Define the $n$-th fibotomic polynomial to be the product of the monic irredicible factors of the $n$-th Fibonacci polynomial which are not factors of any Fibonacci polynomial of smaller degree. In this paper, we prove a number of properties of the fibotomic polynomials. This includes determining the discriminant of the fibotomic polynomials and the resultant of pairs of fibotomic polynomials. Furthermore, we completely determine the factorization form of the fibotomic polynomials in prime fields. Results are also generalized for the bivariate homogenous fibotomic polynomials.
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 satisfying remarkable recurrence relations.
In this paper, we investigated properties of Tribonacci-Lucas polynomials which generalized Tribonacci-Lucas numbers. From this generalization, we also obtain some new algebraic properties on these numbers and polynomials as Binet formula, summation, binomial sum and generating function.
The purpose of this paper is to generalize this relation of symmetry between the power sum polynomials and the generalized Euler polynomials to the relation between the power sum polynomials and the generalized higher-order Euler polynomials.
The aim of this paper is to study the degenerate Bell numbers and polynomials which are degenerate version of the Bell numbers and polynomials. we derive some new identities and properties of those numbers and polynomials that are associated with the degenerate Stirling numbers of the both kinds.
In this paper, we establish more properties of generalized poly-Euler polynomials with three parameters and we investigate a kind of symmetrized generalization of poly- Euler polynomials. Moreover, we introduce a more general form of multi poly-Euler polynomials and obtain some identities parallel to those of the generalized poly-Euler polynomials.
In this article, we study several probabilistic properties of polynomials defined over the ring of $p$-adic integers under the Haar measure. First, we calculate the probability that a monic polynomial is separable, generalizing a result of Polak. Second, we introduce the notion of two polynomials being strongly coprime and calculate the probability of two monic polynomials {being} strongly coprime. Finally, we explain how our method can be used to extrapolate other probabilistic properties of polynomials over the ring of $p$-adic integers from polynomials defined over the integers modulo power…
In this paper, we study the formulae for a product of two product Euler polynomials. From this study, we derive some formulae for the integral of the product of two or more Ruler polynomials.
Many authors studied mix type polynomials and their properties. For their numerous uses in Statistics, Combinatorial analysis and Number theory. In the present article, we define a generating function of mix type Bell based Apostol type polynomials and numbers of order a. Furthermore, we derive some elementary properties and identities of Bell based Apostol type polynomials of order a such as correlation formulas, Implicit summation formulas, derivative formulas and their special cases. Also, we derive some relation with Stirling numbers.
Recently, the higher-order q-Euler polynomials and multiple q-Euler zeta functions are introduced by T. Kim ([8, 9]). In this paper, we investigate some symmetric properties of the multiple q-Euler zeta function and derive various identities concerning the higher-order q-Euler polynomials from the symmetric properties of the multiple q-Euler zeta functions.
For each integer $n\geq 1$ we consider the unique polynomials $P, Q\in\mathbb{Q}[x]$ of smallest degree $n$ that are solutions of the equation $P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$. We derive numerous properties of these polynomials and their derivatives, including explicit expansions, differential equations, recurrence relations, generating functions, resultants, discriminants, and irreducibility results. We also consider some related polynomials and their properties.
We show that polynomials associated with automatic sequences satisfy a certain recurrence relation when evaluated at a root of unity, which generalizes a result of Brillhart, Lomont and Morton on the Rudin--Shapiro polynomials. We study the minimal order of such a relation and the integrality of its coefficients.
The modified Bernoulli numbers \begin{equation*} B_{n}^{*} = \sum_{r=0}^{n} \binom{n+r}{2r} \frac{B_{r}}{n+r}, \quad n > 0 \end{equation*} introduced by D. Zagier in 1998 were recently extended to the polynomial case by replacing $B_{r}$ by the Bernoulli polynomials $B_{r}(x)$. Arithmetic properties of the coefficients of these polynomials are established. In particular, the 2-adic valuation of the modified Bernoulli numbers is determined. A variety of analytic, umbral, and asymptotic methods is used to analyze these polynomials.
In this paper, we show that various kinds of integer polynomials with prescribed properties of their roots have positive density. For example, we prove that almost all integer polynomials have exactly one or two roots with maximal modulus. We also show that for any positive integer $n$ and any set of $n$ distinct points symmetric with respect to the real line, there is a positive density of integer polynomials of degree $n$, height at most $H$ and Galois group $S_n$ whose roots are close to the given $n$ points.
In this paper, we study some properties of associated sequences of special polynomials. From the properties of associated sequences of polynomials, we derive some interesting identities of special polynomials.
In this paper we investigate some properties for the q-Euler numbers ans polymials. From these properties we give some identities on the Bernstein polymials and q-Euler polynpmials.
