Related papers: A generalization of the binomial coefficients
This is the central article of a series of three papers on cross product bialgebras. We present a universal theory of bialgebra factorizations (or cross product bialgebras) with cocycles and dual cocycles. We also provide an equivalent…
For any given finite abelian group, we give factorizations of the group determinant in the group algebra of any subgroup. The factorizations are an extension of Dedekind's theorem. The extension leads to a generalization of Dedekind's…
We investigate the coefficients generated by expressing the falling factorial $(xy)_k$ as a linear combination of falling factorial products $(x)_l (y)_m$ for $l,m =1,...,k$. Algebraic and combinatoric properties of these coefficients are…
In this paper we consider linear combinations of two trivariate homogeneous polynomials of second degree. We formulate and solve two problems: i) Characterization of polynomials for which all linear combinations are factorizable. ii) How…
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 main result concerns a bicategorical factorization system on the bicategory $\mathrm{Cat}$ of categories and functors. Each functor $A\xra{f} B$ factors up to isomorphism as $A\xra{j}E\xra{p}B$ where $j$ is what we call an ultimate…
In this short note, we establish some identities containing sums of binomials with coefficients satisfying third order linear recursive relations. As a result and in particular, we obtain general forms of earlier identities involving…
We show that the Christoffel function (CF) factorizes (or can be disintegrated) as the product of two Christoffel functions, one associated with the marginal and the another related to the conditional distribution, in the spirit of "the CF…
Generalizations of some known results on the best, best linear and best one-sided approxima- tions by trigonometric polynomials of the classes of 2\pi - periodic functions presented in the form of convolutions to the case of set-valued…
Roman logarithmic binomial formula analogue has been found . It is presented here also for the case of fibonomial coefficients which recently have been given a combinatorial interpretation by the present author.
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
This is an introductory note concerning the distribution vectors in a unitary representation of a Lie group. We discuss the definition of matrix coefficients associated with a pair of distributions and how one can compute them. Most of the…
We discuss the problem of factorisation of the symmetric Macdonald polynomials and present the obtained results for the cases of 2 and 3 variables.
There are many possible definitions of derivatives, here we present some and present one that we have called generalized that allows us to put some of the others as a particular case of this but, what interests us is to determine that there…
The article describes prime intervals into the prime factorization of the middle binomial coefficient. Prime factors and prime powers are distributed in layers. Each layer consists of non-repeated prime numbers which are chosen (not…
We discuss a general method to construct correlated binomial distributions by imposing several consistent relations on the joint probability function. We obtain self-consistency relations for the conditional correlations and conditional…
In this study, the new algebraic properties related to bivariate Fibonacci polynomials has been given. We present the partial derivatives of these polynomials in the form of convolution of bivariate Fibonacci polynomials. Also, we define a…
In this paper we study the genralized q-Euler numbers and polynomials. From our results, we derive some interesting congruences related tothe generalized q-Euler numbers.
Factorization of numbers with the help of Gauss sums relies on an intimate relationship between the maxima of these functions and the factors. Indeed, when we restrict ourselves to integer arguments of the Gauss sum we profit from a…
We discuss various aspects of representation of a polynomial as a sum of monomials (for example, uniqueness of such representation and related estimations).