DR

The paper studies how real numbers can be strongly approximated by special values of the generalized cosine and sine functions, $\mathcal{C}_r(x)$ and $\mathcal{S}_r(x)$, via the sets $B$, $C$, and $D$ built from $\log\mathcal{C}_r(x)$, $\log\mathcal{S}_r(x)$, and Dirichlet $\eta$/$\beta$ values. It establishes key integral identities linking these special values to trigonometric integrals: $\int_{0}^{x} t^{r}\tan(\pi t) dt = -\frac{\log \mathcal{C}_{r+1}(x)}{\pi}$ and $\int_{0}^{x} t^{r}\cot\left(\frac{\pi t}{2}\right) dt = \frac{2^{r}}{\pi} \log S_{r}\left(\frac{x}{2}\right)$, then builds approximation schemes using bounded linear functionals and Jackson-type rational polynomial approximations. The main contributions are (i) strong approximation of real numbers by rational-linear combinations of $\frac{\log \mathcal{C}_r(x)}{\pi}$ and $\frac{\log \mathcal{S}_r(x)}{\pi}$, (ii) a parallel approximation property for the Dirichlet eta and beta values in $D$ with coefficients derived from derivatives of rational polynomials, and (iii) explicit integral representations that generalize prior results of Alkan and Lupu-Wu, highlighting deep connections between generalized trigonometric functions and zeta/beta functions.

Abstract

Kurokawa and Koyama's multiple cosine function $\mathcal{C}_{r}(x)$ and Kurokawa's multiple sine function $S_{r}(x)$ are generalizations of the classical cosine and sine functions from their infinite product representations, respectively. For any fixed $x\in[0,\frac{1}{2})$, let $$B=\left\{\frac{\log\mathcal{C}_{r}(x)}π~~\bigg|~~r=1,2,3,\ldots\right\}$$ and $$C=\left\{\frac{\log S_r(x)}π~~\bigg|~~r=1,2,3,\ldots\right\}$$ be the sets of special values of $\mathcal{C}_{r}(x)$ and $S_{r}(x)$ at $x$, respectively. In this paper, we will show that the real numbers can be strongly approximated by linear combinations of elements in $B$ and $C$ respectively, with rational coefficients. Furthermore, let $$D=\left\{\frac{ζ_{E}(3)}{π^2},\frac{ζ_{E}(5)}{π^4}, \ldots, \frac{ζ_{E}(2k+1)}{π^{2k}},\ldots; \frac{β(4)}{π^3},\frac{β(6)}{π^5}, \ldots, \frac{β(2k+2)}{π^{2k+1}},\ldots\right\}$$ be the set of special values of Dirichlet's eta and beta functions. We will prove that the set $D$ has a similar approximation property, where the coefficients are values of the derivatives of rational polynomials. Our approaches are inspired by recent works of Alkan (Proc. Amer. Math. Soc. 143: 3743--3752, 2015) and Lupu-Wu (J. Math. Anal. Appl. 545: Article ID 129144, 2025) as applications of the trigonometric integrals.

Approximations by special values of multiple cosine and sine functions

TL;DR

The paper studies how real numbers can be strongly approximated by special values of the generalized cosine and sine functions, and , via the sets , , and built from , , and Dirichlet / values. It establishes key integral identities linking these special values to trigonometric integrals: and , then builds approximation schemes using bounded linear functionals and Jackson-type rational polynomial approximations. The main contributions are (i) strong approximation of real numbers by rational-linear combinations of and , (ii) a parallel approximation property for the Dirichlet eta and beta values in with coefficients derived from derivatives of rational polynomials, and (iii) explicit integral representations that generalize prior results of Alkan and Lupu-Wu, highlighting deep connections between generalized trigonometric functions and zeta/beta functions.

Abstract

Kurokawa and Koyama's multiple cosine function and Kurokawa's multiple sine function are generalizations of the classical cosine and sine functions from their infinite product representations, respectively. For any fixed , let and be the sets of special values of and at , respectively. In this paper, we will show that the real numbers can be strongly approximated by linear combinations of elements in and respectively, with rational coefficients. Furthermore, let be the set of special values of Dirichlet's eta and beta functions. We will prove that the set has a similar approximation property, where the coefficients are values of the derivatives of rational polynomials. Our approaches are inspired by recent works of Alkan (Proc. Amer. Math. Soc. 143: 3743--3752, 2015) and Lupu-Wu (J. Math. Anal. Appl. 545: Article ID 129144, 2025) as applications of the trigonometric integrals.
Paper Structure (4 sections, 10 theorems, 84 equations)

This paper contains 4 sections, 10 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. The integral $\int_0^x t^{r}\tan(\pi t)dt$
  3. The integral $\int_0^1 t^{r}\log\left(\cos\frac{\pi t}{4}\right)dt$
  4. The integral $\int_0^x t^{r}\cot\left(\frac{\pi t}{2}\right)dt$

Key Result

Theorem 1.1

Let $k_{0}$ and $q$ be positive integers, let $\alpha$ be a real number. For any given integer $n\geq 3$, there exists an integer $r\leq n$ and a rational polynomial $P_{n}(t)=t^{2k_{0}}(1-t)^{2k_{0}}s_{r}(t)$ with degree $4k_{0}+r$ depending on $\alpha,n,q$ and $k_{0}$, satisfying when $r\geq 2$, where for all $k_{0}\leq k \leq 2k_{0}+\lfloor \frac{r}{2}\rfloor$.

Theorems & Definitions (15)

  • Theorem 1.1: LW
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 5 more