Some Hermite polynomial identities and their combinatorics

Markov Type Polynomial Inequality for Some Generalized Hermite Weight

i=0 (n − i)‖p‖∞, p(x) ∈ Πn, 1 ≤ k ≤ n are typical examples of inequalities connecting norms of a polynomial and its derivatives. In (1.1) the equality holds only at x = ±1 and only when p(x) = cTn(x), where Tn(x) is the Chebyshev polynomial of the first kind of a degree n c © 2011 Mathematical Institute, Slovak Academy of Sciences. 2010 Mathemat i c s Sub j e c t C l a s s i f i c a t i on: 33C...

Some Polynomial Identities that Imply Commutativity of Rings

In this paper, we establish some commutativity theorems for certain rings with polynomial constraints as follows: Let R be an associative ring, and for all x, y ∈ R, and fixed non-negative integers m > 1, n ≥ 0, r > 0, s ≥ 0, t ≥ 0, p ≥ 0, q ≥ 0 such that P (x, y) = ±Q(x, y), where P (x, y) = ys[x, y]yt and Q(x, y) = xp[xm, yn]ryq. First,it is shown that a semiprime ring R is commutative if and...

Polynomial identities of some minimal varieties with low PI - exponent

The combinatorics of associated Hermite polynomials

We develop a combinatorial model of the associated Hermite polynomials and their moments, and prove their orthogonality with a sign-reversing involution. We find combinatorial interpretations of the moments as complete matchings, connected complete matchings, oscillating tableaux, and rooted maps and show weight-preserving bijections between these objects. Several identities, linearization form...

Combinatorics of Polynomial Iterations

A complete description of the iterated monodromy groups of postcritically finite backward polynomial iterations is given in terms of their actions on rooted trees and automata generating them. We describe an iterative algorithm for finding kneading automata associated with post-critically finite topological polynomials and discuss some open questions about iterated monodromy groups of polynomials.