Modular periodicity of binomial coefficients

模二项式及其在周期序列中的应用

Modular binomials with an application to periodic sequences

本文利用二项式系数模素数幂的新递推关系研究模周期序列原语的演化，证明可归结为常数序列情形，并将其应用于分析维耶鲁《调式之书》中序列的原语动力学。

Periodicity properties of coefficients of half integral weight modular forms

In this paper we prove a theorem about the coefficients in a block of a half integral weight modular form. We show that the result of Serre and Stark for weight 1/2 forms does not generalize to higher higher weights. Let f be a half integral weight cusp form of weight at leat 3/2. We consider blocks of coefficients of f and prove that, under some weak assumptions on f, if such a coefficient block is periodic when considered as a function of in the square root of its index, then it must vanish completely. The proof is analytic in nature and uses Shimura's lifting theorem together with estimates…

Modular periodicity of exponential sums of symmetric Boolean functions and some of its consequences

This work brings techniques from the theory of recurrent integer sequences to the problem of balancedness of symmetric Boolean functions. In particular, the periodicity modulo $p$ ($p$ odd prime) of exponential sums of symmetric Boolean functions is considered. Periods modulo $p$, bounds for periods and relations between them are obtained for these exponential sums. The concept of avoiding primes is also introduced. This concept and the bounds presented in this work are used to show that some classes of symmetric Boolean functions are not balanced. In particular, every elementary symmetric Boo…

Modular periodicity of the Euler numbers and a sequence by Arnold

For any positive integer $q$, the sequence of the Euler up/down numbers reduced modulo $q$ was proved to be ultimately periodic by Knuth and Buckholtz. Based on computer simulations, we state for each value of $q$ precise conjectures for the minimal period and for the position at which the sequence starts being periodic. When $q$ is a power of $2$, a sequence defined by Arnold appears, and we formulate a conjecture for a simple computation of this sequence.

随机初始化递推的模周期性

Modular Periodicity of Random Initialized Recurrences

研究所有可能初始值下随机初始化递推的周期结构，发现斐波那契递推与其奇偶变换的镜像对称性，揭示素数到素数幂模的分形自相似性，并定义由卢卡斯比值支配的极小分布。

Periodicities for Taylor coefficients of half-integral weight modular forms

Congruences of Fourier coefficients of modular forms have long been an object of central study. By comparison, the arithmetic of other expansions of modular forms, in particular Taylor expansions around points in the upper-half plane, has been much less studied. Recently, Romik made a conjecture about the periodicity of coefficients around $τ=i$ of the classical Jacobi theta function. Here, we prove this conjecture and generalize the phenomenon observed by Romik to a general class of modular forms of half-integral weight.