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Motivated by many recent constructions of permutation polynomials over $\mathbb{F}_{q^2}$, we study permutation polynomials over $\mathbb{F}_{q^3}$ in terms of their coefficients. Based on the multivariate method and resultant elimination,…

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…

We give a combinatorial proof of a formula giving the partial sums of the $k$-bonacci sequence as alternating sums of powers of two multiplied by binomial coefficients. As a corollary we obtain a formula for the $k$-bonacci numbers.

We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the…

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…

The aim of this expository article is twofold. The first is to introduce several polynomials of one variable as well as two variables defined on the positive integers with values as congruent numbers. The second is to present connections…

In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing…

In the paper, the author finds an explicit formula for computing Bell numbers in terms of Lah numbers and Stirling numbers of the second kind.

We prove several congruences for trinomial coefficients.

This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the…

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…

In the paper it is demonstrated that Bells theorem is an unprovable theorem.

Using an explicit computable expression of ordinary multinomials, we establish three remarkable connections, with the q-generalized Fibonacci sequence, the exponential partial Bell partition polynomials and the density of convolution powers…

The nth r-extended Lah-Bell number is defined as the number of ways a set with $n+r$ elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to…

We prove a version of van der Corput's Lemma for polynomials over the p-adic numbers.

We present a proof of completeness for the implicational propositional calculus, based on a variant of the Lindenbaum procedure.

Departing from a class of infinite series with central binomial coefficients in the numerator and depending on a positive integer parameter, we first extend known identities to all complex parameters. Then we use various methods, including…

The aim of this paper is to give some combinatorial relations linked polynomials generalizing those of Appell type to the partial r-Bell polynomials. We give an inverse relation, recurrence relations involving some family of polynomials and…

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…

The paper [GLZ] "L-functions of Carlitz modules, resultantal varieties and rooted binary trees" is devoted to a description of some resultantal varieties related to L-functions of Carlitz modules. It contains a conjecture that some of these…