The polynomial analogue of a theorem of Renyi

New polynomial analogues of Jacobi's triple product and Lebesgue's identities

In a recent paper by the authors, a bounded version of Goellnitz's (big) partition theorem was established. Here we show among other things how this theorem leads to nontrivial new polynomial analogues of certain fundamental identities of Jacobi and Lebesgue. We also derive a two parameter extension of Jacobi's famous triple product identity.

A note on q-analogue of Boole polynomials

In this paper, we consider the q-extensions of Boole polynomials. From those polynomials, we derive some new and interesting properties and identities related to special polynomials.

An overpartition analogue of the $q$-binomial coefficients

We define an overpartition analogue of Gaussian polynomials (also known as $q$-binomial coefficients) as a generating function for the number of overpartitions fitting inside the $M \times N$ rectangle. We call these new polynomials over Gaussian polynomials or over $q$-binomial coefficients. We investigate basic properties and applications of over $q$-binomial coefficients. In particular, via the recurrences and combinatorial interpretations of over q-binomial coefficients, we prove a Rogers-Ramaujan type partition theorem.

Study on q-analogues of Catala-Daehee numbers and polynomials

Catalan-Daehee numbers and polynomials, generating functions of which can be expressed as p-adic Volkenborn integrals on Zp, were studied previously. The aim of this paper is to introduce q-analogues of the catalan-Daehee numbers and polynomials with the help of p-adic q-integrals von Zp. We derive, among other thing, some explicit expressions for the q-analogues of the Catalan-Daehee numbers and polynomials.

An explicit polynomial analogue of Romanoff's theorem

Given a polynomial $g$ of positive degree over a finite field, we show that the proportion of polynomials of degree $n$, which can be written as $h+g^k$, where $h$ is an irreducible polynomial of degree $n$ and $k$ is a nonnegative integer, has order of magnitude $1/°g$.

关于迪比茨公式和欧拉常数的有限类比及其格雷戈里多项式

On finite analogues of Dobiński's formula and of Euler's constant via Gregory polynomials

本文研究了与欧拉常数e相关的迪比茨公式的有限类比及其贝塞尔型推广，并利用格雷戈里多项式扩展了kaneko-matsusaka-seki关于欧拉常数有限类比的研究，与wilson型类比γ_a^w进行比较。

Continued fractions in function fields: polynomial analogues of McMullen's and Zaremba's conjectures

We examine the polynomial analogues of McMullen's and Zaremba's conjectures on continued fractions with bounded partial quotients. It has already been proved by Blackburn that if the base field is infinite, then the polynomial analogue of Zaremba's conjecture holds; we will prove this again with a different method and examine some known results for finite base fields. Translating to the polynomial setting a result of Mercat, we will prove that the polynomial analogue of McMullen's conjecture holds over infinite algebraic extensions of finite fields and that, over finite fields, it would be a c…

On p-adic analogue of q-Bernstein polynomials and related integrals

In this paper, we give p-adic q-integral representation for the Kim's q-Bernstein polynomials and we give some interesting formulae realted to Carlitz's q-Bernoulli numbers.

Quadratic Symmetric Polynomials and an analogue of the Davenport Constant

In this paper, we define the constant $D(φ, p)$, an analogue for the Davenport constant, for sequences on the finite field $\mathbb{F}_p$, defined via quadratic symmetric polynomials. Next, we state a series of results presenting either the exact value of $D(φ, p)$, or lower and upper bounds for this constant.

$q$-Rational Reduction and $q$-Analogues of Series for $π$

In this paper, we present a $q$-analogue of the polynomial reduction which was originally developed for hypergeometric terms. Using the $q$-Gosper representation, we describe the structure of rational functions that are summable when multiplied with a given $q$-hypergeometric term. The structure theorem enables us to generalize the $q$-polynomial reduction to the rational case, which can be used in the automatic proof and discovery of $q$-identities. As applications, several $q$-analogues of series for $π$ are presented.

