两类线性递推式的求解

本文给出了常系数线性递推式（其中k是正整数，s=[（k＋1）／2]）以及交系数线性递推式的解的计算公式．     

一个求解二项式系数幂和序列递推公式的方法

根据同余理论并利用二项式系数幂和序列在模p下具有周期性的事实,提出一种求解递推公式的方法,从理论上证明其可行性.使得求解和验证二项式系数幂和序列递推公式具有完备的理论基础.     

常系数线性递推关系通项的求解

实例说明利用特征根法和生成函数法可求解常系数线性递推关系.并在此基础上给出常见的常系数线性非齐次递推关系通解的求法.     

一类数列极限的矩阵解法

通过矩阵方法可求一类由常系数线性递推公式所确定的数列的极限.实例演示其递推公式形如xn 1=pxn qxn-1(p,q为非零常数)和xn 1=caxxnn db(c≠0,且ad≠bc)的两类数列{xn}的极限的求法.     

常系数线性递推数列通项公式的矩阵求法

设数列为,若有正整数K和K＋1个实常数使对任意自然数n都成立,则称阶常系数线性递推数列,（l）式称为递推公式．彭咏松先生在文［l」中利用等比数列和线性方程组的一些知识,研究了常系数齐次（ho一O）线性递推数列的通项公式．本文利用矩阵理论讨论了一般的常系数线性速推数列通项公式．则（1）变为：将（2）式反复迭代,则有：当矩阵E－A可逆时,由于从而（3）式变为当时,,于是可见求数列（n｝通项公式的关键就是求矩阵A的n次方幂,利用矩阵理论可解决此问题．下面举例说明（X。）的通项公式的矩阵求法．例至已知X;一O,X。一1,…     

关于一类非常系数线性递推式的解的显式表示

本文给出了两个指标的非常系数的线性递推式的显式解。有关方法,避免了由于解高阶线性代数方程所带来的困难。其结果,为求解组合计数中相应定解问题,提供了一个明确的计算公式。     

线性递推关系的一种求解方法

将线性递推关系表示为矩阵形式,求出相应矩阵的特征多项式,并构造一个更低次多项式r(λ),通过直接计算r(λ)得到一种求解线性递推关系的方法。     

基于特征值理论求分式线性递推数列极限

利用分式线性递推数列与二阶方阵的对应关系,通过求二阶方阵的n次幂,给出了分式线性递推数列的通项表达式.再利用矩阵的特征值与不动点关系,得到了分式线性递推数列敛散性的所有表现形式.     

求解常系数线性齐次递推数列通项的生成函数法

通过引入生成函数,并利用其运算性质以及幂级数展开式,将常系数线性齐次递推数列通项的求解转化为对应特征方程的研究,根据特征根的不同情形,给出了数列通项的一般公式并举例加以应用.     

有限递推法与待定系数法的计算复杂性比较——用于求y"+py'+qy=P_m(x)e~(ax)cosbx(或sinbx)的特解时

分析了在求二阶常系数线性常微分方程y"+py'+qy=P_m(x)e~(ax)cos bx;y"+py'+qy=P_m(x)e~(ax)sin bx的特解时;采用有限递推法或待定系数法的各自计算复杂性.证明了在求上述方程特解时,有限递推法在计算复杂性上优于待定系数法.     

Linear recurrence sequences and periodicity of multidimensional continued fractions

Multidimensional continued fractions generalize classical continued fractions with the aim of providing periodic representations of algebraic irrationalities by means of integer sequences. We provide a characterization for periodicity of Jacobi–Perron algorithm by means of linear recurrence sequences. In particular, we prove that partial quotients of a multidimensional continued fraction are periodic if and only if numerators and denominators of convergents are linear recurrence sequences, generalizing similar results that hold for classical continued fractions.     

Modifications of quasi-definite linear functionals via addition of delta and derivatives of delta Dirac functions

We consider the general theory of the modifications of quasi-definite linear functionals by adding discrete measures. We analyze the existence of the corresponding orthogonal polynomial sequences with respect to such linear functionals. The three-term recurrence relation, lowering and raising operators as well as the second order linear differential equation that the sequences of monic orthogonal polynomials satisfy when the linear functional is semiclassical are also established. A relevant example is considered in details.     

On the nonlinearity of linear recurrence sequences

We obtain an upper bound on exponential sums of a new type with linear recurrence sequences. We apply this bound to estimate the Fourier coefficients, and thus the nonlinearity, of a Boolean function associated with a linear recurrence sequence in a natural way.     

The (qr)-Simon Newcomb problem

A new statistic, the r-major index, is defined for sequences. A linear recurrence is then derived that enumerates sequences by r-descent number and r-major index.     

A simple observation on simple zeros

A lower bound on the number of simple and distinct zeros of elements in a function field defined by linear recurrence sequences is computed.     

On the multiplicity of linear recurrence sequences

We prove a lemma regarding the linear independence of certain vectors and use it to improve on a bound due to Schmidt on the zero-multiplicity of linear recurrence sequences.     

一类高阶线性齐次递归数列的几个模性质

给出一类特殊的高阶线性递归序列的几个模性质.     

An algorithmic decomposition system for second order recurrence relations

A new technique of accurate stable computation of sequences satisfying non-homogeneous three term recurrence relations is presented. A decomposition system consisting of one non-linear and two linear first order recurrence relations is obtained. Forward or backward recursion directions of the linear relations provide additional flexibility in computation. This leads to an integrated system of three algorithms which can accurately compute the desired solution in each region of the solution set to the original second order relations. Applications are well known and numerical examples include Bessel functions of the second kind, Anger-Weber functions, and Lommel functions.     

Semiconjugate factorizations of higher order linear difference equations in rings

We use a new nonlinear method to study linear difference equations with variable coefficients in a non-trivial ring R. If the homogeneous part of the linear equation has a solution in the unit group of a ring with identity (a unitary solution), then we show that the equation decomposes into two linear equations of lower orders. This decomposition, known as a semiconjugate factorization in the nonlinear theory, is based on sequences of ratios of consecutive terms of a unitary solution. Such sequences, which may be called eigensequences, are well suited to variable coefficients; for instance, they provide a natural context for the expression of the Poincaré–Perron theorem. As applications, we obtain new results for linear difference equations with periodic coefficients and for linear recurrences in rings of functions (e.g. the recurrence for the modified Bessel functions).     

Arithmetic functions with linear recurrence sequences

We obtain asymptotic formulas for all the moments of certain arithmetic functions with linear recurrence sequences. We also apply our results to obtain asymptotic formulas for some mean values related to average orders of elements in finite fields.