6 j符号的初等方法（经典，量子，有理，三角和椭圆）

Jacob's ladders, interactions between $ζ$-oscillating systems and $ζ$-analogue of an elementary trigonometric identity

In our previous papers, we have introduced within the theory of the Riemann zeta function the following notions: Jacob's ladders, oscillating systems, $ζ$-factorization, metamorphoses, \dots In this paper we obtain $ζ$-analogue of an elementary trigonometric identity and other interactions between oscillating systems.

An elementary proof of the positivity of the intertwining operator in one-dimensional trigonometric Dunkl theory

This note is devoted to the intertwining operator in the one--dimensional trigonometric Dunkl setting. We obtain a simple integral expression of this operator and deduce its positivity.

Nonnegative Trigonometric Polynomials, Sturms Theorem, and Symbolic Computation

In this paper, we explain a procedure based on a classical result of Sturm that can be used to determine rigorously whether a given trigonometric polynomial is nonnegative in a certain interval or not. Many examples are given. This technique has been employed by the author in several recent works. The procedure often involves tedious computations that are time-consuming and error-prone. Fortunately, symbolic computation software is available to automate the procedure. In this paper, we give the details of its implementation in MAPLE 13. Some who are strongly attached to a more traditional theo…

On generalization of D'Aurizio-Sándor trigonometric inequalities with a parameter

In this work, we generalize the D'Aurizio-Sándor inequalities (\cite{D'Aurizio,Sandor}) using an elementary approach. In particular, our approach provides an alternative proof of the D'Aurizio-Sándor inequalities. Moreover, as an immediate consequence of the generalized D'Aurizio-Sándor inequalities, we establish the D'Aurizio-Sándor-type inequalities for hyperbolic functions.

三元集的Sidon常数

The Sidon constant of sets with three elements

本文解决了三角多项式的一个初等极值问题，得到了三元集{n 0，n1，n2}的Sidon常数的精确值：当n = max（n1-n 0，n2-n 0，n2-n1）/ gcd（n1-n 0，n2-n 0）时，Sidon常数为\sec（π/ 2n）.

A new form of the generalized complete elliptic integrals

Generalized trigonometric functions are applied to the Legendre-Jacobi standard form of complete elliptic integrals, and a new form of the generalized complete elliptic integrals of the Borweins is presented. According to the form, it can be easily shown that these integrals have similar properties to the classical ones. In particular, it is possible to establish a computation formula of the generalized $π$ in terms of the arithmetic-geometric mean, in the classical way as the Gauss-Legendre algorithm for $π$ by Salamin and Brent. Moreover, an elementary new proof of Ramanujan's cubic transfor…