Related papers: Products of extended binomial coefficients and the…

The generalized Fibonacci recurrence $g_n=g_{n-k}+g_{n-m}$ was recently used to demonstrate the theoretically optimal nature of limited senescence in morphologically symmetrically dividing bacteria. Here, we study this recurrence from a…

It is known to be difficult to find out whether a certain multivariable function to be a characteristic function when its corresponding measure is not tirivial to be or not to be a probability measure on R^d. Such results were not obtained…

Many problems in combinatorial geometry can be formulated in terms of curves or surfaces containing many points of a cartesian product. In 2000, Elekes and R\'onyai proved that if the graph of a polynomial contains $cn^2$ points of an…

The generalized *-products, or the $*_N$-products, appear both in the one-loop effective action of noncommutative Yang-Mills theories and in the coupling of a closed string to N open strings on a disk when the D-brane world-volume is…

A generalization of the GNS-representation is investigated that represents partial ^*-algebras as systems of operators acting on a partial inner product space (PIP-space). It is based on possibly indefinite B-weights which are closely…

One of the most common types of functions in mathematics, physics, and engineering is a sum of products, sometimes called a partition function. After "normalization," a sum of products has a natural graphical representation, called a normal…

Let $m,r\in\mathbb{Z}$ and $\omega\in\mathbb{R}$ satisfy $0\leqslant r\leqslant m$ and $\omega\geqslant1$. Our main result is a generalized continued fraction for an expression involving the partial binomial sum $s_m(r) =…

Let P_nk(x) denote the sum of the lowest k+1 terms in the expansion of (1+x)^n. We investigate the irreducibility of P_nk(x) and more general univariate polynomials related to it. Polynomials P_nk(x) naturally arise in Schubert calculus,…

Let $t(N)$ denote the largest number such that $N!$ can be expressed as the product of $N$ integers greater than or equal to $t(N)$. The bound $t(N)/N = 1/e-o(1)$ was apparently established in unpublished work of Erd\H{o}s, Selfridge, and…

Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q$ is a prime power and $n$ be a positive integer. In this paper, we explore the factorization of $f(x^{n})$ over $\mathbb{F}_q$, where $f(x)$ is an irreducible polynomial…

Interpolation polynomials were introduced by Knop--Sahi in type $A$, and Okounkov in type $BC$. They are inhomogeneous polynomials whose top terms are Jack and Macdonald polynomials. Thus the expansion coefficients for the product of two…

We study the combinatorial properties of final types, which are certain non-decreasing sequences of integers, together with the partitions naturally associated with them. As a consequence, we obtain an identity expressing the $n$-nacci…

The factorizations of the polynomial $X^n-1$ and the cyclotomic polynomial $\Phi_n$ over a finite field $\mathbb F_q$ have been studied for a very long time. Explicit factorizations have been given for the case that $\mathrm{rad}(n)\mid…

We establish a novel connection between the central binomial coefficients $\binom{2n}{n}$ and Gould's sequence through the construction of a specialized multivariate polynomial quotient ring. Our ring structure is characterized by ideals…

We consider polynomials of bi-degree $(n,1)$ over the skew field of quaternions where the indeterminates commute with each other and with all coefficients. Polynomials of this type do not generally admit factorizations. We recall a…

We develop first steps in the study of factorizations of elements in ultraproducts of commutative cancellative monoids into irreducible elements. A complete characterization of the (multi-)sets of lengths in such objects is given. As…

We consider sequences of integers defined by a system of linear inequalities with integer coefficients. We show that when the constraints are strong enough to guarantee that all the entries are nonnegative, the generating function for the…

The Concepts of poly-Bernoulli numbers $B_n^{(k)}$, poly-Bernoulli polynomials $B_n^{k}{(t)}$ and the generalized poly-bernoulli numbers $B_{n}^{(k)}(a,b)$ are generalized to $B_{n}^{(k)}(t,a,b,c)$ which is called the generalized…

We construct new continued fraction expansions of Jacobi-type J-fractions in $z$ whose power series expansions generate the ratio of the $q$-Pochhamer symbols, $(a; q)_n / (b; q)_n$, for all integers $n \geq 0$ and where $a,b,q \in…

We prove two new forms of Jacobi-type J-fraction expansions generating the binomial coefficients, $\binom{x+n}{n}$ and $\binom{x}{n}$, over all $n \geq 0$. Within the article we establish new forms of integer congruences for these binomial…