On Entry II.16.12: A continued fraction of Ramanujan

Integrals related with Rogers Ramanujan continued fraction and q-products

We present here a way to evaluate a very wide class of integrals relating Ramanujans continued fraction and q-product. To do this we explore briefily a differential equation, which relates these two functions

罗杰斯-拉马努金连续分数

The Rogers--Ramanujan continued fraction

本文综述了罗杰斯-拉马努金连续分数的性质、值、恒等式及推广，涵盖其历史背景与相关数学结果。

A Ramanujan enigma involving the first Stieltjes constant

We provide a rigorous formulation of Entry 17(v) in Ramanujan's Notebooks and show how this relates to the first Stieltjes constant. A new proposition 4.5 is included to show the close relationship with some analysis presented by Candelpergher in 2017.

Ramanujan Machine 猜想的证明与推广

Proof and generalization of conjectures of Ramanujan Machine

本文通过微分方程和 Petkovšek 算法两种方法证明 Ramanujan Machine 的 38 个猜想，并强推广其中 31 个。

Ramanujan's Inverse Elliptic Arc Approximation

We suggest a continued fraction origin to Ramanujan's approximation to {(a-b)/(a+b)}^2 in terms of the arc length of an ellipse with semiaxes a and b. Moreover, we discuss the asymptotic accuracy of the approximation.

The last of the hypergeometric continued fractions

A contiguous relation for complementry pairs of very well poised balanced ${}_{10}ϕ_9$ basic hypergeometric functions is used to derive an explict expression for the associated continued fraction. This generalizes the continued fraction results associated with both Ramanujan's Entry 40 and Askey-Wilson polynomials which can be recovered as limits. Associated with our continued fraction results there are systems of biorthogonal rational functions that have yet to be derived.

Continued fraction expression of the Mathieu series

In this paper, we represent a continued fraction expression of Mathieu series by a continued fraction formula of Ramanujan. As application, we obtain some new bounds for Mathieu series.

The fastest possible continued fraction approximations of a class of functions

The goal of this paper is to formulate a systematical method for constructing the fastest possible continued fraction approximations of a class of functions. The main tools are the multiple-correction method, the generalized Mortici's lemma and the Mortici-transformation. As applications, we will present some sharp inequalities, and the continued fraction expansions associated to the volume of the unit ball. In addition, we obtain a new continued fraction expansion of Ramanujan for a ratio of the gamma functions, which is showed to be the fastest possible. Finally, three conjectures are propos…

Rogers-Ramanujan and the Baker-Gammel-Wills (Padé) conjecture

In 1961, Baker, Gammel and Wills conjectured that for functions $f$ meromorphic in the unit ball, a subsequence of its diagonal Padé approximants converges uniformly in compact subsets of the ball omitting poles of $f$. There is also apparently a cruder version of the conjecture due to Padé himself, going back to the earlier twentieth century. We show here that for carefully chosen $q$ on the unit circle, the Rogers-Ramanujan continued fraction $$1+\frac{qz|}{|1}+\frac{q^{2}z|}{|1}+\frac{q^{3}z|}{|1}+... $$ provides a counterexample to the conjecture. We also highlight some other interesting p…

Multiple-correction and summation of the rational series

The goal of this work is to formulate a systematical method for looking for the simple closed form or continued fraction representation of a class of rational series. As applications, we obtain the continued fraction representations for the alternating Mathieu series and some rational series. The main tools are multiple-correction and two of Ramanujan's continued fraction formulae involving the quotient of the gamma functions.

Ramanujan七次θ函数恒等式的九次类似

Ninth degree analogue of Ramanujan's septic theta function identity

本文提出了Ramanujan七次θ函数恒等式的九次类似，并应用其笔记簿中的三次方程条目、用θ函数解释三角恒等式，最后计算了五个φ(e^{-π√n})的例子。

Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

Motivated by Lévy's characterization of Brownian motion on the line, we propose an analogue of Brownian motion that has as its state space an arbitrary closed subset of the line that is unbounded above and below: such a process will be a martingale, will have the identity function as its quadratic variation process, and will be ``continuous'' in the sense that its sample paths don't skip over points. We show that there is a unique such process, which turns out to be automatically a reversible Feller-Dynkin Markov process. We find its generator, which is a natural generalization of the operator…

Contiguous relations, continued fractions and orthogonality

We examine a special linear combination of balanced very-well-poised $\tphia$ basic hypergeometric series that is known to satisfy a transformation. We call this $Φ$ and show that it satisfies certain three-term contiguous relations. From two sets of contiguous relations for $Φ$ we obtain fifty-six pairwise linearly independent solutions to a three-term recurrence that generalizes the recurrence for Askey-Wilson polynomials. The associated continued fraction is evaluated using Pincherle's theorem. From this continued fraction we are able to derive a discrete system of biorthogonal rational fun…

The MC algorithm and its applications

By applying the MC algorithm and the Bauer-Muir transformation for continued fractions, in this paper we shall give six examples to show how to establish an infinite set of continued fraction formulas for certain Ramanujan-type series, such as Catalan's constant, the exponential function, etc.

受 Ramanujan 启发的渐近展开

An asymptotic expansion inspired by Ramanujan

本文验证了 Ramanujan 笔记本中关于一个和式渐近于 ln(x)+γ 的论断在 n=2 时正确，并讨论了误差项阶数及 n=1 情况的不同正确推广。

将幂级数展开为连分数的一种简单算法

A simple algorithm for expanding a power series as a continued fraction

本文提出了一种将形式幂级数展开为连分数的简单算法，探讨了其历史渊源及与多位数学家工作的联系，并给出了基于 Flajolet 工作的组合学解释。