Searcharxiv⌕ Search

arXiv subjects

Jean-Christophe Pain

Publications and source records attributed to Jean-Christophe Pain.

At least 19 recordsLinked to original sources
arXiv2026

Inferring partial crystalline order in liquids from electrical resistivity

This work investigates how locally persistent crystal-like ordering in liquids influences the Debye-Waller factor. We have developed a theoretical framework based on liquid-phonon theory which introduces a phonon relaxation time, expressed as the ratio of shear viscosity to infinite-frequency shear modulus. These values are obtained using the Yukawa one-component plasma model. Within this framework, we establish expressions for the heat capacity at constant pressure and the Debye-Waller factor for the liquid state. These expressions explicitly introduce additional temperature dependence arising from the finite phonon lifetime. Anharmonicity is accounted for within the quasi-particle approximation. We compare our heat capacity results with values measured by Gathers for aluminum and copper, finding good agreement when assuming partial local crystal-type order. Comparisons with experimental heat capacities serve to validate the approach prior to its application to the study of electrical resistivity, the principal objective of this work. Using liquid-phonon Debye-Waller factors in the methodology developed earlier in [Phys. Rev. E 102, 053209 (2020)] for electrical resistivity in dense matter, and comparing with experimental resistivities from Gathers, we elucidate the character of the locally persisting crystal order in liquid aluminum and liquid copper. These results indicate that the electrical resistivity measurements can serve as a valuable probe for determining both the extent and the nature of crystalline order in the liquid state.

cond-mat.soft↗
arXiv2026

An analytic derivation of a generating function for $k$-alternating permutations

We study the inversion enumerator of permutations whose descent set is fixed to be the set of multiples of a fixed integer $k \ge 2$. For each $n$, let $\mathfrak{S}_n^{(k)}$ denote the set of permutations of $\{1,\dots,n\}$ whose descent set is exactly $\{i : k \mid i\}$, and define the polynomial $$ a_n^{(k)}(q)=\sum_{\sigma \in \mathfrak{S}_n^{(k)}} q^{\mathrm{inv}(\sigma)}. $$ We prove that the associated $q$-exponential generating function $$ F_k(t;q)=\sum_{n\ge 0}\frac{t^n}{[n]_q!}\,a_n^{(k)}(q), $$ where $[n]_q!$ denotes the $q$-factorial, admits an explicit closed form as a ratio of two $k$-periodically truncated $q$-exponential series. The proof is purely analytic and is based on a functional equation satisfied by $F_k(t;q)$, obtained via a decomposition of the $q$-exponential series into residue classes modulo $k$. Coefficient extraction yields a convolution identity involving Gaussian binomial coefficients, which uniquely determines the inversion enumerator. This provides an analytic alternative to classical inclusion--exclusion and structural combinatorial arguments for permutation classes with periodic descent constraints.

math.CO↗
arXiv2026

Analytic and combinatorial approaches to a weighted Catalan sum

We analyze a weighted convolution of Catalan numbers $$ \sum_{k=0}^{n} \binom{2k}{k}\binom{2(n-k)}{n-k} a^k = \sum_{k=0}^{n} (k+1)(n-k+1) C_k C_{n-k} a^k, $$ emphasizing its combinatorial, analytic, and probabilistic aspects. We derive a compact closed form in terms of the Gauss hypergeometric function ${}_2F_1(-n,1/2;1;1-a)$, valid for all complex values of the parameter $a$. The sum admits a natural interpretation in terms of return probabilities of independent simple random walks, linking weighted convolutions of central binomial coefficients to classical probability theory. Furthermore, a refinement via Narayana numbers highlights the contribution of peak distributions in pairs of Dyck paths, providing a finer combinatorial perspective. An integral representation is also proposed, suggesting a connection with orthogonal polynomials and spectral measures. Our approach illustrates how analytic and probabilistic techniques complement combinatorial reasoning in evaluating complex sums.

