A Faster Product for Pi and a New Integral for ln(Pi/2)

Iterated integrals on products of one variable multiple polylogarithms

In this paper we show that the iterated integrals on products of one variable multiple polylogarithms from 0 to 1 are actually multiple zeta values if they are convergent. In the divergent case, we define regularized iterated integrals from 0 to 1. By the same method, we show that the regularized iterated integrals are also multiple zeta values. As an application, we give new series representations for multiple zeta values and calculate some interesting examples of iterated integrals.

On the integral of the product of four and more Bernoulli polynomials

In 1958, L.J. Mordell provided the formula for the integral of the product of two Bernoulli polynomials, he also remarked: "The integrals containing the product of more than two Bernoulli polynomials do not appear to lead to simple results." In this paper, we provide explicit formulas for the integral of the product of $r$ Bernoulli polynomials, where $r$ is any positive integer. Many authors' results in this direction, including Nörlund, Mordell, Carlitz, Agoh and Dilcher are special cases of the formulas given in this paper.

A study on the q-Euler numbers and the fermionic q-integrals of the product of several type $q$-Bernstein polynomials on Zp

In this paper, we investigate some interesting properties of q-Berstein polynomials realted to q-Euler numbers by using the fermionic q-integral on Zp.

An Infinite Product for e^gamma via Hypergeometric Formulas for Euler's Constant, gamma

We derive hypergeometric formulas for Euler's constant, gamma. A "by-product" of Thomae's transformation is an infinite product for e^gamma involving the binomial coefficients. Alternate, non-hypergeometric proofs use a double integral for gamma, the beta integral, and an integral for the digamma function. (I thank P. Sebah, S. Zlobin, and W. Zudilin for valuable suggestions used in the paper.)

三重积积分与Rankin-Selberg L-函数

Triple product integrals and Rankin-Selberg L-functions

本文证明了一个互反公式，将权0和1/2自守形式的三重积积分谱平均与权0自守形式的经典Rankin-Selberg积分相关联。

Integral elements of K-theory and products of modular curves II

We discuss the relationship between different notions of "integrality" in motivic cohomology/K-theory which arise in the Beilinson and Bloch-Kato conjectures, and prove their equivalence in some cases for products of curves (used in the authors' previous paper in this series), as well as obtaining a general result, first proved by Jannsen (unpublished), which reduces their equivalence to standard conjectures in arithmetic algebraic geometry.

On the integral of products of higher-order Bernoulli and Euler polynomials

In this paper, we derive a formula on the integral of products of the higher-order Euler polynomials. By the same way, similar relations are obtained for $l$ higher-order Bernoulli polynomials and $r$ higher-order Euler polynomials. Moreover, we establish the connection between the results and the generalized Dedekind sums and Hardy--Berndt sums. Finally, the Laplace transform of Euler polynomials is given.

On reciprocity formula of character Dedekind sums and the integral of products of Bernoulli polynomials

We give a simple proof for the reciprocity formulas of character Dedekind sums associated with two primitive characters, whose modulus need not to be same, by utilizing the character analogue of the Euler-MacLaurin summation formula. Moreover, we extend known results on the integral of products of Bernoulli polynomials by considering the integral \[ \int\limits_{0}^{x}B_{n_{1}}(b_{1}z+y_{1})... B_{n_{r}}(b_{r}z+y_{r}) dz, \] where $b_{l}$ $(b_{l}\neq 0)$ and $y_{l}$ $(1\leq l\leq r)$ are real numbers. As a consequence of this integral we establish a connection between the reciprocity relations…

$p$-adic model theory, $p$-adic integrals, Euler products, and zeta functions of groups

We give a survey of Denef's rationality theorem on $p$-adic integrals, its uniform in $p$ versions, the relevant model theory, and a number of applications to counting subgroups of finitely generated nilpotent groups and conjugacy classes in congruence quotients of Chevalley groups over rings of integers of local fields. We then state results on analytic properties of Euler products of such $p$-adic integrals over all $p$, and an application to counting conjugacy classes in congruence quotients of certain algebraic groups over the rationals. We then briefly discuss zeta functions arising from …

θ乘积与椭圆积分矩的周期关系

Period Relations for Theta Products and Elliptic Integral Moments

该研究开发周期多项式方法，解释完全椭圆积分矩间已知有理关系，为模形式临界L-值的相关关系提供概念性说明。

Tensor Product $L$-Functions On Metaplectic Covering Groups of $GL_r$

In this note we compute some local unramified integrals defined on metaplectic covering groups of $GL$. These local integrals which were introduced by Suzuki, represent the standard tensor product $L$ function $L(π^{(n)}\times τ^{(n)},s)$ and extend the well known local integrals which represent $L(π\times τ,s)$. The computation is done using a certain "generating function" which extends a similar function introduced by the author in a previous paper. In the last section we discuss the Conjectures of Suzuki and introduce a global integral which unfolds to the above local integrals. This last p…