T. Amdeberhan, L. Medina, Victor H. Moll
Let t[n] be a sequence that satisfies a first order homogeneous recurrence t[n] = Q[n]*t[n-1], where Q is a polynomial with integer coefficients. The asymptotic behavior of the p-adic valuation of t[n] is described under the assumption that all the roots of Q in Z/pZ have nonvanishing derivative.
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On the $p$-adic valuation of third order linear recurrence sequences
本文通过推广Marques和Lengyel的猜想，研究了三阶线性递推序列的$p$-进赋值，证明了若猜想对特定序列成立，则丢番图方程$x_n=m!$的解集有限且可被有效计算。
Lacunary recurrences and 2-adic properties of Eisenstein series
研究Eisenstein级数$G_k$用$G_4$和$G_6$多项式表示时系数的2-adic赋值，通过缺项递归证明其最小2-adic赋值的精确公式，并应用于Faber多项式的不可约性。
We define the Narayana sequence $\{a_n\}_{n\geq 0}$ as the one satisfying the linear recurrence relation $a_n = a_{n-1}+a_{n-3}$ for $n\geq 3$, with initial values $a_0 = 0$ and $a_1 = a_2=1$. In this paper, we fully characterize the $3$-adic valuation of ${a_n}$ and use this to determine all Narayana numbers that are factorials.
Schreier sets have been an object of study since first introduced in 1930 by Jozef Schreier to construct a counterexample to a conjecture of Banach. In 1974 George Andrews found interesting connections between these sets and Fibonacci number, and since then more results of a combinatorial flavor were proven by Chu, Beanland, and Finch-Smith. In parallel Iannucci introduced the concept of a small divisor and characterized all natural numbers whose small divisors are in arithmetic progression, results which were generalized by Chentouf and Chu. Then combining these two ideas, Chu introduced the …
On the size of a linear combination of two linear recurrence sequences over function fields
本文研究函数域上两个非退化线性递推序列之差的赋值上界，证明在适当条件下存在可有效计算的常数，使得该赋值不超过其中一个序列的赋值加常数。
Explicit valuation of elliptic nets for elliptic curves with complex multiplication
针对具有复乘的椭圆曲线，给出了椭圆网多项式在点乘倍下的公共赋值公式，并应用于证明椭圆可除序列满足递推关系。
We prove that for a positive integer a the integer sequence P(n) satisfying for all n, -infty<n<infty, the recurrence P(n)=a+P(n-phi(a)), phi(a) the Euler function, generates in increasing order all integers P(n) coprime to a.The finite Fourier expansion of P(n) is given in terms of a, n, and the phi(a)-th roots of unity. Properties of the sequence are derived.
We study elements of second order linear recurrence sequences $(G_n)_{n= 0}^{\infty}$ of polynomials in $\mathbb{C}[x]$ which are decomposable, i.e. representable as $G_n=g\circ h$ for some $g, h\in \mathbb{C}[x]$ satisfying $\operatorname{deg}g,\operatorname{deg}h>1$. Under certain assumptions, and provided that $h$ is not of particular type, we show that $\operatorname{deg}g$ may be bounded by a constant independent of $n$, depending only on the sequence.
