Rank–Crank-type PDEs and generalized Lambert series identities

We show how Rank–Crank-type PDEs for higher order Appell functions due to Zwegers may be obtained from a generalized Lambert series identity due to the first author. Special cases are the Rank–Crank PDE due to Atkin and the third author and a PDE for a level 5 Appell function also found by the third author. These two special PDEs are related to generalized Lambert series identities due to Watson, and Jackson, respectively. The first author’s Lambert series identity is a common generalization. We also show how Atkin and Swinnerton-Dyer’s proof using elliptic functions can be extended to prove these generalized Lambert series identities.     

Lambert series and Ramanujan

Lambert series are of frequent occurrence in Ramanujan's work on elliptic functions, theta functions and mock theta functions. In the present article an attempt has been made to give a critical and up-to-date account of the significant role played by Lambert series and its generalizations in further development and a better understanding of the works of Ramanujan in the above and allied areas.     

Seven equivalent binomial sums

Seven binomial sums including four of Ruehr (1980) are shown to be equipollent by means of the Lambert series on binomial coefficients.     

Closed form expressions for Appell polynomials

The Ramanujan Journal - Hafner and Stopple proved a conjecture of Zagier on the asymptotic expansion of a Lambert series involving Ramanujan’s tau function with the main term involving the...     

An Identity Relating a Theta Function to a Sum of Lambert Series

We derive an identity connecting a theta function and a sumof Lambert series. As a consequence of this identity, we deducea number of results of Jacobi, Dirichlet, Lorenz, Ramanujanand Rademacher.     

On three identities of Ramanujan

The Ramanujan Journal - In this paper, we represent the generating function of the rank function as a summation of four parts—a constant, two Lambert series and a product. Applying it to...     

Some new transformations for Bailey pairs and WP-Bailey pairs

We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.     

RANK DIFFERENCES FOR OVERPARTITIONS

In 1954, Atkin and Swinnerton-Dyer proved Dyson's conjectureson the rank of a partition by establishing formulae for thegenerating functions for rank differences in arithmetic progressions.In this paper, we prove formulae for the generating functionsfor rank differences for overpartitions. These are in termsof modular functions and generalized Lambert series.     

An Identity of Ramanujan and the Representation of Integers as Sums of Triangular Numbers

Let k be a positive number and t _k(n) denote the number of representations of n as a sum of k triangular numbers. In this paper, we will calculate t _2k (n) in the spirit of Ramanujan. We first use the complex theory of elliptic functions to prove a theta function identity. Then from this identity we derive two Lambert series identities, one of them is a well-known identity of Ramanujan. Using a variant form of Ramanujan's identity, we study two classes of Lambert series and derive some theta function identities related to these Lambert series . We calculate t ₁₂(n), t ₁₆(n), t ₂₀(n), t ₂₄(n), and t ₂₈(n) using these Lambert series identities. We also re-derive a recent result of H. H. Chan and K. S. Chua [6] about t ₃₂(n). In addition, we derive some identities involving the Ramanujan function (n), the divisor function ₁₁(n), and t ₂₄(n). Our methods do not depend upon the theory of modular forms and are somewhat more transparent.     

A new characteristic of Möbius transformations by use of polygons having type A

In the hyperbolic plane Möbius transformations can be characterized by Lambert quadrilaterals, i.e., a continuous bijection which maps Lambert quadrilaterals to Lambert quadrilaterals must be Möbius. In this paper we generalize this result to the case of polygons with n sides having type A, that is, having exactly two non-right interior angle.     

A q-series expansion formula and the Askey–Wilson polynomials

Previously, we proved a q-series expansion formula which allows us to recover many important classical results for q-series. Based on this formula, we derive a new q-formula in this paper, which clearly includes infinitely many q-identities. This new formula is used to give a new proof of the orthogonality relation for the Askey–Wilson polynomials. A curious q-transformation formula is proved, and many applications of this transformation to Hecke type series are given. Some Lambert series identities are also derived.     

