Identities and inequalities for the cosine and sine functions

Evaluation of some non-elementary integrals involving sine, cosine, exponential and logarithmic integrals: Part II

The non-elementary integrals $\mbox{Si}_{β,α}=\int [\sin{(λx^β)}/(λx^α)] dx,β\ge1,α>β+1$ and $\mbox{Ci}_{β,α}=\int [\cos{(λx^β)}/(λx^α)] dx, β\ge1, α>2β+1$, where $\{β,α\}\in\mathbb{R}$, are evaluated in terms of the hypergeometric function $_{2}F_3$. On the other hand, the exponential integral $\mbox{Ei}_{β,α}=\int (e^{λx^β}/x^α) dx, β\ge1, α>β+1$ is expressed in terms of $_{2}F_2$. The method used to evaluate these integrals consists of expanding the integrand as a Taylor series and integrating the series term by term.

Eigenfunctions of the Cosine and Sine Transforms

A description of eigensubspaces of the cosine and sine operators is presented. The spectrum of each of these two operator consists of two eigenvalues (1,\,-1) and their eigensubspaces are infinite--dimensional. There are many possible bases for these subspaces, but most popular are bases constructed from the Hermite functions. We present other "bases" which are not discrete orthogonal sequences of vectors, but continuous orthogonal chains of vectors. Our work can be considered a continuation and further development of results in \textit{Self-reciprocal functions} by Hardy and Titchmarsh: Quart…

General observations on series whose terms proceed as the sines and cosines of multiples of angles

Translated from the Latin original, "Observationes generales circa series, quarum termini secundum sinus vel cosinus angulorum multiplorum progrediuntur" (1777). E655 in the Enestrom index. Euler looks at the binomial expansion $(1+x)^n$ for $x=\cos ϕ+i\sinϕ$. He does $n=$ positive and negative integers, and $n={1/2}, -{1/2}$, and particular values of $ϕ$.

正弦和余弦曲线幂次之间的渐近面积

Asymptotic areas between powers of sine and cosine curves

该研究受Dombrowski和Dresden工作启发，通过明确确定被积函数零点去掉绝对值，利用渐近分析计算伸缩三角和极限，求出特定积分极限精确值并给出相关递归公式。

On Fourier re-expansions

We study an extension to Fourier transforms of the old problem on absolute convergence of the re-expansion in the sine (cosine) Fourier series of an absolutely convergent cosine (sine) Fourier series. The results are obtained by revealing certain relations between the Fourier transforms and their Hilbert transforms.

A New Family of Nonnegative Sine Polynomials

We present a new family of sine polynomials that are nonnegative for all $x$ in $[0,π]$. We also characterize all nonnegative sine polynomials of degree 3 and all nonnegative cosine polynomials of degree 2. In the latest version, typos in (1.4) and (1.6) are corrected (with >= replaced by <=).

与傅里叶余弦-正弦变换和Kontorovich-Lebedev变换相关的三角加权广义卷积算子

Trigonometric weighted generalized convolution operator associated with Fourier cosine-sine and Kontorovich-Lebedev transformations

本研究引入带三角权重γ=sin y、关联傅里叶余弦-正弦与Kontorovich-Lebedev变换的广义卷积算子，证明其在双参数Lebesgue空间的有界性，给出加权L_p范数估计并得到相关卷积积分微分方程的L_1可解条件。

Estimates for the norms of products of sines and cosines

In this paper we prove asymptotic formulas for the $L^p$ norms of $P_n(θ)=\prod_{k=1}^n (1-e^{ikθ})$ and $Q_n(θ)=\prod_{k=1}^n (1+e^{ikθ})$. These products can be expressed using $\prod_{k=1}^n \sin\Big(\frac{kθ}{2}\Big)$ and $\prod_{k=1}^n \cos\Big(\frac{kθ}{2}\Big)$ respectively. We prove an estimate for $P_n$ at a point near where its maximum occurs. Finally, we give an asymptotic formula for the maximum of the Fourier coefficients of $Q_n$.

The sine kernel, two corresponding operator identities, and random matrices

In the present paper, we consider the integral operator, which acts in Hilbert space and has sine kernel. This operator generates two operator identities and two corresponding canonical differential systems. We find the asymptotics of the corresponding resolvent and Hamiltonians. We use both the method of operator identities and the theory of random matrices.

Generalised definitions of certain functions and their uses

Generalised definitions of exponential, trigonometric sine and cosine and hyperbolic sine and cosine functions are given. In the lowest order, these functions correspond to ordinary exponential, trigonometric sine etc. Some of the properties of the generalised functions are discussed. Importance of these functions and their possible applications are also considered.

Evaluation of some non-elementary integrals involving sine, cosine, exponential and logarithmic integrals: Part I

The non-elementary integrals $\text{Si}_{β,α}=\int [\sin{(λx^β)}/(λx^α)] dx,β\ge1,α\leβ+1$ and $\text{Ci}_{β,α}=\int [\cos{(λx^β)}/(λx^α)] dx, β\ge1, α\le2β+1$, where $\{β,α\}\in\mathbb{R}$, are evaluated in terms of the hypergeometric functions $_{1}F_2$ and $_{2}F_3$, and their asymptotic expressions for $|x|\gg1$ are also derived. The integrals of the form $\int [\sin^n{(λx^β)}/(λx^α)] dx$ and $\int [\cos^n{(λx^β)}/(λx^α)] dx$, where $n$ is a positive integer, are expressed in terms $\text{Si}_{β,α}$ and $\text{Ci}_{β,α}$, and then evaluated. $\text{Si}_{β,α}$ and $\text{Ci}_{β,α}$ are also…

