Expansions of the solutions of the general Heun equation governed by two-term recurrence relations for coefficients

We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions. We present several expansions using functions, the forms of which differ from those applied before. In general, the coefficients of the expansions obey three-term recurrence relations. However, there exist certain choices of the parameters for which the recurrence relations become two-term. The coefficients of the expansions are then explicitly expressed in terms of the gamma functions. Discussing the termination of the presented series, we show that the finite-sum solutions of the general Heun equation in terms of generally irreducible hypergeometric functions have a representation through a single generalized hypergeometric function. Consequently, the power-series expansion of the Heun function for any such case is governed by a two-term recurrence relation

On two derivative sequences from scaled geometric mean sequence terms.

The so called geometric mean sequence recurrence, with additional scaling variable, produces a sequence for which the general term has a known closed form. Two types of derivative sequence—comprising products of such sequence terms—are examined. In particular, the general term closed forms formulated are shown to depend strongly on a mix of three existing sequences, from which sequence growth rates are deduced and other results given.N/

On Match Lengths, Zero Entropy and Large Deviations - with Application to Sliding Window Lempel-Ziv Algorithm

The Sliding Window Lempel-Ziv (SWLZ) algorithm that makes use of recurrence times and match lengths has been studied from various perspectives in information theory literature. In this paper, we undertake a finer study of these quantities under two different scenarios, i) \emph{zero entropy} sources that are characterized by strong long-term memory, and ii) the processes with weak memory as described through various mixing conditions. For zero entropy sources, a general statement on match length is obtained. It is used in the proof of almost sure optimality of Fixed Shift Variant of Lempel-Ziv (FSLZ) and SWLZ algorithms given in literature. Through an example of stationary and ergodic processes generated by an irrational rotation we establish that for a window of size $n_w$, a compression ratio given by $O(\frac{\log n_w}{{n_w}^a})$ where $a$ depends on $n_w$ and approaches 1 as $n_w \rightarrow \infty$, is obtained under the application of FSLZ and SWLZ algorithms. Also, we give a general expression for the compression ratio for a class of stationary and ergodic processes with zero entropy. Next, we extend the study of Ornstein and Weiss on the asymptotic behavior of the \emph{normalized} version of recurrence times and establish the \emph{large deviation property} (LDP) for a class of mixing processes. Also, an estimator of entropy based on recurrence times is proposed for which large deviation principle is proved for sources satisfying similar mixing conditions.Comment: accepted to appear in IEEE Transactions on Information Theor

Structure of Certain Chebyshev-type Polynomials in Onsager's Algebra Representation

In this report, we present a systematic account of mathematical structures of certain special polynomials arisen from the energy study of the superintegrable $N$-state chiral Potts model with a finite number of sizes. The polynomials of low-lying sectors are represented in two different forms, one of which is directly related to the energy description of superintegrable chiral Potts \ZZ_N-spin chain via the representation theory of Onsager's algebra. Both two types of polynomials satisfy some $(N+1)$-term recurrence relations, and $N$th order differential equations; polynomials of one kind reveal certain Chebyshev-like properties. Here we provide a rigorous mathematical argument for cases $N=2, 3$, and further raise some mathematical conjectures on those special polynomials for a general $N$.Comment: 18 pages, Latex ; Typos corrected, Small changes for clearer presentatio

Spectral decomposition and matrix-valued orthogonal polynomials

The relation between the spectral decomposition of a self-adjoint operator which is realizable as a higher order recurrence operator and matrix-valued orthogonal polynomials is investigated. A general construction of such operators from scalar-valued orthogonal polynomials is presented. Two examples of matrix-valued orthogonal polynomials with explicit orthogonality relations and three-term recurrence relation are presented, which both can be considered as $2\times 2$-matrix-valued analogues of subfamilies of Askey-Wilson polynomials.Comment: 15 page

Revisiting revisitation in computer interaction: organic bookmark management.

According to Milic-Frayling et al. (2004), there are two general ways of user browsing i.e. search (finding a website where the user has never visited before) and revisitation (returning to a website where the user has visited in the past). The issue of search is relevant to search engine technology, whilst revisitation concerns web usage and browser history mechanisms. The support for revisitation is normally through a set of functional built-in icons e.g. History, Back, Forward and Bookmarks. Nevertheless, for returning web users, they normally find it is easier and faster to re-launch an online search again, rather than spending time to find a particular web site from their personal bookmark and history records. Tauscher and Greenberg (1997) showed that revisiting web pages forms up to 58% of the recurrence rate of web browsing. Cockburn and McKenzie (2001) also stated that 81% of web pages have been previously visited by the user. According to Obendorf et al. (2007), revisitation can be divided into four classifications based on time: short-term (72.6% revisits within an hour), medium-term (12% revisits within a day and 7.8% revisits within a week), and long-term (7.6% revisits longer than a week)