Related papers: Powerful Fibonacci polynomials over finite fields

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…

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…

We continue our study on relationships between Bernoulli polynomials and balancing (Lucas-balancing) polynomials. From these polynomial relations, we deduce new combinatorial identities with Fibonacci (Lucas) and Bernoulli numbers.…

Let $(G_k)_{k\in\mathbb Z}$ be any sequence obeying the recurrence relation of the Fibonacci numbers. We derive formulas for $\sum_{j=1}^n{G_{j + t}^6}$ and $\sum_{j=1}^n{(-1)^{j - 1}G_{j + t}^5(G_{j + t - 1} + G_{j + t + 1})}$, thereby…

Let $ \{F_n\}_{n\ge 0} $ be the sequence of Fibonacci numbers and let $p$ be a prime. For an integer $c$ we write $m_{F,p}(c)$ for the number of distinct representations of $c$ as $F_k-p^\ell$ with $k\ge 2$ and $\ell\ge 0$. We prove that…

Denote by s_F(n) the minimal number of Fibonacci numbers needed to write n as a sum of Fibonacci numbers. We obtain the extremal minimal and maximal orders of magnitude of s_F(n^h)/s_F(n) for any h>= 2. We use this to show that for all…

In this article we compute the $q$th power values of the quadratic polynomials $f$ with negative squarefree discriminant such that $q$ is coprime to the class number of the splitting field of $f$ over $\mathbb{Q}$. The theory of unique…

Fibonacci sequence, generated by summing the preceding two terms, is a classical sequence renowned for its elegant properties. In this paper, leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of…

Fibonacci polynomials are generalizations of Fibonacci numbers, so it is natural to consider polynomial versions of the various results for Fibonacci numbers. According to Hong, Pongsriiam, Bulawa, and Lee, the generating function of the…

We study the set of common F_q-rational zeros of systems of multivariate symmetric polynomials with coefficients in a finite field F_q. We establish certain properties on these polynomials which imply that the corresponding set of zeros…

We investigate $k$-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most $k$. Let $\mathbb F$ be a finite field of characteristic $p$. We…

We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.

The Fibonacci numbers are a sequence of integers in which every number after the first two, 0 and 1, is the sum of the two preceding numbers. These numbers are well known and algorithms to compute them are so easy that they are often used…

Divisibility sequences are defined by the property that their elements divide each other whenever their indices do. The divisibility sequences that also satisfy a linear recurrence, like the Fibonacci numbers, are generated by polynomials…

The Fibonacci word is the fixed point beginning with the letter $a$ of morphism $\sigma(a)=ab$, $\sigma(b)=a$ defined over the alphabet $\{a,b\}$. In this paper, we get explicit expression of the number of distinct fractional powers in each…

We give new identities for some symmetric polynomials. As applications of these identities, we obtain some formulas for a higher order analogue of Fibonacci and Lucas numbers.

An integer sequence is called realizable if it is the count of periodic points of some map. The Fibonacci sequence $(F_n)$ does not have this property, and the Fibonacci sequence sampled along the squares $(F_{n^2})$ also does not have this…

Let $(F_n)_{n\geq 0}$ and $(L_n)_{n\geq 0}$ be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of $…

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we develop general theorems on permutation polynomials over finite fields. As a…

(Below, \Box means "perfect square") Let $P$ and $Q$ be non-zero integers. The Lucas sequence $\{U_n(P,Q)\}$ is defined by $U_0=0$, $U_1=1$, $U_n=P U_{n-1}-Q U_{n-2}$, $(n \geq 2)$. Historically, there has been much interest in when the…