A Bernstein function related to Ramanujan’s approximations of exp(n)

Ramanujan’s sequence θ(n),n=0,1,2,…?, is defined by $\frac{e^{n}}{2}=\sum_{j=0}^{n-1}\frac{n^{j}}{j!}+\frac{n^{n}}{n!} \theta(n)$ . It is possible to define, in a simple manner, the function θ(x) for all nonnegative real numbers x. We show that the function $\lambda(x):=x (\theta(x)-\frac{1}{3} )$ is a Bernstein function on [0,∞), that is, λ(x) is nonnegative with completely monotonic derivative on [0,∞). This implies some earlier results concerning complete monotonicity of the function θ(x) on [0,∞).     

A Hardy–Ramanujan–Rademacher-type formula for (r,s)-regular partitions

Let p _r,s (n) denote the number of partitions of a positive integer n into parts containing no multiples of r or s, where r>1 and s>1 are square-free, relatively prime integers. We use classical methods to derive a Hardy?CRamanujan?CRademacher-type infinite series for p _r,s (n).     

A new proof of Ramanujan’s modular equation relating R(q) with R(q 5)

We give a new proof of Ramanujan’s modular identity relating R(q) with R(q ⁵), where R(q) is the famous Rogers–Ramanujan continued fraction. Our formulation is stronger than those of preceding authors; in particular, we give for the first time identities for the expressions appearing in the numerator and the denominator of Ramanujan’s identity. A related identity for R(q) that has partition-theoretic connections is also proved.     

Identities and congruences for Ramanujan’s ω(q)

Recently, the authors constructed generalized Borcherds products where modular forms are given as infinite products arising from weight 1/2 harmonic Maass forms. Here we illustrate the utility of these results in the special case of Ramanujan’s mock theta function ω(q). We obtain identities and congruences modulo 512 involving the coefficients of ω(q).     

On Ramanujan’s function k(q)=r(q)r 2(q 2)

In both his second and lost notebooks, Ramanujan introduced a function, related to the Rogers–Ramanujan continued fraction and its quadratic transformation, and listed several of its properties. We extend these results and develop a systematic theory.     

IN MEMORY OF PROFESSOR KANG FENG (1920-1993)

Professor Kang FENG, Member of the Chinese Academy of Sciences, HonoraryDirector of the Computing Center of the Chinese Academy of Sciences, world famousmathematician, founder and pioneer of Chinese computational mathematics, Editor of《Journal of Computational Mathematics》,《Numberica Mathematics Sinica》(China)     

Some modular relations for Ramanujan’s function k(q)=r(q)r^2(q^2)

In both his second and lost notebooks, Ramanujan introduced and studied a function \(k(q)=r(q)r^2(q^2)\) , where \(r(q)\) is the Rogers–Ramanujan continued fraction. Ramanujan also recorded five beautiful relations between the Rogers–Ramanujan continued fraction \(r(q)\) and the five continued fractions \(r(-q)\) , \(r(q^2)\) , \(r(q^3)\) , \(r(q^4)\) , and \(r(q^5)\) . Motivated by those relations, we present some modular relations between \(k(q)\) and \(k(-q)\) , \(k(-q^2)\) , \(k(q^3)\) , and \(k(q^5)\) in this paper.     

An analogue of Ramanujan’s sum with respect to regular integers (modr)

An integer a is said to be regular (modr) if there exists an integer x such that a ² x≡a (mod r). In this paper we introduce an analogue of Ramanujan’s sum with respect to regular integers (modr) and show that this analogue possesses properties similar to those of the usual Ramanujan’s sum.     

A Note on Base Change,Identities Involving τ(n), and a Congruence of Ramanujan

We use the theory of quadratic base change to derive some new identities involving the Ramanujan -function, and show how the Ramanujan congruence (n) ₁₁(n) (mod 691) follows.     

An information‐theoretic‐based (MaxEnt) approach to social dynamical systems

Jeffreys‐Jaynes’ Predictive Statistics appears to provide a promising approach for the study of general dynamical systems. We describe an application of such theory to the analysis of the dynamics of interacting social groups. For that purpose the said statistical theory is redirected towards the construction of an equivalent stochastic theory. The working of the formalism is illustrated by applying it to a simplified case of opinion forming in a two‐candidates election.     

A new proof of two identities involving Ramanujan’s cubic continued fraction

In this paper, we give a new proof of two identities involving Ramanujan’s cubic continued fraction. These identities are the key ingredients to an analog of Ramanujan’s “Most Beautiful Identity” discovered recently.