In this paper, we investigate the properties of q-Hermite polynomials related to q-Bernstein polynomials. From these properties, we derive some interesting relations between q-Berstein polynomials and q-Hermite polynomials.
In this work we introduce interesting infinite series, related to Ramanujan-Soldner constant. Our method uses general properties of polynomials of binomial type and Lagrange inversion theorem. Also we study properties of the operator 1/dlog, acting on formal power series. In addition, several properties of polynomials of binomial type associated to elementary functions are discussed.
In this paper, we introduce the new fully degenerate poly-Bernoulli numbers and polynomials and investigate some properties of these polynomials and numbers. From our properties, we derive some identities for the fully degenerate poly-Bernoulli numbers and polynomials.
The main purpose of this paper is to present various identities and computation formulas for certain classes of Apostol-type numbers and polynomials. The results of this paper contain not only the $λ$-Apostol-Daehee numbers and polynomials, but also Simsek numbers and polynomials, the Stirling numbers of the first kind, the Daehee numbers, and the Chu-Vandermonde identity. Furthermore, we derive an infinite series representation for the $λ$-Apostol-Daehee polynomials. By using functional equations containing the generating functions for the Cauchy numbers and the Riemann integrals of the gener…
In this paper, we study some properties of umbral calculus related to Appell sequence. From those properties, we derive new and interesting identities of Frobenius-Euler polynomials.
The aim of this paper is to give a new approach to modified $q$-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling numbers and generalized Bernoulli polynomials are derived. Moreover, the generating function, interpolation function of these polynomials of several variables and also the derivatives of these polynomials and their generating function are given. Finally, we get new interesting identities of modified $q$-Bernoulli numbers and $q$-Euler numbers applying $p$-a…
In this paper we study the Fibonacci numbers and derive some interesting properties and recurrence relations. We prove some charecterizations for $F_p$, where $p$ is a prime of a certain type. We also define period of a Fibonacci sequence modulo an integer, $m$ and derive certain interesting properties related to them. Afterwards, we derive some new properties of a class of generalized Fibonacci numbers. In the last part of the paper we introduce some generalized Fibonacci polynomial sequences and we derive some results related to them.
This research is aimed to give a determinantal definition for the $q$-Appell polynomials and show some classical properties as well as find some interesting properties of the mentioned polynomials in the light of the new definition.
In this paper, we study some properties of associated sequaences in umbral calculus. From these properties, we derive new and interesting identities of several kinds of polynomials.
In this paper, we study some properties of Changhee's q-Bernou lli polynomials which are derived from p-adic invariant integral on Zp. By using these properties, we give some interesting identities related to higher- order q-Bernoulli polynomials.
In this paper, we study some symmetric properties of the multiple q-Euler zeta function. From these properties, we derive several identities of symmetry for the (h;q)-extension of higher-order Euler polynomials.
In this paper, by the properties of p-adic invariant integral on Zp, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties of p-adic invariant integral on Zp, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.
Inspired by the framework of operational methods and based on the generating functions of Legendre-Gould Hopper polynomials and Sheffer sequences, we discuss certain new mixed type polynomials and their important properties. We show that the use of operational nature allows the relevant polynomials to be unified and general in nature. It is illustrated how the polynomials, we develop, provide an easy derivation of a wide class of new and known polynomials, and their respective properties.
Eisenstein polynomials, which were defined by Oura, are analogues of the concept of an Eisenstein series. Oura conjectured that there exist some analogous properties between Eisenstein series and Eisenstein polynomials. In this paper, we provide new analogous properties of Eisenstein polynomials and zeta polynomials. These properties are finite analogies of certain properties of Eisenstein series.
In this paper, we investigate some symmetric properties of the multivariate p-adic fermionic integrals on Zp and derive various identities concerning the expansions of q-Euler polynomials from the symmetric properties of the multivariate p-adic fermionic integrals on Zp
Recently, D. S. Kim and T. Kim have studied applications of um- bral calculus associated with p-adic invariant integrals on Zp (see [6]). In this paper, we investigate some interesting properties arising from umbral calculus. These properties are useful in deriving some identities of Bernoulli polynomials.
In this work we continue to study the properties of polynomials of binomial type and their canonical continuations to the complex index by exploring the properties of transformation T:=1/dlog which acts on formal power series $f(x)$ of the form $x+x^2\mathbb{C}[[x]]$.