Taylor's series expansions for real powers of functions containing squares of inverse (hyperbolic) cosine functions, explicit formulas for special partial Bell polynomials, and series representations for powers of circular constant

In the paper, by virtue of expansions of two finite products of finitely many square sums, with the aid of series expansions of composite functions of (hyperbolic) sine and cosine functions with inverse sine and cosine functions, and in the light of properties of partial Bell polynomials, the author establishes Taylor's series expansions of real powers of two functions containing squares of inverse (hyperbolic) cosine functions in terms of the Stirling numbers of the first kind, presents an explicit formula of specific partial Bell polynomials at a sequence of derivatives of a function contain…

(Anti)symmetric multivariate trigonometric functions and corresponding Fourier transforms

Four families of special functions, depending on n variables, are studied. We call them symmetric and antisymmetric multivariate sine and cosine functions. They are given as determinants or antideterminants of matrices, whose matrix elements are sine or cosine functions of one variable each. These functions are eigenfunctions of the Laplace operator, satisfying specific conditions at the boundary of a certain domain F of the n-dimensional Euclidean space. Discrete and continuous orthogonality on F of the functions within each family, allows one to introduce symmetrized and antisymmetrized mult…

多重余弦函数与正弦函数特殊值的逼近

Approximations by special values of multiple cosine and sine functions

该研究证明多重余弦、正弦函数的特殊值集合的有理系数线性组合可强逼近实数，且Dirichlet eta与beta函数的特殊值集合也有类似逼近性质，方法受三角积分相关研究启发。

Nonnegative Trigonometric Polynomials and Sturms Theorem

In an earlier article [3], we presented an algorithm that can be used to rigorously check whether a specific cosine or sine polynomial is nonnegative in a given interval or not. The algorithm proves to be an indispensable tool in establishing some recent results on nonnegative trigonometric polynomials. See, for example, [2], [4] and [5]. It continues to play an essential role in several ongoing projects. The algorithm, however, cannot handle general trigonometric polynomials that involve both cosine and sine terms. Some ad hoc methods to deal with such polynomials have been suggested in [3], …

Derivatives of tangent function and tangent numbers

In the paper, by induction, the Faà di Bruno formula, and some techniques in the theory of complex functions, the author finds explicit formulas for higher order derivatives of the tangent and cotangent functions as well as powers of the sine and cosine functions, obtains explicit formulas for two Bell polynomials of the second kind for successive derivatives of sine and cosine functions, presents curious identities for the sine function, discovers explicit formulas and recurrence relations for the tangent numbers, the Bernoulli numbers, the Genocchi numbers, special values of the Euler polyno…

Parametric kinds of generalized Apostol-Bernoulli polynomials and their properties

The purpose of this paper is to define generalized Apostol--Bernoulli polynomials with including a new cosine and sine parametric type of generating function using the quasi-monomiality properties and trigonometric functions. In this study, the Apostol-Bernoulli polynomials with three variable are defined with two new generating functions cosine and sine parameters. Then, we investigate multiplicative and derivative operators, diffrential equations, some summation formulas and partial differential equations for these polynomials. Moreover, we introduce Gould--Hopper--Apostol--Bernoulli type po…

关于退化三角函数的若干恒等式

Some identities on degenerate trigonometric functions

该研究以初等方法探讨退化三角函数，推导函数间恒等式，并给出退化余切与退化正弦的倍角公式。

Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring

The fundamental set of solutions of the generalized hypergeometric differential equation in the neighborhood of unity has been built by Nørlund in 1955. The behavior of the generalized hypergeometric function in the neighborhood of unity has been described in the beginning of 1990s by Bühring, Srivastava and Saigo. In the first part of this paper we review their results rewriting them in terms of Meijer's $G$-function and explaining the interconnections between them. In the second part we present new formulas and identities for the coefficients that appear in the expansions of Meijer's $G$-fun…

On d-dimensional d-Semimetrics and Simplex-Type Inequalities for High-Dimensional Sine Functions

We show that high-dimensional analogues of the sine function (more precisely, the d-dimensional polar sine and the d-th root of the d-dimensional hypersine) satisfy a simplex-type inequality in a real pre-Hilbert space H. Adopting the language of Deza and Rosenberg, we say that these d-dimensional sine functions are d-semimetrics. We also establish geometric identities for both the d-dimensional polar sine and the d-dimensional hypersine. We then show that when d=1 the underlying functional equation of the corresponding identity characterizes a generalized sine function. Finally, we show that …

Which Urbanik class $L_k$, do the hyperbolic and the generalized logistic characteristic functions belong to?

Selfdecomposable variables obtained from series of Laplace (double exponential) variables are objects of this study. We proved that hyperbolic-sine and hyperbolic-cosine variables are in the difference of the Urbanik classes $L_2$ and $L_3$ while generalized logistic variable is at least in the Urbanik class $L_1$. Hence some ratios of those corresponding selfdecomposable characteristic functions are again selfdecomposable.

A generalization of the Shafer-Fink inequality

In this article we show a tecnique based on the Weierstrass product for the sine and cosine function and the bisection formula for the cotangent function that leads to a generalization of the classical Shafer-Fink inequality $ \frac{3 x}{1+2\sqrt{1+x^2}} < \arctan x < \frac{πx}{1+2\sqrt{1+x^2}}$. Other algebraic approximations are also shown, including one that follows from the Chebyshev expansion for the arctangent function in the interval $[-1,1]$.