math.CO↗
arXiv2026

Reciprocal binomial sums via Beta integrals

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 from a simple integral representation, we provide a derivation of classical identities, including Frisch's formula, with all intermediate transformations rigorously justified. This framework naturally extends to parametric sums, yielding integral representations that lead to closed forms in terms of hypergeometric functions. In particular, we establish connections with terminating ${}_2F_1$ and generalized ${}_3F_2$ series, thereby linking discrete combinatorial sums with the analytic theory of special functions. We further derive explicit finite expansions suitable for symbolic and numerical computation, as well as higher-order extensions involving Pochhammer symbols. In addition, we present new families of identities, including shifted reciprocal sums and weighted sums involving powers of the summation index, which admit unified hypergeometric representations. Overall, the Beta integral method provides a versatile and unifying framework bridging combinatorial identities, integral representations, and hypergeometric analysis, and opens the way to further generalizations in combinatorics and special function theory.

math.CO↗
arXiv2026

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

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, and the connection with harmonic sums and Bernoulli numbers is made completely explicit. We further derive the classical refinement modulo $p^4$ and provide a precise extraction of the $B_{p-3}$ term. Our purpose is not to provide the most concise proofs, but rather to demonstrate, by showing how established results can be recovered, a general method based on complex analysis for deriving congruence properties in number theory.

math.NT↗
arXiv2026

Explicit constructions of mutually unbiased bases via Hadamard matrices

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 the challenging case of 6. Starting from the Hadamard-phase parametrization, we derive explicit analytical conditions for mutual unbiasedness in dimension 4, providing a tractable system of trigonometric constraints on the phase parameters. We then explore a tensor-product construction via Pauli operators, highlighting the algebraic and group-theoretical origin of MUBs in two-qubit systems, and demonstrating how these constructions yield a complete set of 5 MUBs in dimension 4. Extending our approach, we investigate the Fourier-family method in dimension 6, where the absence of a prime-power structure imposes strong rigidity constraints and limits the known constructions to sets of 3 MUBs. We provide a systematic computational framework for testing candidate phase vectors, bridging the gap between analytical insight and numerical exploration. Finally, we generalize the discussion to arbitrary prime-power dimensions, emphasizing the role of finite-field structures, Heisenberg-Weyl operators, and discrete symmetries in generating complete sets of MUBs. Our work offers a transparent, line-by-line verification methodology, highlighting both the geometric and algebraic richness of MUBs, and clarifying why certain dimensions resist full analytical constructions. This study serves as a comprehensive resource for researchers seeking both theoretical understanding and practical construction of MUBs in quantum information science.

quant-ph↗
arXiv2026

Modular Ackermann maps and hierarchical structures

We introduce and study modular truncations of the Ackermann function, formalized as discrete dynamical trajectories on the set of least non-negative residues. These maps form a hierarchy of rapidly increasing compositional complexity indexed by recursion depth. We investigate their structural properties, sensitivity to depth variation, and the induced distributions modulo powers of two. While such hierarchical constructions are superficially motivated by hash-type mixing functions, we analyze how powers of two interact with the recursive structure modulo $2^k$, leading to strong saturation effects in the depth $m=3$ case. Instead of the expected asymptotic equidistribution, an absorption phenomenon occurs where the uniform measure concentrates onto a localized subset of residues, driving the total variation distance from the uniform distribution to 1.

math.CO↗
arXiv2026

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

We establish a structural connection between the classical Olech-Opial inequality and the Wirtinger inequality. Using an integral identity involving the mixed energy term $uu'$, we derive a nonlinear interpolation inequality linking these two results. The optimal constant is characterized by a variational problem whose extremals satisfy an Emden-Fowler equation. An explicit expression of the optimal constant is obtained in terms of the Beta function. This approach provides a natural bridge between mixed-energy integral inequalities, classical spectral estimates, and nonlinear boundary value problems.

math.AP↗
arXiv2026

Eulerian polynomials and the alternating sum of excedances

Tangent numbers $T_{2n-1}$, which enumerate alternating permutations of odd length, play a prominent role in the Taylor series expansion of the tangent function $\tan(x)$. In this work, we adopt a combinatorial approach based on the excedance statistic of permutations, which allows us to interpret the coefficients of the tangent series in a structural and enumerative way. Using this framework, we establish a classical identity that relates the alternating sum of excedances to the hyperbolic tangent function. This perspective highlights deep connections with Eulerian polynomials, provides a combinatorial interpretation of tangent numbers, and links these sequences to Genocchi numbers and related arithmetic properties. The approach not only unifies analytic and combinatorial viewpoints but also opens the way to generalizations to other permutation statistics and families of specialized permutations.

