Related papers: On Ramanujan's cubic composition formula
In this paper we want to prove some formulas listed by S. Ramanujan in his paper "Modular equations and approximations to $\pi$" \cite{24} with an elementary method.
In this paper we present experimental ways of evaluating Ramanujan`s quantities which as someone can see are related with algebraic numbers. The good thing with algebraic numbers is that can be found in a closed form, from there…
We study Ramanujan's cubic continued fraction and explicit evaluations of theta-functions
In this short research note, we aim to establish an interesting extension of a summation due to Ramanujan.The result is derived with the help of an extension of Gauss's summation theorem available in the literature.
A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We…
Explicit formulas involving a generalized Ramanujan sum are derived. An analogue of the prime number theorem is obtained and equivalences of the Riemann hypothesis are shown. Finally, explicit formulas of Bartz are generalized.
We give the complete evaluation of the first derivative of the Ramanujans cubic continued fraction using Elliptic functions. The Elliptic functions are easy to handle and give the results in terms of Gamma functions and radicals from…
A cubic partition consists of partition pairs $(\lambda,\mu)$ such that $\vert\lambda\vert+\vert\mu\vert=n$ where $\mu$ involves only even integers but no restriction is placed on $\lambda$. This paper initiates the notion of generalized…
We obtain an exact formula for the cubic partition function and prove a conjecture by Banerjee, Paule, Radu and Zeng.
We provide an historical account of equivalent conditions for the Riemann Hypothesis arising from the work of Ramanujan and, later, Guy Robin on generalized highly composite numbers. The first part of the paper is on the mathematical…
Several terminating generalizations of Ramanujan's formula for $\frac{1}{\pi}$ with complete WZ proofs are given.
We give new nested radical equations of similar kind to Ramanujan's questions to the Indian Mathematical Society 100 years ago. While many have since considered these from the perspectives of the Notebooks of Ramanujan and from the theory…
In this short note we use the umbral formalism to derive the Ramanujan Master Theorem and discuss its extension to more general cases.
We present a detailed error analysis of Ramanujan's most accurate approximation to the perimeter of an ellipse.
We derive the mean square of the divisor function using only elementary techniques.
It is well known that there is no closed form analytic expression for the perimeter of an ellipse. In 1927, Srinivasa Ramanujan provides two approximations to the perimeter of an ellipse that are amazingly accurate. However, he does not…
In this article we use theoretical and numerical methods to evaluate in a closed-exact form the parameters of Ramanujan type $1/\pi$ formulas.
Using the WZ-method we find some of the easiest Ramanujan's formulae and also some new interesting Ramanujan-like sums.
In this article we give the theoretical background for generating Ramanujan type $1/\pi^{2\nu}$ formulas. As applications of our method we give a general construction of $1/\pi^4$ series and examples of $1/\pi^6$ series. We also study the…
This paper considers a higher-dimensional generalization of the notion of Ramanujan graphs, defined by Lubotzky, Phillips, and Sarnak. Specifically the Ramanujan property is studied for cubical complexes which are uniformized by an ordered…