ABSTRACT: Knop and Sahi introduced a family of non-homogeneous and nonsymmetric polynomials, Gα(x; q, t), indexed by compositions. An explicit formula for the bivariate Knop-Sahi polynomials reveals a connection between these polynomials and q-special functions. In particular, relations among the q-ultraspherical polynomials of Askey and Ismail, the two variable symmetric and non-symmetric Macd...

We report on some initial results of a brute-force search for determining the maximum correlation between degree-d polynomials modulo p and the n-bit mod q function. For various settings of the parameters n, d, p, and q, our results indicate that symmetric polynomials yield the maximum correlation. This contrasts with the previouslyanalyzed settings of parameters, where non-symmetric polynomial...

Spherically symmetric random walks in arbitrary dimension D can be described in terms of Gegenbauer (ultraspherical) polynomials. For example, Legendre polynomials can be used to represent the special case of two-dimensional spherically symmetric random walks. In general, there is a connection between orthogonal polynomials and semibounded one-dimensional random walks; such a random walk can be...

tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Our aim in this paper is to discover special symmetric properties for twisted q-Genocchi polynomials. We are going to find a symmetric identity for twisted q-Genocchi zeta function. From property of the twis...

Standard algorithms for dealing with symmetric polynomials are presented using rewriting techniques. This is for instance the case of the ”fundamental theorem of symmetric polynomials”, which states that any symmetric polynomial is a polynomial in the elementary symmetric ones, and whose proof involves rewriting using a suitable elimination order. This kind approach usually spoils useful featur...

For any orthogonal polynomials system on real line we construct an appropriate oscillator algebra such that the polynomials make up the eigenfunctions system of the oscillator hamiltonian. The general scheme is divided into two types: a symmetric scheme and a non-symmetric scheme. The general approach is illustrated by the examples of the classical orthogonal polynomials: Hermite, Jacobi and La...

Under mild conditions on n, p, we give a lower bound on the number of n-variable balanced symmetric polynomials over finite fields GF (p), where p is a prime number. The existence of nonlinear balanced symmetric polynomials is an immediate corollary of this bound. Furthermore, we prove that X(2, 2`− 1) are balanced and conjecture that these are the only balanced symmetric polynomials over GF (2...

In (1.2) mκ(z) is the monomial symmetric function in the variables z1, . . . , zN , and the sum is over all partitions μ which have the same modulus as κ but are smaller in dominance ordering. The polynomials Pκ possess a host of special properties, and in fact form the natural basis for a class of symmetric multivariable orthogonal polynomials generalizing the classical orthogonal polynomials ...