关于 Euler 反射公式的思考以及 Legendre 加倍公式的一个加法类似物

The Legendre polynomials associated with Bernoulli, Euler, Hermite and Bernstein polynomials

In the present paper, we deal mainly with arithmetic properties of Legendre polynomials by using their orthogonality property. We show that Legendre polynomials are proportional with Bernoulli, Euler, Hermite and Bernstein polynomials.

Legendre Functions, Spherical Rotations, and Multiple Elliptic Integrals

A closed-form formula is derived for the generalized Clebsch-Gordan integral $ \int_{-1}^1 {[}P_ν(x){]}^2P_ν(-x)\D x$, with $ P_ν$ being the Legendre function of arbitrary complex degree $ ν\in\mathbb C$. The finite Hilbert transform of $ P_ν(x)P_ν(-x),-1<x<1$ is evaluated. An analytic proof is provided for a recently conjectured identity $\int_0^1[\mathbf K(\sqrt{1-k^2})]^{3}\D k=6\int_0^1[\mathbf K(k)]^2\mathbf K(\sqrt{1-k^2})k\D k=[Γ(1/4)]^{8}/(128π^2) $ involving complete elliptic integrals of the first kind $ \mathbf K(k)$ and Euler's gamma function $ Γ(z)$.

Closed-form formulae for the derivatives of trigonometric functions at rational multiples of $π$

In this sequel to our recent note it is shown, in a unified manner, by making use of some basic properties of certain special functions, such as the Hurwitz zeta function, Lerch zeta function and Legendre chi function, that the values of all derivatives of four trigonometric functions at rational multiples of $π$ can be expressed in closed form as simple finite sums involving the Bernoulli and Euler polynomials. In addition, some particular cases are considered.

高阶凸函数Bohr-Mollerup定理的推广：一篇导引

A generalization of Bohr-Mollerup's theorem for higher order convex functions: a tutorial

本文以导引形式系统回顾了Bohr-Mollerup定理向高阶凸函数情形的推广理论，展示其解如何继承对数伽马函数的众多经典性质，并给出一个说明性应用。

Mizuno 乘积公式的交替推广与修正 gamma 函数

An alternating Mizuno formula, modified gamma functions, and double sine special values

本文建立了 Mizuno 乘积公式的完整交替推广，引入修正 gamma 函数的交替版本，并推导出 Wallis 乘积、Kurokawa--Wakayama 型公式及多重交替 gamma 函数的性质，最后给出 Shintani 双正弦函数交替模拟的算术二分法。