Symmetric Pascal matrices modulo p

Fractal generalized Pascal matrices

Set of generalized Pascal matrices whose elements are generalized binomial coefficients is considered as an integral object. The special system of generalized Pascal matrices, based on which we are building fractal generalized Pascal matrices, is introduced. Pascal matrix (Pascal triangle) is the Hadamard product of the fractal generalized Pascal matrices. The concept of zero generalized Pascal matrices, an example of which is the Pascal triangle modulo 2, arise in connection with the system of matrices introduced.

On digital sequences associated with Pascal's triangle

We consider the sequence of integers whose $n$th term has base-$p$ expansion given by the $n$th row of Pascal's triangle modulo $p$ (where $p$ is a prime number). We first present and generalize well-known relations concerning this sequence. Then, with the great help of Sloane's On-Line Encyclopedia of Integer Sequences, we show that it appears naturally as a subsequence of a $2$-regular sequence. Its study provides interesting relations and surprisingly involves odious and evil numbers, Nim-sum and even Gray codes. Furthermore, we examine similar sequences emerging from prime numbers involvin…

与Pascal金字塔模素数分形维数相关的完全Bernstein函数

A complete Bernstein function related to the fractal dimension of Pascal's pyramid modulo a prime

本文证明函数$f_r(x)=\log(1+rx)/\log(1+x)$在$0\le r\le 1$时为完全Bernstein函数，在$r\ge 1$时为Stieltjes函数，从而解决了David Bradley的猜想。

Generalized Riordan groups and zero generalized Pascal matrices

The generalized Riordan group consists of infinite lower triangular matrices that correspond to certain operators in the space of formal power series. Each such group contains the matrix (generalized Pascal matrix), elements of which are generalized binomial coefficients. Generalized Pascal matrices with non-negative elements form an infinite-dimensional vector space. The paper gives an idea of groups similar to the generalized Riordan groups, but associated with matrices, which in the space of generalized Pascal matrices correspond to the points at infinity; examples of such matrices are the …

Algorithm for studying polynomial maps and reductions modulo prime number

In our previous paper an effective algorithm for inverting polynomial automorphisms was proposed. Also the class of Pascal finite polynomial automorphisms was introduced. Pascal finite polynomial maps constitute a generalization of exponential automorphisms to positive characteristic. In this note we explore properties of the algorithm while using Segre homotopy and reductions modulo prime number. We give a method of retrieving an inverse of a given polynomial automorphism $F$ with integer coefficients form a finite set of the inverses of its reductions modulo prime numbers. Some examples illu…

The number of nonzero binomial coefficients modulo p^alpha

In 1947 Fine obtained an expression for the number of binomial coefficients on row n of Pascal's triangle that are nonzero modulo p. In this paper we use Kummer's theorem to generalize Fine's theorem to prime powers, expressing the number of nonzero binomial coefficients modulo p^alpha as a sum over certain integer partitions. For fixed alpha, this expression can be rewritten to show explicit dependence on the number of occurrences of each subword in the base-p representation of n.

Evaluations of some determinants of matrices related to the Pascal triangle

We prove several evaluations of determinants of matrices, the entries of which are given by the recurrence $a_{i,j}=a_{i-1,j}+a_{i,j-1}$, or variations thereof. These evaluations were either conjectured or extend conjectures by Roland Bacher [J. Théorie Nombres Bordeaux 13 (2001), to appear; see http://arxiv.org/abs/math.CO/0109013].

关于 $\mathbb{Z}$ 上元素源自二项式系数及 Catalan 数的矩阵的注记：分别考虑原始情形与模 $2$ 情形

A note on matrices over $\mathbb{Z}$ with entries stemming from binomial coefficients and from Catalan numbers once pure and once taken modulo $2$

本文在整数环中整理了 Pascal 矩阵及其模 $2$ 形式，以及由 Catalan 数定义的 Hankel 矩阵在原始与模 $2$ 情形下的共同特征，为研究仅含 $0$ 和 $1$ 元素的矩阵提供了动机。

与帕斯卡三角和斐波那契数列相关的三对角实对称矩阵

Tridiagonal real symmetric matrices with a connection to Pascal's triangle and the Fibonacci sequence

本研究推导了一类三对角实对称矩阵特征多项式的三项递推关系与闭式解，揭示其系数与帕斯卡三角对角元的关联，给出特征值包含关系的充分条件，并提出关联斐波那契数列的开放问题。

Matrix Powers of Column-Justified Pascal Triangles and Fibonacci Sequences

If L, respectively R are matrices with entries binom{i-1,j-1}, respectively binom{i-1,n-j}, it is known that L^2 = I (mod 2), respectively R^3 = I (mod 2), where I is the identity matrix of dimension n > 1 (see P10735-May 1999 issue of the American Mathematical Monthly). We generalize it for any prime p, and give a beautiful connection to Fibonacci numbers.

