Related papers: On a conjecture of Ira Gessel

Let $f,g\in\mathbb{C}[z]\setminus\mathbb{C}$ and $c\in\mathbb{C}[z]$. Suppose that $\mathrm{deg}(c)=1$ if $\mathrm{deg}(f)=\mathrm{deg}(g)=1$. Using the theory of Presburger arithmetic, we prove that the rank-two recurrence set…

In the algebra $\Cal A=\Cal L(PSL(2,\Bbb Z)\otimes B(H)=\Cal L(F_N)\otimes B(H)$, $N$ finite, there exists a bounded subnormal operator $Z$, such that $\Cal A$ is the weak closure of linear span of the set ${(Z^*)^n Z^m| n,m=0,1,2...}$.

We prove Lusztig's conjectures ${\bf P1}$-${\bf P15}$ for the affine Weyl group of type $\tilde{C}_2$ for all choices of positive weight function. Our approach to computing Lusztig's $\mathbf{a}$-function is based on the notion of a…

Let $\mathcal{F}=\{A_1,A_2,\ldots,A_k\}$ be a collection of finite arithmetic progressions, where each $A_d$ is an initial segment of the set $D_d=\{d,2d,3d,\ldots\}$ of consecutive multiples of a positive integer $d$. Let $m(\mathcal{F})$…

We define an enumerative function F(n,k,P,m) which is a generalization of binomial coefficients. Special cases of this function are also power function, factorials, rising factorials and falling factorials. The first section of the paper is…

We work with lattice walks in $\mathbb{Z}^{r+1}$ using step set $\{\pm 1\}^{r+1}$ that finish with $x_{r+1} = 0$. We further impose conditions of avoiding backtracking (i.e. $[v,-v]$) and avoiding consecutive steps (i.e. $[v,v]$) each…

Let ${\mathcal U}(\lambda)$ denote the family of analytic functions $f(z)$, $f(0)=0=f'(0)-1$, in the unit disk $\ID$, which satisfy the condition $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<\lambda $ for some $0<\lambda \leq 1$. The…

The conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions $f_{house(\alpha)}(z)$ associated with the dynamical zeta functions $\zeta_{house(\alpha)}(z)$ of the…

For univalent and normalized functions $f$ the logarithmic coefficients $\gamma_n(f)$ are determined by the formula $\log(f(z)/z)=\sum_{n=1}^{\infty}2\gamma_n(f)z^n$. In the paper \cite{Pon} the authors posed the conjecture that a locally…

We approximate $d$-variate periodic functions in weighted Korobov spaces with general weight parameters using $n$ function values at lattice points. We do not limit $n$ to be a prime number, as in currently available literature, but allow…

Let $\wedge$ be a lattice in $\mathbb{R}^n$ reduced in the sense of Korkine and Zolotareff having a basis of the form $(A_1,0,0,\ldots,0),(a_{2,1},A_2,0,\ldots,0)$, $\ldots,(a_{n,1},a_{n,2},\ldots,a_{n,n-1},A_n)$ where $A_1, A_2,\ldots,A_n$…

We study the transition probability, say $p_A^n(x,y)$, of a one-dimensional random walk on the integer lattice killed when entering into a non-empty finite set $A$. The random walk is assumed to be irreducible and have zero mean and a…

We begin with (densely-defined) fractional linear transformations (FLT) on (some) Banach algebras and their relatives. This leads to Wedderburn's continued fractions (recursively-defined noncommutative polynomials) for any ring. Along the…

The purpose of these notes is to introduce some of the problems the enumeration of lattice walks is dedicated to and familiarize with some of the arguments they can be addressed with. We discuss the enumeration of lattice walks, their…

Gessel and Zeilberger generalized the reflection principle to handle walks confined to Weyl chambers, under some restrictions on the allowable steps. For those models that are invariant under the Weyl group action, they express the counting…

Given an integer mxn matrix A satisfying certain regularity assumptions, we consider the set F(A) of all integer vectors b such that the associated knapsack polytope P(A,b)={x: Ax=b, x>=0} contains an integer point. When m=1 the set F(A) is…

We investigate a functional equation which resembles the functional equation for the generating function of a lattice walk model for the quarter plane. The interesting feature of this equation is that its orbit sum is zero while its…

For $2\le k\le t<s$, the Erd\H{o}s-Rogers function $f^{(k)}_{t,s}(N)$ denotes the largest $m$ such that every $K^{(k)}_s$-free $k$-graph on $N$ vertices contains a $K^{(k)}_t$-free induced subgraph on $m$ vertices. Mubayi and Suk (J. London…

Let $R$ be a noetherian ring of dimension $d$ and let $n$ be an integer so that $n \leq d\leq 2n-3$. Let $(a_1,...,a_{n+1})$ be a unimodular row so that the ideal $J=(a_1,...,a_n)$ has height $n$. Jean Fasel has associated to this row an…

Zeckendorf's Theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. We consider higher-dimensional lattice analogues, where a legal decomposition of a number $n$ is a collection of…