On a Continued Fraction Formula of Wall

We study the combinatorics of a continued fraction formula due to Wall. We also derive the orthogonality of little q-Jacobi polynomials from this formula, as Wall did for little q-Laguerre polynomials.     

Note on a Continued Fraction of Ramanujan

Ramanujan, who loved continued fractions, recorded many of his formulas in his two notebooks. The 25th entry in Chapter 12 of Notebook II is a continued fraction involving a quotient of gamma function products. We are going to give a new proof of Entry 25, which, incidently, is the least difficult of all Ramanujan's continued fraction formulas involving quotients of gamma function products. In our proof, we make use of a hypergeometric formula of his.     

On a New Continued Fraction Expansion with Non-Decreasing Partial Quotients

We investigate metric properties of the digits occurring in a new continued fraction expansion with non-decreasing partial quotients, the so-called Engel continued fraction (ECF) expansion.     

On a New Continued Fraction Expansion with Non-Decreasing Partial Quotients

We investigate metric properties of the digits occurring in a new continued fraction expansion with non-decreasing partial quotients, the so-called Engel continued fraction (ECF) expansion.     

一个柔性边界条件约束下的随机变量的分布

讨论n个受柔性边界条件约束的随机变量的概率分布．理论解显示其概率密度函数随变量值增大而减小,当n趋於无穷大时收敛于Delta函数．在有序统计的理论框架下,同时得到最小值分布的解析解．     

The Spectrum and Density of States of the Almost Mathieu Operator with Frequency Represented by a Continued Fraction with Large Elements

Mathematical Notes -     

Convergence of a Limit Periodic Schur Continued Fraction

Mathematical Notes - In this paper, we show that if the parameters of a Schur continued fraction tend to zero, then the functions to which the even convergents converge inside the unit disk and the...     

随机环境中马氏链的平稳分布(英文)

本文引入了随机环境中马氏链平稳分布的概念. 在合适的条件下, 给出了随机环境中马氏链的平稳分布存在的一些充分条件. 特别地, 讨论了Cogburn链的平稳分布存在性问题. 同时, 构造了一个随机环境中马氏链的例子, 它的平稳分布是存在的.     

On the Expansion of a Continued Fraction of Gordon

We show that when certain infinite products associated with a continued fraction of Basil Gordon are expanded as power series, the sign of the coefficients is periodic, with period 8.     

A Type of Hyperelliptic Continued Fraction

Based on hints of Tschebychev, a continued fraction is described which gives an effective algorithm for calculating the torsion, if finite, of divisors D − D⁻ on a hyperelliptic curve of genus ≥2, where D is an effective divisor of degree 2 and D⁻ denotes the image of D under the hyperelliptic involution. The difficulties involved in extending the algorithm to divisors of degree ≥3 are briefly discussed.     

关于Psi函数的连分式估计及其应用（英文）

建立了Psi函数的连分式估计.作为应用,我们获得了Euler-Mascheroni常数的更高阶估计.     

On the Rogers--Ramanujan Periodic Continued Fraction

In the paper, the convergence properties of the Rogers--Ramanujan continued fraction $$1 + \frac{qz}{1 + \tfrac{q^2 z}{1 + \cdots}}$$ are studied for q = exp (2 π i τ), where τ is a rational number. It is shown that the function H _q to which the fraction converges is a counterexample to the Stahl conjecture (the hyperelliptic version of the well-known Baker--Gammel--Wills conjecture). It is also shown that, for any rational τ, the number of spurious poles of the diagonal Padé approximants of the hyperelliptic function H _q does not exceed one half of its genus.     

On the Expansion of Ramanujan's Continued Fraction

If Ramanujan's continued fraction (or its reciprocal) is expanded as a power series, the sign of the coefficients is (eventually) periodic with period 5. We give combinatorial interpretations for the coefficients from which the result is immediate. We make use of the quintuple product identity, which we prove.     

Engel连分数展式与Huasdorff维数

本文研究了Engel连分数中部分商以某种速度增长的集合,以及Engel连分数展式收敛速度较快的点组成的集合,利用质量分布原理,证明了这些集合的Haus-dorff维数为1.     

Modular Equations for Ramanujan's Cubic Continued Fraction

In this paper we present two new identities providing relations between Ramanujan's cubic continued fraction G(q) and the two continued fractions G(q⁵) and G(q⁷).     

Limiting Behavior of Random Continued Fractions

Let K(a _n /b _n ) be a continued fraction with elements (a _n ,b _n ) picked randomly and independently from $(\mathbb{C}\setminus\{0\})\times\mathbb{C}$ according to some probability distribution μ. We find sufficient conditions on μ for K(a _n /b _n ) to converge with probability 1 or to be restrained with probability 1. More generally, we also consider μ-random sequences {τ _n } of independent Möbius transformations and find sufficient conditions for $\{\tau_{1}\circ\tau_{2}\circ\cdots\circ\tau_{n}\}_{n=1}^{\infty}$ to converge or be restrained with probability 1. The analysis is based on an important paper by Furstenberg.