This paper investigates coefficients of cyclotomic polynomials theoretically and experimentally. We prove the following result. {{\em If $n=p_1\ldots p_k$ where $p_i$ are odd primes and $p_1<p_2<\ldots<p_r<p_1+p_2<p_{r+1}<\ldots<p_t$ with $t\geq 3$ odd, then the numbers $-(r-2),-(r-3),\ldots, r-2, r-1$ are all coefficients of the cyclotomic polynomial $Φ_{2n}$. Furthermore, if $1+p_r<p_1+p_2$ then $1-r$ is also a coefficient of $Φ_{2n}$.} In the experimental part, in two instances we present computational evidence for asymptotic symmetry between distribution of positive and negative coefficien…
We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers $r,n\geq 0$, we conjecture that $L_n^{(-1-n-r)}(x) = \sum_{j=0}^n \binom{n-j+r}{n-j}x^j/j!$ is a $\Q$-irreducible polynomial whose Galois group contains the alternating group on $n$ letters. That this is so for $r=n$ was conjectured in the 50's by Grosswald and proven recently by Filaseta and Trifonov. It follows from recent work of Hajir and Wong that the conjecture is true when $r$ is large with respect to $n\geq 5$. Here we verify it in three situations: i) when …
We provide several properties of the geometric polynomials discussed in earlier works of the authors. Further, the geometric polynomials are used to obtain a closed form evaluation of certain series involving Riemann's zeta function.
In this paper, we present a new type of generating function of generalized Apostol-Bernoulli poly Daehee polynomial (GABPDP). By using the generating function of GABPDP, we discuss some special cases and useful identities of generalized Apostol-Bernoulli poly Daehee polynomials. We also drive implicit summation formulae of it.
We prove several theorems concerning arithmetic properties of Stern polynomials defined in the following way: $B_{0}(t)=0, B_{1}(t)=1, B_{2n}(t)=tB_{n}(t)$, and $B_{2n+1}(t)=B_{n}(t)+B_{n+1}(t)$. We study also the sequence $e(n)=\op{deg}_{t}B_{n}(t)$ and give various of its properties.
Eisenstein polynomials, which were defined by the second author, are analogues of the concept of an Eisenstein series. The second author conjectured that there exist some analogous properties between Eisenstein series and Eisenstein polynomials. In the previous paper, the first author provided new analogous properties of Eisenstein polynomials and zeta polynomials for the Type II case. In this paper, the analogous properties of Eisenstein polynomials and zeta polynomials are shown to also hold for the Type I, Type III, and Type IV cases. These properties are finite analogies of certain propert…
We investigate algebraic and arithmetic properties of a class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. In addition to divisibility and irreducibility results we also consider rational roots. This leads to the study of an infinite class of integer sequences which have interesting properties and satisfy linear recurrence relations.
We continue our study of residual properties of mapping tori of free group endomorphisms. In this paper, we prove that each of these groups are virtually residually (finite $p$)-groups for all but finitely many primes$p$. The method involves further studies of polynomial maps over finite fields and $p$-adic completions of number fields.
We investigate the Stern polynomials defined by $B_0 ( t ) =0,B_1 ( t ) =1$, and for $n \geq 2$ by the recurrence relations $B_{2n}( t) =tB_{n}( t) ,$ $B_{2n+1}( t) =B_n( t) +B_{n+1}( t) $. We prove that all possible rational roots of that polynomials are $0,-1,-1/2,-1/3$. We give complete characterization of $n$ such that $deg( B_n) = deg( B_{n+1}) $ and $deg( B_n) =deg( B_{n+1}) =deg( B_{n+2}) $. Moreover, we present some result concerning reciprocal Stern polynomials.
We introduce a multivariate analogue of Bernoulli polynomials and give their fundamental properties: difference and differential relations, symmetry, explicit formula, inversion formula, multiplication theorem, and binomial type formula. Further, we consider a multivariate analogue of the multiple Bernoulli polynomials and give their fundamental properties.
Numerous results on self-reciprocal polynomials over finite fields have been studied. In this paper we generalize some of these to a-self reciprocal polynomials defined in [4]. We consider some properties of the divisibility of a-reciprocal polynomials and characterize the parity of the number of irreducible factors for a-self reciprocal polynomials over finite fields of odd characteristic.
Let $B_{n}(t)$ be a $n$-th Stern polynomial and let $e(n)=\op{deg}B_{n}(t)$ be its degree. In this note we continue our study started in \cite{Ul} of the arithmetic properties of the sequence of Stern polynomials and the sequence $\{e(n)\}_{n=1}^{\infty}$. We also study the sequence $d(n)=\op{ord}_{t=0}B_{n}(t)$. Among other things we prove that $d(n)=ν(n)$, where $ν(n)$ is the maximal power of 2 which dividies the number $n$. We also count the number of the solutions of the equations $e(m)=i$ and $e(m)-d(m)=i$ in the interval $[1,2^{n}]$. We also obtain an interesting closed expression for a …
The general number field sieve (GNFS) is the most efficient algorithm known for factoring large integers. It consists of several stages, the first one being polynomial selection. The quality of the chosen polynomials in polynomial selection can be modelled in terms of size and root properties. In this paper, we describe some algorithms for selecting polynomials with very good root properties.
