Jean‐Christophe Pain

activity

20222026

most citedElectrical conductivities and low frequency opacities in the warm dense matter regime

2 citations · 3 across the 55 of their papers we have counts for

collaborators

55 papers

math.CO2026

Reciprocal binomial sums via Beta integrals

Jean-Christophe Pain

We develop a systematic and fully explicit approach to the evaluation of binomial sums involving reciprocals of binomial coefficients based on Beta integral techniques. Starting fr…

math.CO2026

Analytic and combinatorial approaches to a weighted Catalan sum

Jean-Christophe Pain

We analyze a weighted convolution of Catalan numbers $k = 0 \sum n (k 2 k) (n - k 2 ( n - k )) a^{k} = k = 0 \sum n (k + 1) (n - k + 1) C_{k} C_{n - k} a^{k},$ emphasizing its combin…

math.NT2026

A proof of Wolstenholme's theorem and congruence properties via an Egorychev-type integral

Jean-Christophe Pain

We present a detailed proof of Wolstenholme's theorem using an Egorychev-type contour integral and an exponential change of variables. All formal series manipulations are justified…

quant-ph2026

Explicit constructions of mutually unbiased bases via Hadamard matrices

Jean-Christophe Pain

We present a detailed computational and algebraic study of Mutually Unbiased Bases (MUBs) in finite-dimensional Hilbert spaces, with a particular focus on dimensions 2, 3, 4, and t…

math.CO2026

Eulerian polynomials and the alternating sum of excedances

Jean-Christophe Pain

Tangent numbers $T_{2 n - 1}$ , which enumerate alternating permutations of odd length, play a prominent role in the Taylor series expansion of the tangent function $tan (x)$ . In this…

math.AP2026

A Variational link between the Olech-Opial inequality, the Wirtinger inequality, and Emden-Fowler equations

Jean-Christophe Pain

We establish a structural connection between the classical Olech-Opial inequality and the Wirtinger inequality. Using an integral identity involving the mixed energy term $u u^{'}$ , we…

math.PR2026

Toeplitz matrices from permutation displacements and the triangular kernel

Jean-Christophe Pain

Toeplitz matrices arise naturally in harmonic analysis, operator theory, and numerical analysis. In this note we investigate Toeplitz matrices whose coefficients depend on the matr…

math.NT2026

An involutive perspective on Eisenstein's proof of quadratic reciprocity

Jean-Christophe Pain

We revisit Eisenstein's geometric proof of quadratic reciprocity and make explicit the involutive symmetry underlying Eisenstein's lattice-point argument. Building on Gauss's lemma…

math.CO2026

Geometric constructions for Steinitz-type bounds in dimension two

Jean-Christophe Pain

We investigate inequalities for partial sums of complex numbers with bounded modulus and zero total sum, a topic referred to as "polygonal confinement". Starting from Steinitz's cl…

math.GM2026

An Ahmed-like integral

Jean-Christophe Pain

The so-called Ahmed integral $\int_{0}^{1} \frac{arctan ( 2 + x ^{2} )}{( 1 + x ^{2} ) 2 + x ^{2}} d x = \frac{5 π ^{2}}{96},$ has attracted considerable intere…

math.CA2026

Cumulative Riemann sums, distribution functions, and a universal inequality

Jean-Christophe Pain

We study discrete expressions of the form $T_{n} (g) = i = 1 \sum n a_{i} g (S_{i}), S_{i} = j = 1 \sum i a_{j},$ where $a_{i} > 0$ and $\sum_{i = 1}^{n} a_{i} = 1$ . If $g : [0, 1] \to R$ is a…

math.CO2026

Sum rules for permutations with fixed points involving Stirling numbers of the first kind

Jean-Christophe Pain

We propose sum rules for permutations $p_{n} (k)$ of the ensemble ${1, 2, \dots, n}$ with $k$ fixed points, in the form of partial sums of their moments. The corresponding…

math.CA2026

Elliptic integral identities derived from Coxeter's integrals

Jean-Christophe Pain

We revisit the classical integrals introduced by Coxeter, not to recalculate their well-known exact values, but to use them as a tool to derive elliptic integral identities. By emb…

math.CO2026

Analysis of moments and cumulants in alternating sign matrices

Jean-Christophe Pain

In this work, we study the discrete observables $E_{k} = i, j = 1 \sum n (i - j)^{k} A_{i, j}$ associated with $n \times n$ alternating sign matrices $A = (A_{i, j})$ . This work develops e…

