Related papers: Polynomials of Binomial Type and Lucas' Theorem
We give elementary proofs of some congruence criteria to compute binomial coefficients in modulo a prime. These criteria are analogues to the symmetry property of binomial coefficients. We give extended version of Lucas Theorem by using…
We derive formulas for characterizing bounded orthogonally additive polynomials in two ways. Firstly, we prove that certain formulas for orthogonally additive polynomials derived in \cite{Kusa} actually characterize them. Secondly, by…
We give necessary and sufficient existence criteria, and methods for finding, continuous solutions of linear equations whose coefficients are polynomials.
We derive a collection of identities for bivariate Fibonacci and Lucas polynomials using essentially a matrix approach as well as properties of such polynomials when the variables $x$ and $y$ are replaced by polynomials. A wealth of…
Our paper deals about identities involving Bell polynomials. Some identities on Bell polynomials derived using generating function and successive derivatives of binomial type sequences. We give some relations between Bell polynomials and…
This work is a study of polynomial compositions having a fixed number of terms. We outline a recursive method to describe these characterizations, give some particular results and discuss the general case. In the final sections, some…
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
Using generating functions, we derive many identities involving balancing and Lucas-balancing polynomials. By relating these polynomials to Chebyshev polynomials of the first and second kind, and Fibonacci and Lucas numbers, we offer some…
In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds…
In recent years, the notion of characteristic polynomial of representations of Lie algebras has been widely studied. This paper provides more properties of these characteristic polynomials. For simple Lie algebras, we characterize the…
In their study of a binomial sum related to Wolstenholme's theorem, Chamberland and Dilcher prove that the corresponding sequence modulo primes $p$ satisfies congruences that are analogous to Lucas' theorem for the binomial coefficients…
We show the existence of and explicitly construct generic polynomials for various groups, over fields of positive characteristic. The methods we develop apply to a broad class of connected linear algebraic groups defined over finite fields…
A class of bilinear permutation polynomials over a finite field of characteristic 2 was constructed in a recursive manner recently which involved some other constructions as special cases. We determine the compositional inverses of them…
In this article, we survey the the recent literature surrounding the geometry of complex polynomials. Specific areas surveyed are i) Generalizations of the Gauss--Lucas Theorem, ii) Geometry of Polynomials Level Sets, and iii) Shape…
The paper studies constructions of irreducible polynomials over finite fields using polynomial composition method.
The real type of a finite family of univariate polynomials characterizes the combined sign behavior of the polynomials over the real line. We derive an explicit formula for the number of real types subject to given degree bounds. For the…
In this paper we generalize to bivariate polynomials of Fibonacci and Lucas, properties obtained for Chebyshev polynomials. We prove that the coordinates of the bivariate polynomials over appropriate basis are families of integers…
Permutation polynomials over finite fields have taken an important role in vast areas in mathematics as well as engineering. Recently, Tu et al. gave some classes of complete permutation polynomials over finite fields of even…
We propose several techniques to construct complete permutation polynomials of finite fields by virtue of complete permutations of subfields. In some special cases, any complete permutation polynomials over a finite field can be used to…
In this paper we introduce a new sequence of polynomials, which follow the same recursive rule of the well-known Lucas-Lehmer integer sequence. We show the most important properties of this sequence, relating them to the Chebyshev…