Lucas定理模$p^2$

Lucas' theorem modulo $p^2$

本文刻画了Lucas定理在模$p^2$下成立的数字对条件，并利用帕斯卡三角形的对称性给出自然表达。

Proofs of power sum and binomial coefficient congruences via Pascal's identity

A frequently cited theorem says that for n > 0 and prime p, the sum of the first p n-th powers is congruent to -1 modulo p if p-1 divides n, and to 0 otherwise. We survey the main ingredients in several known proofs. Then we give an elementary proof, using an identity for power sums proven by Pascal in 1654. An application is a simple proof of a congruence for certain sums of binomial coefficients, due to Hermite and Bachmann.

The fractal nature of the Fibonomial triangle

It is well known that Pascal's triangle exhibits fractal behavior when reduced modulo a prime. We show that the triangle of Fibonomial coefficients has a similar nature modulo two. Specifically, for any $m \ge 0$, the subtriangle consisting of the first $3 \cdot 2^m$ rows is duplicated on the left and right sides of the next $3 \cdot 2^m$ rows, with an inverted triangle of zeros in between. We give three proofs of this fact. The first uses a combinatorial interpretation of the Fibonomials due to Sagan and Savage. The second employs an analogue of Lucas' congruence for the parity of binomial co…

Extended Bernoulli and Stirling matrices and related combinatorial identities

In this paper we establish plenty of number theoretic and combinatoric identities involving generalized Bernoulli and Stirling numbers of both kinds. These formulas are deduced from Pascal type matrix representations of Bernoulli and Stirling numbers. For this we define and factorize a modified Pascal matrix corresponding to Bernoulli and Stirling cases.

模 $p$ 的二项式系数和与指数为 $p^n$ 的群

Sums of binomial coefficients modulo $p$ and groups of exponent $p^n$

本文利用矩阵方法证明模素数 $p$ 的二项式系数和同余方程，并据此构造指数为 $p^n$ 的群，从而说明 $p=2$ 在 Heisenberg 群相关问题上并非真正例外。

Determinants related to Dirichlet characters modulo 2, 4 and 8 of binomial coefficients and the algebra of recurrence matrices

Using recurrence matrices, defined and described with some details, we study a few determinants related to evaluations of binomial coefficients on Dirichlet characters modulo 2, 4 and 8.

On the order of unimodular matrices modulo integers

Assuming the Generalized Riemann Hypothesis, we prove the following: If b is an integer greater than one, then the multiplicative order of b modulo N is larger than N^(1-ε) for all N in a density one subset of the integers. If A is a hyperbolic unimodular matrix with integer coefficients, then the order of A modulo p is greater than p^(1-ε) for all p in a density one subset of the primes. Moreover, the order of A modulo N is greater than N^(1-ε) for all N in a density one subset of the integers.

Matrices autosimilaires

We describe a class of matrices whose determinants are trivial to compute. A nice example of such a matrix is given by considering the symmetric matrix with entries {i+j choose i} (mod 2) in {0,1}, 0 <= i,j < n the binomial coefficients modulo 2 represented by {0,1}. The determinant of the above matrix is then closely related to the Thue-Morse sequence counting digits modulo 2 of binary integers.

Polynomial equations for matrices over integers modulo a prime power and the cokernel of a random matrix

Given a prime $p$ and a positive integer $k$, let $\mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ be the ring of $n \times n$ matrices over $\mathbb{Z}/p^{k}\mathbb{Z}$. We consider the number of solutions $X \in \mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ to the polynomial equation $P(X) = 0$, where $P(t)$ is a monic polynomial in $(\mathbb{Z}/p^{k}\mathbb{Z})[t]$ whose reduction modulo $p$ is square-free over the finite field $\mathbb{F}_{p}$ of $p$ elements. Noting that $P(X) = 0$ if and only if $\mathrm{cok}(P(X)) \simeq (\mathbb{Z}/p^{k}\mathbb{Z})^{n}$, we give a conjectural generalization …

帕斯卡矩阵、椭圆曲线上的点计数与普罗拉特球面函数

Pascal's Matrix, Point Counting on Elliptic Curves and Prolate Spheroidal Functions

研究帕斯卡矩阵的特征向量与普罗拉特球面函数的关系，通过椭圆曲线点计数验证生成函数的性质。

Hypergeometric Integrals Modulo $p$ and Hasse--Witt Matrices

We consider the KZ differential equations over $\mathbb C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $\mathbb F_p$. We study the space of polynomial solutions of these differential equations over $\mathbb F_p$, constructed in a previous work by V. Schechtman and the second author. Using Hasse-Witt matrices we identify the space of these polynomial solutions over $\mathbb F_p$ with the space dual to a certain subspace of regular differentials on an associated curve. We also relate these poly…

Rotation Symmetries of Sequential Matrices with Applications to the Jacobi Symbol

Suppose that $p$ is an odd prime and $\genfrac{(}{)}{}{}{\cdot}{p}$ denotes the Legendre symbol modulo $p$. If $p$ is has the form $p= n^2+1$ then one easily verifies that $\genfrac{(}{)}{}{}{a}{p} = \genfrac{(}{)}{}{}{-a}{p}$ for all $a\in \mathbb Z/p\mathbb Z$. We identify various symmetry properties of sequential matrices over $\mathbb Z/(n^2+1)\mathbb Z$ regardless of whether $n^2+1$ is prime. We deduce from these results a collection of symmetries involving Jacobi symbol modulo $n^2+1$ which generalize our above observation on the Legendre symbol.