math.CO↗
arXiv2026

Toeplitz matrices from permutation displacements and the triangular kernel

Toeplitz matrices arise naturally in harmonic analysis, operator theory, and numerical analysis. In this note we investigate Toeplitz matrices whose coefficients depend on the matrix size through a scaled kernel $a_k=f(k/n)$. We show that the empirical mean of their eigenvalues converges to a weighted integral of $f$, where the weight $1-|x|$ reflects the density of diagonals in Toeplitz matrices. We then introduce a combinatorial construction associating a Toeplitz matrix to a permutation via its displacement counts. For a uniformly random permutation, the expected matrix converges to the Toeplitz matrix generated by the triangular kernel $1-|x|$. Interestingly, the triangular kernel also appears as the covariance function of the integrated Brownian motion, providing a probabilistic interpretation of the same operator. Finally, we analyze the integral operator with kernel $(1-|x-y|)$ on $[0,1]$ and determine its eigenfunctions and eigenvalues explicitly. This operator describes the limiting spectral structure associated with the averaged Toeplitz matrices arising from permutation displacements. These results highlight a natural bridge between Toeplitz matrix theory, permutation statistics, and classical integral operators.

math.PR↗
arXiv2026

Geometric constructions for Steinitz-type bounds in dimension two

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 classical result, we provide detailed constructions yielding explicit bounds, including $\sqrt{5}$, $\sqrt{3}$, $2$, and $\sqrt{2}$, depending on geometric constraints or weighted settings. The proofs are fully detailed with step-by-step constructions of permutations, highlighting the combinatorial and geometric intuition. We conclude with conjectures on optimal universal constants and directions for future research.

math.CO↗
arXiv2026

An involutive perspective on Eisenstein's proof of quadratic reciprocity

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, we interpret the Legendre symbols as counts of lattice points in a finite rectangle and construct a simple fixed-point-free involution corresponding to the central symmetry of the rectangle, which exchanges points above and below the line $qx=py$. This reformulation highlights the involutive symmetry and places the classical proof in the spirit of Zagier-type involutive arguments. The approach shows how the reciprocity law emerges from an elementary combinatorial pairing principle.

math.NT↗
arXiv2026

An Ahmed-like integral

The so-called Ahmed integral $$ \int_{0}^{1}\frac{\arctan\left(\sqrt{2+x^{2}}\right)}{(1+x^{2})\sqrt{2+x^{2}}}\,\mathrm{d} x=\frac{5\pi^{2}}{96}, $$ has attracted considerable interest since its appearance in the "American Mathematical Monthly" in 2001. Several proofs and extensions have been proposed, including a probabilistic multivariate approach introduced by Pla based on powers of the Gaussian integral. In this note, we extend Pla's method to the fifth power of the Gaussian integral. By expressing this power as a sequence of iterated integrals and performing successive reductions, we obtain a new integral identity closely related to Ahmed's integral. In particular, we prove that $$ \int_{0}^{1}\frac{\arctan\!\left(\sqrt{\displaystyle\frac{2+x^2}{4+x^2}}\right)}{(1+x^{2})\sqrt{2+x^{2}}}\,\mathrm{d} x=\frac{\pi^2}{30}. $$ The derivation suggests that Pla's technique can systematically generate a family of Ahmed-type integrals associated with higher powers of the Gaussian integral.

math.GM↗
arXiv2026

Cumulative Riemann sums, distribution functions, and a universal inequality

We study discrete expressions of the form $$ T_n(g)=\sum_{i=1}^n a_i g(S_i), \qquad S_i=\sum_{j=1}^i a_j, $$ where $a_i>0$ and $\sum_{i=1}^n a_i=1$. If $g:[0,1]\to\mathbb{R}$ is a decreasing integrable function, we have $$ \sum_{i=1}^n a_i g(S_i) \le \int_0^1 g(x)\,dx, $$ from which classical inequalities can be obtained, for instance for the choice $g(x)=1-x^k$. Although elementary, this inequality admits a natural interpretation in terms of Riemann sums, Abel summation, and the probability integral transform. The aim of this paper is to present a unified perspective emphasizing that the discrete inequality is a consequence of a distribution-free continuous identity. Beyond the specific example, we establish a general discrete theorem for monotone functions and discuss connections with majorization theory and Karamata's inequality.

