Related papers: Generalized Eulerian Triangles and Some Special Pr…
We show that the Catalan-Schroeder convolution recurrences and their higher order generalizations can be solved using Riordan arrays and the Catalan numbers. We investigate the Hankel transforms of many of the recurrence solutions, and…
Generalized Pl\"ucker numbers are defined to count certain types of tangent lines of generic degree $d$ complex projective hypersurfaces. They can be computed by identifying them as coefficients of GL(2)-equivariant cohomology classes of…
This paper is concerned with the generalized Euler polynomial matrix $\E^{(\alpha)}(x)$ and the Euler matrix $\E$. Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for $\E^{(\alpha)}(x)$ and…
We study the higher-order Euler polynomials and give the corresponding monic orthogonal polynomials, which are Meixner-Pollaczek polynomials with certain arguments and constant factors. Moreover, through a general connection between moments…
In his study of generalised permutahedra, Postnikov considered the mixed volumes of hypersimplices, giving rise to the family of mixed Eulerian numbers. It comprises usual Eulerian numbers, binomial coefficients, Catalan numbers, and the…
Generalized Euler polynomials ${{\alpha }_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{p}_{n}}}\left( m \right){{x}^{m}}$, where ${{p}_{n}}\left( x \right)$ is the polynomial of degree $n$, are the…
The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order.…
Random orthogonal matrices play an important role in probability and statistics, arising in multivariate analysis, directional statistics, and models of physical systems, among other areas. Calculations involving random orthogonal matrices…
We show how to represent various families of Laguerre polynomials by the three-dimensional Riordan arrays, and use the fundamental theorem of Riordan arrays to obtain the corresponding exponential generating functions.
We investigate several families of multiple orthogonal polynomials associated with weights for which the moment generating functions are hypergeometric series with slightly varying parameters. The weights are supported on the unit interval,…
We generalize well-known Catalan-type integrals for Euler's constant to values of the generalized-Euler-constant function and its derivatives. Using generating functions appeared in these integral representations we give new Vacca and…
A family of multivariate orthogonal polynomials generalizing the standard (univariate) Charlier polynomials is shown to arise in the matrix elements of the unitary representation of the Euclidean group E(d) on oscillator states. These…
In a previous paper, Rahmani introduced a new family of p-Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. Motivated by this paper and as a degenerate version of those numbers and polynomials, we introduce…
A new family of polynomials, called cumulant polynomial sequence, and its extensions to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients of these polynomials are cumulants, but depending…
The "2-variable general-$\lambda$-matrix polynomials (2VG$\lambda$MP)" is a new family of matrix polynomials, introduced and studied in this article. These matrix polynomials are constructed using umbral and symbolic methods. We delve into…
We present a parametric family of Riordan arrays which are obtained by multiplying any Riordan array with a generalized Pascal array. In particular, we focus on some interesting properties of one-parameter Catalan triangles. We obtain…
In this paper, by using some families of special numbers and polynomials with their generating functions, we give various properties of these numbers and polynomials. These numbers are related to the well-known numbers and polynomials,…
Summation formulas, such as the Euler-Maclaurin expansion or Gregory's quadrature, have found many applications in mathematics, ranging from accelerating series, to evaluating fractional sums and analyzing asymptotics, among others. We show…
A variety of descent and major-index statistics have been defined for symmetric groups, hyperoctahedral groups, and their generalizations. Typically associated to pairs of such statistics is an Euler--Mahonian distribution, a bivariate…
Using the language of Riordan arrays, we study a one-parameter family of orthogonal polynomials that we call the restricted Chebyshev-Boubaker polynomials. We characterize these polynomials in terms of the three term recurrences that they…