Euler matrices and their algebraic properties revisited

This paper is concerned with the generalized Euler polynomial matrix $\E^{(α)}(x)$ and the Euler matrix $\E$. Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for $\E^{(α)}(x)$ and determine the inverse matrix of $\E$. We establish some explicit expressions for the Euler polynomial matrix $\E(x)$, which involving the generalized Pascal, Fibonacci and Lucas matrices, respectively. From these formulae we get some new interesting identities involving Fibonacci and Lucas numbers. Also, we provide some factorizations of the Euler polynomial matrix in …

不被素数整除的二项式系数计数的周期性最小值

Periodic minimum in the count of binomial coefficients not divisible by a prime

针对不被素数整除的二项式系数计数的周期函数最小值难以确定的问题，研究提出通用求解方法，解决了B. Wilson此前的相关猜想。

Hyperelliptic integrals modulo $p$ and Cartier-Manin matrices

The hypergeometric solutions of the KZ equations were constructed almost 30 years ago. The polynomial solutions of the KZ equations over the finite field $F_p$ with a prime number $p$ of elements were constructed recently. In this paper we consider the example of the KZ equations whose hypergeometric solutions are given by hyperelliptic integrals of genus $g$. It is known that in this case the total $2g$-dimensional space of holomorphic solutions is given by the hyperelliptic integrals. We show that the recent construction of the polynomial solutions over the field $F_p$ in this case gives onl…

A lower bound for periods of matrices

For a nonsingular integer matrix A, we study the growth of the order of A modulo N. We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or all eigenvalues are powers of a single unit in a real quadratic field. For exceptional matrices, it is easily seen that there are arbitrarily large values of N for which the order of A modulo N is logarithmically small. In contrast, we show that if the matrix is not exceptional, then the order of A modulo N goes to infinity faster than any constant multiple …

Binomial-coefficient multiples of irrationals

Denote by $x$ a random infinite path in the graph of Pascal's triangle (left and right turns are selected independently with fixed probabilities) and by $d_n(x)$ the binomial coefficient at the $n$'th level along the path $x$. Then for a dense $G_δ$ set of $θ$ in the unit interval, $\{d_n(x)θ\}$ is almost surely dense but not uniformly distributed modulo 1.

Counting Reducible Matrices, Polynomials, and Surface and Free Group Automorphisms

We give upper bounds on the numbers of various classes of polynomials reducible over the integers and over integers modulo a prime and on the number of matrices in SL(n), GL(n) and Sp(2n) with reducible characteristic polynomials, and on polynomials with non-generic Galois groups. We use our result to show that a random (in the appropriate sense) element of the mapping class group of a closed surface is pseudo-Anosov, and that a random automorphism of a free group is strongly irreducible (aka irreducible with irreducible powers). We also give a necessary condition for all powers of an algebrai…

Computing Hasse-Witt matrices of hyperelliptic curves in average polynomial time

We present an efficient algorithm to compute the Hasse-Witt matrix of a hyperelliptic curve C/Q modulo all primes of good reduction up to a given bound N, based on the average polynomial-time algorithm recently introduced by Harvey. An implementation for hyperelliptic curves of genus 2 and 3 is more than an order of magnitude faster than alternative methods for N = 2^26.

Computing L-Polynomials of Picard curves from Cartier-Manin matrices

We study the sequence of zeta functions $Z(C_p,T)$ of a generic Picard curve $C:y^3=f(x)$ defined over $\mathbb{Q}$ at primes $p$ of good reduction for $C$. We define a degree 9 polynomial $ψ_f\in \mathbb{Q}[x]$ such that the splitting field of $ψ_f(x^3/2)$ is the $2$-torsion field of the Jacobian of $C$. We prove that, for all but a density zero subset of primes, the zeta function $Z(C_p,T)$ is uniquely determined by the Cartier-Manin matrix $A_p$ of $C$ modulo $p$ and the splitting behavior modulo $p$ of $f$ and $ψ_f$; we also show that for primes $\equiv 1 \pmod{3}$ the matrix $A_p$ suffice…

取值于戴德金整环的随机矩阵余核的普适性

Universality for Cokernels of Dedekind Domain Valued Random Matrices

该研究利用Wood的矩方法，针对元素取值于带有限商的可数戴德金整环的随机矩阵，提出并证明了一个可得到余核分布普适性结果的条件，拓展了此前整数矩阵场景下的相关结论。

Hasse--Witt 矩阵的同余式与 $p$-adic KZ 方程的解

Congruences for Hasse--Witt matrices and solutions of $p$-adic KZ equations

本文证明 Laurent 多项式组 Hasse--Witt 矩阵的一般 Dwork 型同余式，并借此研究 KZ 方程模 $p^s$ 多项式解的算术与 $p$-adic 解析性质，进而证明超椭圆曲线族对应的 $p$-adic KZ 联络存在秩 $g$ 的不变子丛。