Zobrazeno 1 - 10
of 237
pro vyhledávání: '"Vignat, Christophe"'
We address a class of definite integrals known as Berndt-type integrals, highlighting their role as specialized instances within the integral representation framework of the Barnes-zeta function. Building upon the foundational insights of Xu and Zhao
Externí odkaz:
http://arxiv.org/abs/2407.02365
We contribute to the zoo of dubious identities established by J.M. and P.B. Borwein in their 1992 paper, "Strange Series and High Precision Fraud" with five new entries, each of a different variety than the last. Some of these identities are again a
Externí odkaz:
http://arxiv.org/abs/2307.05565
Autor:
Salminen, Paavo, Vignat, Christophe
In this note we deduce well known modular identities for Jacobi theta functions using the spectral representations associated with the real valued Brownian motion taking values on $[-1,+1]$. We consider two cases: (i) reflection at $-1$ and $+1$, (ii
Externí odkaz:
http://arxiv.org/abs/2303.05942
We give a formal extension of Ramanujan's master theorem using operational methods. The resulting identity transforms the computation of a product of integrals on the half-line to the computation of a Laplace transform. Since the identity is purely f
Externí odkaz:
http://arxiv.org/abs/2209.03399
Dirichlet Series Under Standard Convolutions: Variations on Ramanujan's Identity for Odd Zeta Values
Inspired by a famous identity of Ramanujan, we propose a general formula linearizing the convolution of Dirichlet series as the sum of Dirichlet series with modified weights; its specialization produces new identities and recovers several identities
Externí odkaz:
http://arxiv.org/abs/2107.06457
Autor:
Chavan, Sarth, Vignat, Christophe
A detailed study of a double integral representation of the Catalan's constant allows us to identify a duality identity for the Stieltjes transform on which it is based. This duality identity is then extended to an arbitrary dimensional integral and
Externí odkaz:
http://arxiv.org/abs/2105.11771
Autor:
Dilcher, Karl, Vignat, Christophe
Koecher in 1980 derived a method for obtaining identities for the Riemann zeta function at odd positive integers, including a classical result for $\zeta(3)$ due to Markov and rediscovered by Ap\'ery. In this paper we extend Koecher's method to a ver
Externí odkaz:
http://arxiv.org/abs/2010.15424
We establish the triple integral evaluation \[ \int_{1}^{\infty} \int_{0}^{1} \int_{0}^{1} \frac{dz \, dy \, dx}{x(x+y)(x+y+z)} = \frac{5}{24} \zeta(3), \] as well as the equivalent polylogarithmic double sum \[ \sum_{k=1}^{\infty} \sum_{j=k}^{\infty
Externí odkaz:
http://arxiv.org/abs/2004.06232
Publikováno v:
Open Mathematics, Vol 21, Iss 1, Pp 50-73 (2023)
The method of brackets is a symbolic approach to the computation of integrals over Rn{{\mathbb{R}}}^{n} based on a deep result by Ramanujan. Its usefulness to obtain new and difficult integrals has been demonstrated many times in the last few years.
Externí odkaz:
https://doaj.org/article/d23f2eef1672419b81651dbd759075b2
Autor:
Wakhare, Tanay, Vignat, Christophe
We extend some results recently obtained by Dan Romik about the Taylor coefficients of the theta function $\theta_{3}\left(1\right)$ to the case $\theta_{3}\left(q\right)$ of an arbitrary value of the elliptic modulus $k.$ These results are obtained
Externí odkaz:
http://arxiv.org/abs/1909.01508