The Green polynomials via vertex operators

An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Green's polynomials associated with upper partitions of length ≤3 and the diagonal lengths ≤3; (3) a Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4) an iterative formula is derived for the bitrace of the finite general linear group G and the Iwahori-Hecke algebra of type A on the permutation module of G by its Borel subgroup.     

Recursions associated to trapezoid,symmetric and rotation symmetric functions over Galois fields

Rotation symmetric Boolean functions are invariant under circular translation of indices. These functions have very rich cryptographic properties and have been used in different cryptosystems. Recently, Thomas Cusick proved that exponential sums of rotation symmetric Boolean functions satisfy homogeneous linear recurrences with integer coefficients. In this work, a generalization of this result is proved over any Galois field. That is, exponential sums over Galois fields of some rotation symmetric polynomials satisfy linear recurrences with integer coefficients. In the particular case of ${\mathbb{F}}_2$, an elementary method is used to obtain explicit recurrences for exponential sums of some of these functions. The concept of trapezoid Boolean function is also introduced and it is showed that the linear recurrences that exponential sums of trapezoid Boolean functions satisfy are the same as the ones satisfied by exponential sums of the corresponding rotations symmetric Boolean functions. Finally, it is proved that exponential sums of trapezoid and symmetric polynomials also satisfy linear recurrences with integer coefficients over any Galois field ${\mathbb{F}}_q$. Moreover, the Discrete Fourier Transform matrix and some Complex Hadamard matrices appear as examples in some of our explicit formulas of these recurrences.     

Extension of Laguerre polynomials with negative arguments

We consider the irreducibility of polynomial $L_n^{\left(\alpha \right)}\left(x\right)$ where $\alpha$ is a negative integer. We observe that the constant term of $L_n^{\left(\alpha \right)}\left(x\right)$ vanishes if and only if $n\ge \left|\alpha \right|=?\alpha$. Therefore we assume that $\alpha =?n?s?1$ where $s$ is a non-negative integer. Let $g\left(x\right)={(?1)}^n{L}_n^{(?n?s?1)}\left(x\right)=\sum _{j=0}^n{a}_j\frac{x^j}{j!}$ and more general polynomial, let $G\left(x\right)=\sum _{j=0}^n{a}_j{b}_j\frac{x^j}{j!}$ where $b_j$ with $0\le j\le n$ are integers such that $\left|b_0\right|=\left|b_n\right|=1$. Schur was the first to prove the irreducibility of $g\left(x\right)$ for $s=0$. It has been proved that $g\left(x\right)$ is irreducible for $0\le s\le 60$. In this paper, by a different method, we prove: Apart from finitely many explicitly given possibilities, either $G\left(x\right)$ is irreducible or $G\left(x\right)$ is linear factor times irreducible polynomial. This is a consequence of the estimate $s>1.9k$ whenever $G\left(x\right)$ has a factor of degree $k\ge 2$ and $(n,k,s)\ne (10,5,4)$. This sharpens earlier estimates of Shorey and Tijdeman and Nair and Shorey.     

Multifold factorizations of cyclic groups into subsets

Let $(G,+)$ be an abelian group. A finite multiset A over G is said to give a λ-fold factorization of G if there exists a multiset B over G such that each element of G occurs λ times in the multiset $A+B:=\{a+b:a\in A,b\in B\}$. In this article, restricting G to a cyclic group, we will provide sufficient conditions on a given multiset A under which the exact value or an upper bound of the minimum multiplicity λ of a factorization of G can be given by introducing a concept of ‘lcm-closure’. Furthermore, a couple of properties on a given factor A will be shown when A has a prime or prime power order (cardinality). A relation to multifold factorizations of the set of integers will be also glanced at a general perspective.