Related papers: Combinatorial Properties of Mills Ratio
We give new definitions for the determinant over commutative ring $K$, noncommutative ring $\mathbf{K}$, noncommutative ring $\mathcal{K}$ with associative powers, over noncommutative nonassociative ring $\mathfrak{K}$, and study their…
We consider a quaternionic analogue of the univariate complex Hermite polynomials and study some of their analytic properties in some detail. We obtain their integral representation as well as the operational formulas of exponential and…
Combinatorial $t$-designs have been an interesting topic in combinatorics for decades. It was recently reported that the image sets of a fixed size of certain special polynomials may constitute a $t$-design. Till now only a small amount of…
This paper is a short survey of the recent results on examples of periodic two-dimensional continued fractions (in Klein's model). In the last part of this paper we formulate some questions, problems and conjectures on geometrical…
In this article we obtain a general polynomial identity in $k$ variables, where $k\geq 2$ is an arbitrary positive integer. We use this identity to give a closed-form expression for the entries of the powers of a $k \times k$ matrix.…
This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith…
We introduce a class of doubly indexed real Hermite polynomials and we deal with their related properties like the associated recurrence formulae, Runge's addition formula, generating function and Nielsen's identity.
We investigate some arithmetic properties of the q-Fibonacci numbers and the q-Pell numbers.
We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co- multiplication…
The causal set theory d'Alembertian has rational coefficients for which alternating expressions are known. Here, a combinatorial interpretation of these numbers is given.
Several combinatorial identities are presented, involving Stirling functions of the second kind with a complex variable. The identities involve also Stirling numbers of the first kind, binomial coefficients and harmonic numbers.
We establish some identities in law for the convolution of a beta prime distribution with itself, involving the square root of beta distributions. The proof of these identities relies on transformations on generalized hypergeometric series…
In this work we show how to get advantage from the Riemann--Hilbert analysis in order to obtain information about the matrix orthogonal polynomials and functions of second kind associated with a weight matrix. We deduce properties for the…
We study combinatorial modulus on boundaries of hyperbolic Coxeter groups. We give new examples of hyperbolic groups whose boundary satisfies a combinatorial version of the Loewner property, and prove Cannon's conjecture for Coxeter groups.…
We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebraic number theory and the geometry of numbers. We give…
We give several families of polynomials which are related by Eulerian summation operators. They satisfy interesting combinatorial properties like being integer-valued at integral points. This involves nearby-symmetries and a recursion for…
We investigate multiple Charlier polynomials and in particular we will use the (nearest neighbor) recurrence relation to find the asymptotic behavior of the ratio of two multiple Charlier polynomials. This result is then used to obtain the…
In this work the interplay between matrix biorthogonal polynomials with respect to a matrix of linear functionals, the $k$-th associated matrix polynomials and the second kind matrix functions, is studied in terms of quasideterminants. A…
In their study of a quartic integral, Boros and Moll discovered a special class of Jacobi polynomials, which we call the Boros-Moll polynomials. Kauers and Paule proved the conjecture of Moll that these polynomials are log-concave. In this…
Orthogonal polynomials are of fundamental importance in many fields of mathematics and science, therefore the study of a particular family is always relevant. In this manuscript, we present a survey of some general results of the Hermite…