半直线上的独立随机环境中可逗留的随机游动

主要研究了在随机环境独立的情况下,右半直线上随机环境中可逗留的随机游动的常返性和非常返性.     

The analytic solution for the power series expansion of Heun function

The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric ₂F₁${}_2{F}_1$, ₁F₁${}_1{F}_1$ and ₀F₁${}_0{F}_1$ functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice systems in statistical mechanics, solution of the Schrödinger equation of quantum mechanics, and addition of three quantum spins.     

On the number of hypercubic bipartitions of an integer

For n≤2^k$n\le 2^k$ we study the maximum number of edges of an induced subgraph on n$n$ vertices of the k$k$-dimensional hypercube _Qk$Q_k$. In the process we revisit a well-known divide-and-conquer maximin recurrence f(n)=max(min(_n1,_n2)+f(_n1)+f(_n2))$f\left(n\right)=max(min(n_1,n_2)+f\left(n_1\right)+f\left(n_2\right))$ where the maximum is taken over all proper bipartitions n=_n1+_n2$n=n_1+n_2$. We first use known results to present a characterization of those bipartitions n=_n1+_n2$n=n_1+n_2$ that yield the maximum f(n)=min(_n1,_n2)+f(_n1)+f(_n2)$f\left(n\right)=min(n_1,n_2)+f\left(n_1\right)+f\left(n_2\right)$. Then we use this characterization to present the main result of this article, namely, for a given n∈N$n\in \mathbb{N}$, the determination of the number h(n)$h\left(n\right)$ of these bipartitions that yield the said maximum f(n)$f\left(n\right)$. We present recursive formulae for h(n)$h\left(n\right)$, a generating function h(x)$h\left(x\right)$, and an explicit formula for h(n)$h\left(n\right)$ in terms of a special representation of n$n$.     

Closed forms for 4-parameter families of integrals

We compute closed forms for two multiparameter families of definite integrals, thus obtaining combinatorial formulas. As a consequence, a surprising formula is derived between a definite integral and an improper integral for the same parametric function.     

On the solution of a third order linear homogeneous difference equation with variable coefficents

A closed form solution of a third order linear homogeneous difference equation with variable coefficients is presented. From it, the solution for the special case of an equation with constant coefficients is also obtained.     

Sums of primes and quadratic linear recurrence sequences

Let u be a sequence of positive integers which grows essentially as a geometric progression. We give a criterion on u in terms of its distribution modulo d, d = 1, 2,..., under which the set of positive integers expressible by the sum of a prime number and an element of u has a positive lower density. This criterion is then checked for some second order linear recurrence sequences. It follows, for instance, that the set of positive integers of the form p ＋ [（2 ＋ √3）n], where p is a prime number and n is a positive integer, has a positive lower density. This generalizes a recent result of Enoch Lee. In passing, we show that the periods of linear recurrence sequences of order m modulo a prime number p cannot be ＂too small＂ for most prime numbers p.     

Complete set of periodic solutions of a discontinuous recurrence equation

A nonlinear three-term recurrence relation arising from seeking the steady states of a cellular neural network with bang‐bang control is studied. We show that each solution is periodic and its prime period can be determined by three of its consecutive terms. By means of this periodicity property, we may then solve the steady state problem which to our knowledge is not solved by other means.