arXiv subjects

Christophe Vignat

Publications and source records attributed to Christophe Vignat.

At least 19 recordsLinked to original sources

arXiv2025

Integrals involving arbitrary powers of the arcsine, with applications to infinite series

Using appropriate power series evaluations, we determine all moments of arbitrary positive powers of the arcsine. As consequences we evaluate several doubly infinite classes of power series involving central binomial coefficients and generalized multiple harmonic sums. By specializing the variable involved, we then evaluate classes of numerical sequences, mostly in terms of powers of $\pi$. Finally, we obtain limit expressions for arbitrary powers of $\pi$.

math.NT↗

arXiv2025

Powers of the arcsine and infinite classes of series involving central binomial coefficients

A general integral expression to transform power series is applied to $\arcsin{x}$ and its positive integer powers. We concentrate on the first to the fourth powers and obtain infinite classes of new power series involving central binomial coefficients. Specializing the variable to appropriate simple values leads to different classes of series expansions for $\pi$ and some of its positive integer powers. We also discuss several limit expressions and connections with hypergeometric series.

math.CA↗

arXiv2025

Further Classes of Series Involving Central Binomial Coefficients

Departing from a class of infinite series with central binomial coefficients in the numerator and depending on a positive integer parameter, we first extend known identities to all complex parameters. Then we use various methods, including exponential Bell polynomials and integral representations, to further extend these results. Throughout the paper, we make extensive use of the gamma and polygamma functions and their properties.

math.NT↗

arXiv2025

The joy of multisection: elementary approaches to Ramanujan's lacunary identities for Bernoulli numbers

An identity by Ramanujan related to the multisection of Bernoulli numbers is revisited. Two alternative approaches are proposed, both relying on the multisection technique. A geometric approach reveals the role played by the symmetries of the summation domain in the complex plane induced by the multisection technique. The second approach, based on generating functions, allows us to extend Ramanujan's identity to other special functions such as Eisenstein's series.

math.NT↗

arXiv2025

Bernoulli and Euler Partitions

Exact rational partitions are presented for Bernoulli and Euler numbers as novel sums involving Faulhaber and Sali\'e coefficients.

math.CO↗

arXiv2024

Berndt-type Integrals: Unveiling Connections with Barnes Zeta and Jacobi Elliptic Functions

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, who adeptly evaluate these integrals using rational linear combinations of Lambert-type series and derive closed-form expressions involving products of $\Gamma^4(1/4)$ and $\pi^{-1}$, we uncover direct evaluations of the Barnes-zeta function. Moreover, our inquiry leads us to establish connections between Berndt-type integrals and Jacobi elliptic functions, as well as moment polynomials investigated by Lomont and Brillhart, a relationship elucidated through the seminal contributions of Kuznetsov. In this manner, we extend and integrate these diverse mathematical threads, unveiling deeper insights into the intrinsic connections and broader implications of Berndt-type integrals in special function and integration theory.

math.CA↗

arXiv2023

Dubious Identities: A Visit to the Borwein Zoo

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 high precision fraud and picking out the true from the bogus can be a challenging task with many unexpected twists along the way.

math.HO↗

arXiv2023

Probabilistic aspects of Jacobi theta functions

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) killing at $-1$ and $+1$. It is seen that these two representations give, in a sense, most compact forms of the modular theta-function identities. We study also discrete Gaussian distributions generated by theta functions, and derive, in particular, addition formulas for discrete Gaussian variables.

math.PR↗

arXiv2022

Evaluation of multivariate integrals based on a duality identity for the Stieltjes transform

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 several special cases are deduced. On the way, we also highlight a relationship with some multivariate generalizations of the Riemann zeta function.

math.NT↗

arXiv2022

An Operational Calculus Generalization of Ramanujan's Master Theorem

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 formal, we show consistency of this operational approach with various standard calculus results, followed by several examples to illustrate the power of the extension. We then briefly discuss the connection between Ramanujan's master theorem and identities of Hardy and Carr before extending the latter identities in the same way we extended Ramanujan's. Finally, we generalize our results, producing additional interesting identities as a corollary.

