Related papers: Multivariate Rogers-Szeg\"o polynomials and flags …

We study the generalized Galois numbers which count flags of length r in N-dimensional vector spaces over finite fields. We prove that the coefficients of those polynomials are asymptotically Gaussian normally distributed as N becomes…

We study Schubert polynomials using geometry of infinite-dimensional flag varieties and degeneracy loci. Applications include Graham-positivity of coefficients appearing in equivariant coproduct formulas and expansions of back-stable and…

Following our earlier work, where doubly indexed and irreducible over Q two-variable Laguerre polynomials were introduced, we prove for such polynomials some recurrence formulas and obtain a generating function. In addition, we show how…

Let $H_n(t)$ denote the classical Rogers-Szeg\"o polynomial, and let $\tH_n(t_1, \ldots, t_l)$ denote the homogeneous Rogers-Szeg\"o polynomial in $l$ variables, with indeterminate $q$. There is a classical product formula for…

Using the theory of functions of several complex variables, we prove that if an analytic function in several variables satisfies a system of $q$-partial differential equations, then, it can be expanded in terms of the product of the…

The Rogers-Szeg\"o polynomials are natural q-analogues of Newton binomials. In general they have no closed expression. We consider some exceptional cases which are products of a factor with a closed formula and another one with nice values…

We study generalizations of Schur functors from categories consisting of flags of vector spaces. We give different descriptions of the category of such functors in terms of representations of certain combinatorial categories and infinite…

Let $R$ be a commutative ring, $f \in R[X_1,\ldots,X_k]$ a multivariate polynomial, and $G$ a finite subgroup of the group of units of $R$ satisfying a certain constraint, which always holds if $R$ is a field. Then, we evaluate $\sum…

We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of…

Based on results by Brugall\'e and Mikhalkin, Fomin and Mikhalkin give formulas for computing classical Severi degrees $N^{d, \delta}$ using long-edge graphs. In 2012, Block, Colley and Kennedy considered the logarithmic version of a…

Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial…

Using sequences of finite length with positive integer elements and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their $q$-analog form via combinatorial proofs. Using the…

In this paper we establish the coefficient bodies for a wide class of families of inverse functions. We also completely describe those functions that provide boundary points of that bodies in small dimensions. As an application we get sharp…

By introducing a generalized notion of multiple zeta values associated with an arbitrary finite subset $S\subset \mathbb{P}^1(\mathbb{C})$ and studying their transformation properties under rational functions, we show that multiple…

$q$-Analogues of the coefficients of $x^a$ in the expansion of $\prod_{j=1}^N (1+x+...+x^j)^{L_j}$ are proposed. Useful properties, such as recursion relations, symmetries and limiting theorems of the ``$q$-supernomial coefficients'' are…

In this paper we consider a semi-classical variation of the weight related to the little $q$-Laguerre polynomials and obtain a second order second degree discrete equation for the recurrence coefficients in the three-term recurrence…

Linear recursions of degree $k$ are determined by evaluating the sequence of Generalized Fibonacci Polynomials, $\{F_{k,n}(t_1,...,t_k)\}$ (isobaric reflects of the complete symmetric polynomials) at the integer vectors $(t_1,...,t_k)$. If…

We consider multiple orthogonal polynomials associated with the exponential cubic weight e^{-x^3} over two contours in the complex plane. We study the basic properties of these polynomials, including the Rodrigues formula and…

We study the row-space partition and the pivot partition on the matrix space $\mathbb{F}_q^{n \times m}$. We show that both these partitions are reflexive and that the row-space partition is self-dual. Moreover, using various combinatorial…

We investigate the summability of the coefficients of $m$-homogeneous polynomials and $m$-linear mappings defined on $\ell_{p}$-spaces. In our research we obtain results on the summability of the coefficients of $m$-linear mappings defined…