Related papers: Fibonacci self-reciprocal polynomials and Fibonacc…

We give a direct and intuitive proof (via sliding some columns up and down) of the following interesting fact: if we write out the Chebyshev polynomials in a chart and take the sums of coefficients along certain diagonals, we obtain the…

We determine all permutation trinomials of type $x^{2p^s+r}+x^{p^s+r}+\lambda x^r$ over the finite field $\mathbb{F}_{p^t}$ when $(2p^s+r)^4<p^t$. This partially extends a previous result by Bhattacharya and Sarkar in the case $p=2$, $r=1$.

A modified Lagrange Polynomial is introduced for polynomial extrapolation, which can be used to estimate the equally spaced values of a polynomial function. As an example of its application, this article presents a prime-generating…

In this paper we show that for any prime number $p$ not equal to $11$ or $19$, the Tribonacci number $T_{p-1}$ is divisible by $p$ if and only if $p$ is of the form $x^2+11y^2$. We first use class field theory on the Galois closure of the…

Four recursive constructions of permutation polynomials over $\gf(q^2)$ with those over $\gf(q)$ are developed and applied to a few famous classes of permutation polynomials. They produce infinitely many new permutation polynomials over…

Let $\mu_{q+1}$ denote the set of $(q+1)$-th roots of unity in $\mathbb{F}_{q^2 }$. We construct permutation polynomials over $\mathbb{F}_{q^2}$ by using rational functions of any degree that induce bijections either on $\mu_{q+1}$ or…

We derive some q-analogs of Euler-Cassini-type identities and of recurrence formulas for powers of Fibonacci polynomials.

We define generalized bivariate polynomials, from which upon specification of initial conditions the bivariate Fibonacci and Lucas polynomials are obtained. Using essentially a matrix approach we derive identities and inequalities that in…

The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials {n} in variables s and t given by {0} = 0, {1} = 1, and {n} = s{n-1}+t{n-2} for n ge 2. The latter are defined…

In this paper, we investigate the umbral representation of the Fubini polynomials $F_{x}^{n}:=F_{n}(x)$ to derive some properties involving these polynomials. For any prime number $p$ and any polynomial $f$ with integer coefficients, we…

We obtain new bounds of exponential sums modulo a prime $p$ with sparse polynomials $a_0x^{n_0} + \cdots + a_{\nu}x^{n_\nu}$. The bounds depend on various greatest common divisors of exponents $n_0, \ldots, n_\nu$ and their differences. In…

We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.

We determine all permutation polynomials among several families of polynomials over $\mathbb{F}_{q^3}$ for arbitrary prime powers $q$. We obtain some new families of permutation polynomials over $\mathbb{F}_{q^3}$ with simple coefficients…

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 define a variant of Fibonacci-like sequences that we call prime Fibonacci sequences, where one takes the sum of the previous two terms and returns the smallest odd prime divisor of that sum as the next term. We prove that…

We prove that if x^m + c*x^n permutes the prime field GF(p), where m>n>0 and c is in GF(p)^*, then gcd(m-n,p-1) > sqrt{p} - 1. Conversely, we prove that if q>=4 and m>n>0 are fixed and satisfy gcd(m-n,q-1) > 2q*(log log q)/(log q), then…

We explore a combinatorial bijection between two seemingly unrelated topics: the roots of irreducible polynomials of degree $m$ over a finite field $F_p$ for a prime number $p$ and the number of points that are periodic of order $m$ for a…

In this paper, a computationally simple and explicit method of constructing recursive sequence of primitive polynomials of degree $n2^k (k = 1, 2, 3,\ldots)$ over $\mathbb{F}_{q}$ is given.

We prove some separation results for the roots of the generalized Fibonacci polynomials and their absolute values

Let $p$ be a prime and $q$ a power of $p$. For $n\ge 0$, let $g_{n,q}\in\Bbb F_p[{\tt x}]$ be the polynomial defined by the functional equation $\sum_{a\in\Bbb F_q}({\tt x}+a)^n=g_{n,q}({\tt x}^q-{\tt x})$. When is $g_{n,q}$ a permutation…