On Generalized Hardy Sums <Emphasis Type="Italic">s</Emphasis><Subscript>5</Subscript>(<Emphasis Type="Italic">h,k</Emphasis>)

The aim of this paper is to study generalized Hardy sums s ₅(h, k). By using mediants and the adjacent difference of Farey fractions, we establish a relationship between s ₅(h, k) and Farey fractions. Using generalized Dedekind sums and a generalized periodic Bernoulli function, we define generalized Hardy sums s _5,p (h,k). A relationship between s _5,p (h, k) and the Hurwitz zeta function is established. By using the definitions of Lambert series and cotπz, we establish a relationship between s ₅(h,k) and Lambert series.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1434–1440, October, 2004.     

Stochastic service systems,random interval graphs and search algorithms

We consider several stochastic service systems, and study the asymptotic behavior of the moments of various quantities that have application to models for random interval graphs and algorithms for searching for an idle server or for an vacant or occupied waiting station. In some cases the moments turn out to involve Lambert series for the generating functions for the sums of powers of divisors of positive integers. For these cases we are able to obtain complete asymptotic expansions for the moments of the quantities in question. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 421–442, 2014     

On the arithmetic product of combinatorial species

We introduce two new binary operations on combinatorial species; the arithmetic product and the modified arithmetic product. The arithmetic product gives combinatorial meaning to the product of Dirichlet series and to the Lambert series in the context of species. It allows us to introduce the notion of multiplicative species, a lifting to the combinatorial level of the classical notion of multiplicative arithmetic function. Interesting combinatorial constructions are introduced; cloned assemblies of structures, hyper-cloned trees, enriched rectangles, etc. Recent research of Cameron, Gewurz and Merola, about the product action in the context of oligomorphic groups, motivated the introduction of the modified arithmetic product. By using the modified arithmetic product we obtain new enumerative results. We also generalize and simplify some results of Canfield, and Pittel, related to the enumerations of tuples of partitions with the restrictions met.     

Randomized projection method for estimating angular distributions of polarized radiation based on numerical statistical modeling

To study the intensity of radiation transmitted through a layer of substance, a Monte Carlo algorithm is developed based on the expansion of the corresponding angular distribution density in terms of orthonormalized polynomials with a “Lambert” weight. The algorithm is optimized so as to simplify the computations as much as possible. Even a small effect of polarization and the deviation of the angular distribution from the Lambert one can be estimated rather accurately by applying the algorithm.     

LambertW函数对一阶复时滞动力系统的稳定性分析

Lambert W函数具有的一些性质以及现今成熟的数学软件Maple等使得它能很好地应用于时滞微分方程的稳定性判别中.通过应用Lambert W函数对一阶复系数时滞微分方程渐近稳定性的判别命题,分析了一类参数反馈控制复系数时滞微分方程的稳定性,得到了更加精细的结果.相比已往的方法,新方法更简单、计算更方便并能快速有效的给出判定结果.     

Lambert's W, infinite divisibility and Poisson mixtures

The Lambert W function is shown to be the Laplace exponent of a positive infinitely divisible law (i.e. W is a Bernstein function) called the standard Lambert law. This law is a generalized gamma convolution. At least three Poisson mixture families are defined in terms of W. One of these is the generalized Poisson laws which are shown to be generalized negative-binomial convolutions. Mixing with positive stable laws yields further generalizations.     

An analytic approach to projectile motion in a linear resisting medium

The time of flight, range and the angle which maximizes the range of a projectile in a linear resisting medium are expressed in analytic form in terms of the recently defined Lambert W function. From the closed-form solutions a number of results characteristic to the motion of the projectile in a linear resisting medium are analytically confirmed, and asymptotic and approximate expressions valid within the weak and strong damping limits are developed. The problem provides an accessible account of the increasingly applicable Lambert W function and highlights much of the important mathematics associated with this simple yet rapidly emerging ‘implicitly’ elementary function.     

A note on Chrystal’s equation

The exact, explicit form of the transcendental solution of Chrystal’s equation, a first order nonlinear ordinary differential equation (ODE) of degree two, is derived in terms of the Lambert W-function. It is shown that this case of the general solution is dual-valued over a finite interval and that, for a special case of the coefficients, its zeros involve the Golden ratio. Additionally, a number of applications involving special cases of this ODE are noted and the main properties of the Lambert W-function are briefly reviewed.     

Computing the Lambert W function in arbitrary-precision complex interval arithmetic

Numerical Algorithms - We describe an algorithm to evaluate all the complex branches of the Lambert W function with rigorous error bounds in arbitrary-precision interval arithmetic or ball...