math.CA↗
arXiv2026

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

We propose sum rules for permutations $p_n(k)$ of the ensemble $\left\{1,2,\cdots,n\right\}$ with $k$ fixed points, in the form of partial sums of their moments. The corresponding identities involve Stirling numbers of the first kind $s(q,r)$. Using a formula due to Vassilev-Missana and the Schl\"omlich expression of Stirling numbers, we also deduce sum rules for binomial coefficients. Connections with Bell numbers $B_n$ are outlined.

math.CO↗
arXiv2026

Elliptic integral identities derived from Coxeter's integrals

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 embedding Coxeter's first integral into a one-parameter family $$ I(\lambda)=\int_{0}^{\pi/2} \arccos\!\left(\frac{\cos\theta}{1+\lambda\cos\theta}\right)\,d\theta, $$ and differentiating with respect to the parameter \(\lambda\), we show that the derivative $I'(\lambda)$ can be expressed as an elliptic-type integral. Integrating $I'(\lambda)$ between 0 and 2 yields the identity $$ \int_0^2 \int_0^{\pi/2} \frac{\cos^2\theta} {(1+s\cos\theta)\sqrt{(1+s\cos\theta)^2-\cos^2\theta}} \,d\theta\, ds=A-B=\frac{\pi^2}{12}, $$ where $A$ and $B$ are the first two so-called Coxeter integrals $$ A = \int_0^{\pi/2} \arccos\!\left(\frac{\cos\theta}{1+2\cos\theta}\right) d\theta, $$ and $$ B = \int_0^{\pi/2} \arccos\!\left(\frac{1}{1+2\cos\theta}\right) d\theta. $$ The derivative $I'(\lambda)$ can be expressed in terms of incomplete elliptic integrals of the first kind $F$ and of the third kind $\Pi$. This approach establishes a direct connection between classical Coxeter integrals and elliptic functions. The method highlights how well-known trigonometric integrals can serve as a bridge to explore properties and relations of elliptic integrals, offering new analytic insights beyond the original Coxeter evaluations.

math.CA↗
arXiv2026

Analysis of moments and cumulants in alternating sign matrices

In this work, we study the discrete observables $$E_k = \sum_{i,j=1}^n (i-j)^k A_{i,j}$$ associated with $n\times n$ alternating sign matrices $A = (A_{i,j})$. This work develops exact formulas for expectations using Bernoulli polynomials, exponential generating functions, expansions in $1/n$ linked to Riemann zeta functions, and cumulants up to fourth order via integrable kernel methods. All intermediate calculations, expansions, and pedagogical details are provided to illustrate the interplay between combinatorial sums, analytic expansions, and integrable structures in alternating sign matrices.

math.CO↗
arXiv2026

Integral and arithmetic structures of alternating (zigzag) numbers $A_n$

The alternating (zigzag) numbers $A_n$, counting the ascending alternating permutations of $\left\{1,\cdots,n\right\}$ and defined by the exponential generating function $\tan x+\sec x$, admit several classical combinatorial and analytic representations. In this work we unify and extend three complementary structures of $A_n$. First, starting from the Stirling number expansion of zigzag numbers, we derive a contour integral representation, as well as a positive Laplace-type integral representation $$ A_n = 2^n \int_0^\infty e^{-y} f_n(y)\, dy, \qquad f_n(y) := \sum_{k=0}^{n} (-1)^k S(n,k) \left(\frac{y}{2}\right)^k, $$ where the kernel $f_n(y)$ is the polynomial generating function of Stirling numbers. A continuous interpolation of the discrete product (falling factorial) is introduced subsequently. This provides a direct analytic bridge between set partitions and Laplace asymptotics. Second, using the partial fraction expansion of $\tan$, we obtain the well-known hyperbolic integral representation $$ A_{2n+1}=\frac{1}{\pi}\int_0^\infty\frac{y^{2n+1}}{\sinh(y/2)}\,dy, $$ equivalently expressed in classical $\cosh$ form for $A_{2n}$. This representation interprets zigzag numbers as spectral moments associated with half-integer poles. The connection with Fourier analysis and Mellin transforms is also outlined. Finally, combining spectral expansions with Stirling identities, we derive congruence relations modulo primes for $A_n$. These results exhibit a dual analytic-combinatorial structure of zigzag numbers, linking partition expansions, trigonometric spectra, and arithmetic properties.

math.CO↗