math.PR2026

A probabilistic journey through the Newton-Girard identities

Jean-Christophe Pain

This article presents a pedagogical probabilistic exploration of the Newton-Girard identities. We show that the coefficients in these classical relations between power sums and ele…

math.CO2026

Integral and arithmetic structures of alternating (zigzag) numbers $A_{n}$

Jean-Christophe Pain

The alternating (zigzag) numbers $A_{n}$ , counting the ascending alternating permutations of ${1, \dots, n}$ and defined by the exponential generating function $\tan x+\s…

math.CO2026

Touchard's identity and a detailed determination of the radius of convergence of the Catalan series

Jean-Christophe Pain

While the value of the radius of convergence of the generating series of the Catalan numbers is well-known, obtaining it solely from recurrence relations is less immediate. It is s…

math.CO2026

A combinatorial proof of Jacobi's elliptic identity via alternating permutations

Jean-christophe Pain

We provide a unified combinatorial framework connecting Entringer numbers, Dumont-Viennot snakes, and elliptically weighted continued fractions, which gives a structural interpreta…

math.CO2026

Steganographic information hiding via symmetric numerical semigroups

Jean-Christophe Pain

We introduce a steganographic information hiding scheme based on structural properties of numerical semigroups arising from the Frobenius coin problem. Instead of encoding data thr…

math.GM2026

Three expressions of the $n$ -th prime number: discrete sieving, spectral analysis and probabilistic dynamics

Jean-Christophe Pain

The search for a closed-form expression of the $n$ -th prime number, $p_{n}$ , has long oscillated between the rigid determinism of analytic functions and the apparent randomness of lo…

math.NT2026

Generalized relations between arithmetic functions

Jean-Christophe Pain

The aim of this article is to present in a self-contained way identities arising in elementary number theory, among which the following one: $$ \sum_{d\mid n}\frac{μ^2(d)}{φ(d)\,d^…

math.CO2025

Identities for permutations with fixed points

Jean-Christophe Pain

We present identities for permutations with fixed points. The formulas are based on successive derivations or integrations of the determinant of a particular matrix.

math.NT2025

Asymptotic equivalents of partial sums of the reciprocals of prime numbers via the von Mangoldt function

Jean-Christophe Pain

In this paper, we discuss an alternative approach to determine an asymptotic equivalent of the partial sum of the reciprocals of prime numbers. This well-known result, related to M…

math.NT2025

A sum rule for derangements

Jean-Christophe Pain

We propose a sum rule for derangements. Three different proofs are provided. The first one involves integral representations and the second one relies on the Hermite identity for t…

Reciprocal binomial sums via Beta integrals

Analytic and combinatorial approaches to a weighted Catalan sum

A proof of Wolstenholme's theorem and congruence properties via an Egorychev-type integral

Explicit constructions of mutually unbiased bases via Hadamard matrices

Eulerian polynomials and the alternating sum of excedances

A Variational link between the Olech-Opial inequality, the Wirtinger inequality, and Emden-Fowler equations

Toeplitz matrices from permutation displacements and the triangular kernel

An involutive perspective on Eisenstein's proof of quadratic reciprocity

Geometric constructions for Steinitz-type bounds in dimension two

An Ahmed-like integral

Cumulative Riemann sums, distribution functions, and a universal inequality

Sum rules for permutations with fixed points involving Stirling numbers of the first kind

Elliptic integral identities derived from Coxeter's integrals

Analysis of moments and cumulants in alternating sign matrices

A probabilistic journey through the Newton-Girard identities

Integral and arithmetic structures of alternating (zigzag) numbers An​

Touchard's identity and a detailed determination of the radius of convergence of the Catalan series

A combinatorial proof of Jacobi's elliptic identity via alternating permutations

Steganographic information hiding via symmetric numerical semigroups

Three expressions of the n-th prime number: discrete sieving, spectral analysis and probabilistic dynamics

Generalized relations between arithmetic functions

Identities for permutations with fixed points

Asymptotic equivalents of partial sums of the reciprocals of prime numbers via the von Mangoldt function

A sum rule for derangements

Integral and arithmetic structures of alternating (zigzag) numbers $A_{n}$

Three expressions of the $n$ -th prime number: discrete sieving, spectral analysis and probabilistic dynamics