An analogue of Ruzsa's conjecture for polynomials over finite fields

In 1971, Ruzsa conjectured that if $f:\ \mathbb{N}\rightarrow\mathbb{Z}$ with $f(n+k)\equiv f(n)$ mod $k$ for every $n,k\in\mathbb{N}$ and $f(n)=O(θ^n)$ with $θ<e$ then $f$ is a polynomial. In this paper, we investigate the analogous problem for the ring of polynomials over a finite field.

An Analogue of Weil's Converse Theorem for Harmonic Maass Forms of Polynomial Growth

We construct a family of harmonic Maass forms of polynomial growth of any level corresponding to any cusp whose shadows are Eisenstein series of integral weight. We further consider Dirichlet series attached to a harmonic Maass form of polynomial growth, study its analytic properties, and prove an analogue of Weil's converse theorem.

A new family of q-analogue of Genocchi numbers and polynomials of higher order

In this work, we consider the generating function of Kim's q-Euler polynomials and introduce new generalization of q-Genocchi polynomials and numbers of higher order. Also, we give surprising identities for studying in Analytic Numbers Theory and especially in Mathematical Physics. Moreover, by applying q-Mellin transformation to generating function of q-Genocchi polynomials of higher order and so we define q-Hurwitz-Zeta type function which interpolates of this polynomials at negative integers.

Polynomial analogue of the Smarandache function

In the integer case, the Smarandache function of a positive integer $n$ is defined to be the smallest positive integer $k$ such that $n$ divides the factorial $k!$. In this paper, we first define a natural order for polynomials in $\mathbb{F}_q[t]$ over a finite field $\mathbb{F}_q$ and then define the Smarandache function of a non-zero polynomial $f \in \mathbb{F}_q[t]$, denoted by $S(f)$, to be the smallest polynomial $g$ such that $f$ divides the Carlitz factorial of $g$. In particular, we establish an analogue of a problem of Erd{\H o}s, which implies that for almost all polynomials $f$, $…

Analogues of the $3x + 1$ Problem in Polynomial Rings of Characteristic 2

The Collatz conjecture (also known as the $3x+1$ problem) concerns the behavior of the discrete dynamical system on the positive integers defined by iteration of the so-called $3x + 1$ function. We investigate analogous dynamical systems in rings of functions of algebraic curves over $\mathbb{F}_2$ . We prove in this setting a generalized analogue of a theorem of Terras concerning the asymptotic distribution of stopping times. We also present experimental data on the behavior of these dynamical systems.

An overpartition analogue of $q$-binomial coefficients, II: combinatorial proofs and $(q,t)$-log concavity

In a previous paper, we studied an overpartition analogue of Gaussian polynomials as the generating function for overpartitions fitting inside an $m \times n$ rectangle. Here, we add one more parameter counting the number of overlined parts, obtaining a two-parameter generalization $\overline{m+n \brack n}_{q,t}$ of Gaussian polynomials, which is also a $(q,t)$-analogue of Delannoy numbers. First we obtain finite versions of classical $q$-series identities such as the $q$-binomial theorem and the Lebesgue identity, as well as two-variable generalizations of classical identities involving Gauss…

Polynomial analogues of restricted multicolor b-ary partition functions

Given an integer base $b\geq 2$, a number $ρ\geq 1$ of colors, and a finite sequence $Λ=(λ_1,\ldots,λ_ρ)$ of positive integers, we introduce the concept of a $Λ$-restricted $ρ$-colored $b$-ary partition of an integer $n\geq 1$. We also define a sequence of polynomials in $λ_1+\cdots+λ_ρ$ variables, and prove that the $n$th polynomial characterizes all $Λ$-restricted $ρ$-colored $b$-ary partitions of $n$. In the process we define a recurrence relation for the polynomials in question, obtain explicit formulas and identify a factorization theorem.

