Double Integrals for Euler's Constant and ln(4/Pi) and an Analog of Hadjicostas's Formula

Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent

The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for $ζ(2)$ and $ζ(3),$ and those of the second author for Euler's constant $γ$ and its alternating analog $\ln(4/π),$ and on the other hand the infinite products of the first author for $e$, and of the second author for $π$ and $e^γ.$ We obtain new double integral and infinite product representations of many classical constants, as well as a generalization to Lerch's transcendent of Hadjicostas's double integral formula for the Riemann zeta function, and logarithmic series for the digamma a…

A signed analog of Euler's reduction formula for the double zeta function

The double zeta function is a function of two arguments defined by a double Dirichlet series, and was first studied by Euler in response to a letter from Goldbach in 1742. By calculating many examples, Euler inferred a closed form evaluation of the double zeta function in terms of values of the Riemann zeta function, in the case when the two arguments are positive integers with opposite parity. Here, we consider a signed analog of Euler's evaluation: namely a reduction formula for the signed double zeta function that reduces to Euler's evaluation when the signs are specialized to 1. This formu…

New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi)

In math.CA/0211148 we observed that $\ln(4/ π)$ is an "alternating" analog of Euler's constant $γ$. Here we use the binary expansion of an integer to give a rational series for $\ln(4/ π)$ analogous to Vacca's series for $γ$. Also, using a known generalization of Vacca's series to other bases, we accelerate Addison's series for $γ$. Along the way, we give new proofs of Vacca's and Addison's formulas.

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

Reflections on Euler's reflection formula and an additive analogue of Legendre's duplication formula

本文研究 gamma 函数的反射与加倍关系，给出 Euler 反射公式新证明，证明 Landau 结果的最优性并推广至复数情形，还借助超几何函数理论推广 Schlömilch 的加法类似公式。

On $q$-analogs of zeta functions associated with a pair of $q$-analogs of Bernoulli numbers and polynomials

In this paper, we use two different approaches to introduce $q$-analogs of Riemann's zeta function and prove that their values at even integers are related to the $q$-Bernoulli and $q$ Euler's numbers introduced by Ismail and Mansour [Analysis and Applications, {\bf{17}}, 6, 2019, 853--895].

Euler and Laplace integral representations of GKZ hypergeometric functions

We introduce an interpolation between Euler integral and Laplace integral: Euler-Laplace integral. We establish a combinatorial method of constructing a basis of the rapid decay homology group associated to Euler-Laplace integral with a nice intersection property. This construction yields a remarkable expansion formula of cohomology intersection numbers in terms of GKZ hypergeometric series. As an application, we obtain closed formulas of the quadratic relations of Aomoto-Gelfand hypergeometric functions and their confluent analogue in terms of bipartite graphs.