36 papers
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…
Analytic and combinatorial approaches to a weighted Catalan sum
Jean-Christophe Pain
We analyze a weighted convolution of Catalan numbers emphasizing its combin…
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…
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…
Eulerian polynomials and the alternating sum of excedances
Jean-Christophe Pain
Tangent numbers , which enumerate alternating permutations of odd length, play a prominent role in the Taylor series expansion of the tangent function . In this…
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 , we…