math.CA↗

arXiv2022

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 derived earlier in the literature, such as the convolution of squares of Bernoulli numbers by A. Dixit and collaborators, or the convolution of Bernoulli numbers by Y. Komori and collaborators.

math.NT↗

arXiv2021

Structural properties of multiple zeta values

We study some classical identities for multiple zeta values and show that they still hold for zeta functions built on the zeros of an arbitrary function. We introduce the complementary zeta function of a system, which naturally occurs when lifting identities for multiple zeta values to identities for quasisymmetric functions.

math.NT↗

arXiv2020

On a result of Koecher concerning Markov-Apéry type formulas for the Riemann zeta function

Koecher in 1980 derived a method for obtaining identities for the Riemann zeta function at odd positive integers, including a classical result for $ζ(3)$ due to Markov and rediscovered by Apéry. In this paper we extend Koecher's method to a very general setting and prove two more specific but still rather general results. As applications we obtain infinite classes of identities for alternating Euler sums, further Markov-Apéry type identities, and identities for even powers of $π$

math.NT↗

arXiv2020

A triple integral analog of a multiple zeta value

We establish the triple integral evaluation \[ \int_{1}^{\infty} \int_{0}^{1} \int_{0}^{1} \frac{dz \, dy \, dx}{x(x+y)(x+y+z)} = \frac{5}{24} ζ(3), \] as well as the equivalent polylogarithmic double sum \[ \sum_{k=1}^{\infty} \sum_{j=k}^{\infty} \frac{(-1)^{k-1}}{k^{2}} \, \frac{1}{j \, 2^{j}} = \frac{13}{24} ζ(3). \] This double sum is related to, but less approachable than, similar sums studied by Ramanujan. It is also reminiscent of Euler's formula $ζ(2,1) = ζ(3)$, which is the simplest instance of duality of multiple polylogarithms. We review this duality and apply it to derive a companion identity. We also discuss approaches based on computer algebra. All of our approaches ultimately require the introduction of polylogarithms and nontrivial relations between them. It remains an open challenge to relate the triple integral or the double sum to $ζ(3)$ directly.

math.NT↗

arXiv2020

Taylor coefficients of the Jacobi $θ_{3}\left( q \right)$ function

We extend some results recently obtained by Dan Romik about the Taylor coefficients of the theta function $θ_{3}\left(1\right)$ to the case $θ_{3}\left(q\right)$ of an arbitrary value of the elliptic modulus $k.$ These results are obtained by carefully studying the properties of the cumulants associated to a $θ_{3}$ (or discrete normal) distributed random variable. This article also states some congruence conjectures about integers sequences that generalize the one studied by D. Romik.

math.NT↗

arXiv2019

Identities for Bernoulli polynomials related to multiple Tornheim zeta functions

We show that each member of a doubly infinite sequence of highly nonlinear expressions of Bernoulli polynomials, which can be seen as linear combinations of certain higher-order convolutions, is a multiple of a specific product of linear factors. The special case of Bernoulli numbers has important applications in the study of multiple Tornheim zeta functions. The proof of the main result relies on properties of Eulerian polynomials and higher-order Bernoulli polynomials.

math.NT↗

arXiv2019

Analytic Continuation for Multiple Zeta Values using Symbolic Representations

We introduce a symbolic representation of $r$-fold harmonic sums at negative indices. This representation allows us to recover and extend some recent results by Duchamp et al., such as recurrence relations and generating functions for these sums. This approach is also applied to the study of the family of extended Bernoulli polynomials, which appear in the computation of harmonic sums at negative indices. It also allows us to reinterpret the Raabe analytic continuation of the multiple zeta function as both a constant term extension of Faulhaber's formula, and as the result of a natural renormalization procedure for Faulhaber's formula.

math.NT↗

arXiv2018

Connection Coefficients for Higher-order Bernoulli and Euler Polynomials: A Random Walk Approach

We consider the use of random walks as an approach to obtain connection coefficients for higher-order Bernoulli and Euler polynomials. In particular, we consider the cases of a $1$-dimensional linear reflected Brownian motion and of a $3$-dimensional Bessel process. Considering the successive hitting times of two, three, and four fixed levels by these random walks yields non-trivial identities that involve higher-order Bernoulli and Euler polynomials.

math.NT↗