Values of symmetric polynomials and a truncated analogue of the Riemann zeta function

For each positive integer n, we determine the set of symmetric functions f for which the congruence f(p/1,p/2,...,p/(p-1)) \equiv 0 mod p^n holds for all sufficiently large primes p. Our determination is conditional on a conjecture regarding the modulo p independence of Bernoulli numbers. In a recent work the author introduced a new truncated analogue of the multiple zeta function and investigated a class of relations among values of this function at positive integers. The question answered in the present work is equivalent to the determination of the relations satisfied by values of the corre…

关于毕达哥拉斯三元组的Berggren定理的多项式类比

A polynomial analogue of Berggren's theorem on Pythagorean triples

该研究在特征≠2的域上的一元多项式环中建立了Berggren定理的多项式类比，并由此推导出多项式环上毕达哥拉斯二次型对应正交群的若干结构定理。

A potential analogue of Schinzel's hypothesis for polynomials with coefficients in Fq[t]

The Schinzel hypothesis essentially claims that finitely many irreducible polynomials in one variable over Z simultaneously assume infinitely many prime values unless there is an obvious reason why this is impossible. We prove that under a restriction on the characteristic of the finite field Fq and a smoothness assumption, finitely many irreducible polynomials in one variable over the ring Fq[t] assume simultaneous prime values after a sufficiently large extension of the field of constants.

Modular forms and q-analogues of modified double zeta values

We present explicit formulas for Hecke eigenforms as linear combinations of q-analogues of modified double zeta values. As an application, we obtain period polynomial relations and sum formulas for these modified double zeta values. These relations have similar shapes as the period polynomial relations of Gangl, Kaneko and Zagier and the usual sum formulas for classical double zeta values.

费马问题的一个动力系统类比

A Dynamical Analogue of a Question of Fermat

本文研究二次多项式迭代轨道中的连续平方数问题，构造了含三个连续平方数的单参数族，并证明存在x^2+c形式的多项式其轨道含四个连续平方数，同时在标准猜想下给出周期点轨道仅含有理平方数的充要条件。

Multivariate Bernoulli 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.

Analogues of the Ramanujan--Mordell Theorem

The Ramanujan--Mordell Theorem for sums of an even number of squares is extended to other quadratic forms and quadratic polynomials.

On quantitative analogues of the Goldbach and twin prime conjectures over F_q[t]

We study the number of ways to decompose a monic polynomial in F_q[t] of degree n as a sum of two monic irreducible polynomials in F_q[t]. Our principal result is an asymptotic formula for the number of such representations in the case when q is large compared to n. In its range of validity, this formula agrees with what is suggested by heuristic arguments from the rational setting. We also present similar results towards an analogue of the twin prime conjecture.

q-Ehrhart polynomials of Gorenstein polytopes, Bernoulli umbra and related Dirichlet series

This article considers some q-analogues of classical results concerning the Ehrhart polynomials of Gorenstein polytopes, namely properties of their q-Ehrhart polynomial with respect to a good linear form. Another theme is a specific linear form Ψ (involving Carlitz' q-analogues of Bernoulli numbers) on the space of polynomials, for which one shows interesting behaviour on these q-Ehrhart polynomials. A third point is devoted to some related zeta-like functions associated with polynomials

Inequities in the Shanks-Renyi prime number race over function fields

Fix a prime $p >2$ and a finite field $\mathbb{F}_{q}$ with $q$ elements, where $q$ is a power of $p$. Let $m$ be a monic polynomial in the polynomial ring $\mathbb{F}_{q}[T]$ such that $deg(m)$ is large. Fix an integer $r\geq 2$, and let $a_1,\dots,a_r$ be distinct residue classes modulo $m$ that are relatively prime to $m$. In this paper, we derive an asymptotic formula for the natural density $δ_{m;a_1,\dots,a_r}$ of the set of all positive integers $X$ such that $\sum\limits_{N=1}^{X} π_{q}(a_1,m,N) > \sum\limits_{N=1}^{X} π_{q}(a_2,m,N) > \dots > \sum\limits_{N=1}^{X} π_{q}(a_r,m,N)$, whe…

Lagrange Inversion Theorem for Dirichlet series

We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [4].

超有限域上多项式环的高次互反律及 Grunwald–Wang 定理的类比

Higher reciprocity law and An analogue of the Grunwald--Wang theorem for the ring of polynomials over an ultra-finite field

本文研究有限域非主超积上的多项式环，建立显式高次互反律，并据此证明特征 0 与特征 p 下 Grunwald–Wang 定理的类比。