Let wλ(x) := (1−x2)λ−1/2 and P (λ) n be the ultraspherical polynomials with respect to wλ(x). Then we denote by E (λ) n+1 the Stieltjes polynomials with respect to wλ(x) satisfying ∫ 1 −1 wλ(x)P (λ) n (x)E (λ) n+1(x)x dx { = 0, 0 ≤ m < n+ 1, = 0, m = n+ 1. In this paper, we show uniform convergence of the Hermite–Fejér interpolation polynomials Hn+1[·] and H2n+1[·] based on the zeros of the Sti...

We give an exact characterization of permutation polynomials mod ulo n w w a polynomial P x a a x adx d with integral coe cients is a permutation polynomial modulo n if and only if a is odd a a a is even and a a a is even We also characterize polynomials de ning latin squares mod ulo n w but prove that polynomial multipermutations that is a pair of polynomials de ning a pair of orthogonal latin...

Krawtchouk polynomials play an important role in coding theory and are also useful in graph theory and number theory. Although the basic properties of these polynomials are known to some extent, there is, to my knowledge, no detailed development available. My aim in writing this article is to fill in this gap. Notation In the following we will use capital letters for (algebraic) polynomials, fo...

We investigate the zeros of a family of hypergeometric polynomials 2F1(−n,−x; a; t), n ∈ N that are known as the Meixner polynomials for certain values of the parameters a and t. When a = −N, N ∈ N and t = p , the polynomials Kn(x; p,N) = (−N)n2F1(−n,−x;−N; p ), n = 0, 1, . . .N, 0 < p < 1 are referred to as Krawtchouk polynomials. We prove results for the zero location of the orthogonal polyno...

Let p be a prime and let φ ∈ Zp[x1, x2, . . . , xp] be a symmetric polynomial, where Zp is the field of p elements. A sequence T in Zp of length p is called a φ-zero sequence if φ(T ) = 0; a sequence in Zp is called a φ-zero free sequence if it does not contain any φ-zero subsequence. Define g(φ, Zp) to be the smallest integer l such that every sequence in Zp of length l contains a φ-zero seque...

We study asymptotics of characters of the symmetric groups on a fixed conjugacy class. It was proved by Kerov that such a character can be expressed as a polynomial in free cumulants of the Young diagram (certain functionals describing the shape of the Young diagram). We show that for each genus there exists a universal symmetric polynomial which gives the coefficients of the part of Kerov char...

A theorem due to Tokuyama expresses Schur polynomials in terms of GelfandTsetlin patterns, providing a deformation of the Weyl character formula and two other classical results, Stanley’s formula for the Schur q-polynomials and Gelfand’s parametrization for the Schur polynomials. We generalize Tokuyama’s formula to the Hall-Littlewood polynomials by extending Tokuyama’s statistics. Our result, ...

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...

This behavior has been seen in some notable cases. Kirillov [3] shows that elementary symmetric polynomials in noncommuting variables commute (and, in some cases, all Schur functions) when elementary symmetric polynomials of degree at most three commute when restricted to at most three of the variables. Generalizing this, Blasiak and Fomin [1] give a wider theory for rules of three of generatin...

Using the action of the Yang–Baxter elements of the Hecke algebra on polynomials, we define two bases of polynomials in n variables. The Hall–Littlewood polynomials are a subfamily of one of them. For q = 0, these bases specialize to the two families of classical Key polynomials (i.e., Demazure characters for type A). We give a scalar product for which the two bases are adjoint to each other.