Let $(x_n)_{n\geq0}$ be a linear recurrence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In [`The quotient set of $k$-generalised Fibonacci numbers is dense in $\mathbb{Q}_p$', \emph{Bull. Aust. Math. Soc.} \textbf{96} (2017), 24-29], Sanna posed an open question to classify primes $p$ for which the quotient set of $(x_n)_{n\geq0}$ is dense in $\mathbb{Q}_p$. In this article, we find a sufficient condition for denseness of the quotient set of the $k$th-order linea…
Linear recurrence sequences and palindromic concatenations of two repdigits in base $β$
研究在特定条件下，满足三阶线性递推的序列中，能表示为基 $\beta$ 中两个重复数字回文拼接的项仅有有限个。
An upper bound for valuations of elliptic nets at primes of bad reduction
针对短Weierstrass方程定义的椭圆曲线，研究人员给出了椭圆网多项式相关量在坏约化素处赋值的一致上界，该网多项式可用于椭圆曲线计数问题研究。
The quotient problem for linear recurrence sequences
研究线性递推序列商集的有限性问题，通过多项式和线性递推关系证明相关结论，利用Schmidt子空间定理等方法。
An elliptic divisibility sequence is an integer recurrence sequence associated to an elliptic curve over the rationals together with a rational point on that curve. In this paper we present a higher-dimensional analogue over arbitrary base fields. Suppose E is an elliptic curve over a field K, and P_1, ..., P_n are points on E defined over K. To this information we associate an n-dimensional array of values in K satisfying a nonlinear recurrence relation. Arrays satisfying this relation are called elliptic nets. We demonstrate an explicit bijection between the set of elliptic nets and the set …
Let $K$ be a field of characteristic zero and suppose that $f:\mathbb{N}\to K$ satisfies a recurrence of the form $$f(n)\ =\ \sum_{i=1}^d P_i(n) f(n-i),$$ for $n$ sufficiently large, where $P_1(z),...,P_d(z)$ are polynomials in $K[z]$. Given that $P_d(z)$ is a nonzero constant polynomial, we show that the set of $n\in \mathbb{N}$ for which $f(n)=0$ is a union of finitely many arithmetic progressions and a finite set. This generalizes the Skolem-Mahler-Lech theorem, which assumes that $f(n)$ satisfies a linear recurrence. We discuss examples and connections to the set of zero coefficients of a …
The only recurrence sequence satisfying the Fibonacci recurrence and realizable as the number of periodic points of a map is (a multiple of) the Lucas sequence.
Log-concavity of rows of triangular arrays satisfying a certain super-recurrence
本文研究满足超递推关系的三角阵列，给出其行具有对数凹性的充分条件，统一证明多类经典组合序列的对数凹性并证实广义 Lah 数猜想，还引入双参数 Eulerian 数推广并讨论其性质。
Let $K$ be a homogeneous self-similar set satisfying the strong separation condition. This paper is concerned with the quantitative recurrence properties of the natural map $T: K\rightarrow K$ induced by the shift. Let $μ$ be the natural self-similar measure supported on $K$. For a positive function $φ$ defined on $\mathbb{N}$, we show that the $μ$-measure of the following set \begin{equation*} R(φ):=\{x\in K: |T^n x-x|<φ(n) \; \text{for infinitely many} \; n\in\mathbb{N}\} \end{equation*} is null or full according to convergence or divergence of a certain series. Moreover, a similar dichotomy…
$p$-Adic quotient sets: linear recurrence sequences with reducible characteristic polynomials
本文针对特征多项式在有理数域上可约的一类线性递推序列，解答了Sanna提出的判断其$p$-进商集在$p$-进数域中是否稠密的公开问题，补充了此前不可约情形下的相关研究结论。
Zeros of Taylor polynomials formed by a three-term recurrence
本文研究由满足三项递推关系的实数序列作系数所构成的 Taylor 多项式，证明在显式条件下高次多项式零点集中于由生成函数分母零点模确定的圆的一侧。
In 1927, Artin conjectured that any integer other than -1 or a perfect square generates the multiplicative group $\mathbb{Z}/p\mathbb{Z}^\times$ for infinitely many $p$. In \cite{MoSt}, Moree and Stevenhagen considered a two-variable version of this problem, and proved a positive density result conditionally to the generalized Riemann Hypothesis by adapting a proof by Hooley for the original conjecture (\cite{Ho}). In this article, we prove an unconditional lower bound for this two-variable problem. In particular, we prove an estimate for the number of distinct primes which divide one of the f…
For integers $k \geq 2$ and $n \neq 0$, let $v_k(n)$ denotes the greatest nonnegative integer $e$ such that $k^e$ divides $n$. Moreover, let $u_n$ be a nondegenerate Lucas sequence satisfying $u_0 = 0$, $u_1 = 1$, and $u_{n + 2} = a u_{n + 1} + b u_n$, for some integers $a$ and $b$. Shu and Yao showed that for any prime number $p$ the sequence $v_p(u_{n + 1})$ is $p$-regular, while Medina and Rowland found the rank of $v_p(F_{n + 1})$, where $F_n$ is the $n$-th Fibonacci number. We prove that if $k$ and $b$ are relatively prime then $v_k(u_{n + 1})$ is a $k$-regular sequence, and for $k$ a pri…
On the sequence $\mathrm{gcd}(a^n-1,b^n-1)$
研究整数序列 $g_n=\gcd(a^n-1,b^n-1)$ 的整除性质，证明其满足常系数线性递推当且仅当 $a$ 与 $b$ 乘法相关，并给出局部结构公式，推进了整数 Ailon-Rudnick 猜想。
Let K be a field with a valuation satisfying the following conditions: both K and the residue field k have characteristic zero; the value group is not 2-divisible; there exists a maximal subfield F in the valuation ring such that Gal(\bar{F}/F) and Gal(\bar{k}/k) have the same 2-cohomological dimension and this dimension is finite. Then Hilbert's Tenth Problem has a negative answer for any function field of a variety over K. In particular, this result proves undecidability for varieties over C((T)).
Zeckendorf's theorem states that every positive integer can be written uniquely as the sum of non-consecutive shifted Fibonacci numbers $\{F_n\}$, where we take $F_1=1$ and $F_2=2$. This has been generalized for any Positive Linear Recurrence Sequence (PLRS), which informally is a sequence satisfying a homogeneous linear recurrence with a positive leading coefficient and non-negative integer coefficients. These decompositions are generalizations of base $B$ decompositions. In this and the follow-up paper, we provide two approaches to investigate linear recurrences with leading coefficient zero…
The Somos 4 sequences are a family of sequences satisfying a fourth order bilinear recurrence relation. In recent work, one of us has proved that the general term in such sequences can be expressed in terms of the Weierstrass sigma function for an associated elliptic curve. Here we derive the analogous family of sequences associated with an hyperelliptic curve of genus two defined by the affine model $y^2=4x^5+c_4 x^4+ ... +c_1 x+c_0$. We show that the recurrence sequences associated with such curves satisfy bilinear recurrences of order 8. The proof requires an addition formula which involves…
For every positive integer $n$ and every $δ\in [0,1]$, let $B(n, δ)$ denote the probabilistic model in which a random set $A \subseteq \{1, \dots, n\}$ is constructed by choosing independently every element of $\{1, \dots, n\}$ with probability $δ$. Moreover, let $(u_k)_{k \geq 0}$ be an integer sequence satisfying $u_k = a_1 u_{k - 1} + a_2 u_{k - 2}$, for every integer $k \geq 2$, where $u_0 = 0$, $u_1 \neq 0$, and $a_1, a_2$ are fixed nonzero integers; and let $α$ and $β$, with $|α| \geq |β|$, be the two roots of the polynomial $X^2 - a_1 X - a_2$. Also, assume that $α/ β$ is not a root of …
In this paper we continue to investigate the properties of those sequences $\{a_n\}$ satisfying the condition $\sum_{k=0}^n\binom nk(-1)^ka_k=\pm a_n$ $(n\ge 0)$. As applications we deduce new recurrence relations and congruences for Bernoulli and Euler numbers.
Zeckendorf's theorem states that every positive integer can be written uniquely as the sum of non-consecutive shifted Fibonacci numbers $\{F_n\}$, where we take $F_1=1$ and $F_2=2$. This has been generalized for any Positive Linear Recurrence Sequence (PLRS), which informally is a sequence satisfying a homogeneous linear recurrence with a positive leading coefficient and non-negative integer coefficients. In this and the preceding paper we provide two approaches to investigate linear recurrences with leading coefficient zero, followed by non-negative integer coefficients, with differences betw…
An infinite sequence $\langle{u_n}\rangle_{n\in\mathbb{N}}$ of real numbers is holonomic (also known as P-recursive or P-finite) if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is said to be positive if each $u_n \geq 0$, and minimal if, given any other linearly independent sequence $\langle{v_n}\rangle_{n \in\mathbb{N}}$ satisfying the same recurrence relation, the ratio $u_n/v_n$ converges to $0$. In this paper, we focus on holonomic sequences satisfying a second-order recurrence $g_3(n)u_n = g_2(n)u_{n-1} + g_1(n)u_{n-2}$, where each coefficient $g…
Log-Concavity and Infinite Log-Concavity of Linear Recurrent Sequences with Linear Coefficients via Companion Matrix Methods
本文研究了满足线性递推关系的实序列的对数凹性，通过伴随矩阵形式分析了对数凹算子的二次形式，并推导了对数凹性的充分条件。
We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the $j^{th}$ derivatives of a sequence generating function and sets of generalized coefficients satisfying a non-triangular recurrence relation in two variables. The generalized transformation coefficients share a number of analogous properties with the Stirling numbers of the second kind and the known harmonic number expansions of the unsigned Stirling numbers of the first kind. We prove a number of propertie…
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