Related papers: A WZ proof of Ramanujan's Formula for Pi
We prove a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of $\pi$, and conjecture a type of supercongruences associated to them. We support this conjecture by checking all the cases for many…
We give an elementary proof for new strict upper and lower bounds for the correction term in Ramanujan's approximation for the factorial function
Using a modular equation of level $3$ and degree $23$ due to Chan and Liaw, we prove the fastest convergent rational Ramanujan-type series for $1/\pi$ of level $3$.
In this paper, we derive a unified generalization of Ramanujan's transformation identities for the theta function $f(a,b)$, originally appearing in Ramanujan's Notebooks, Parts~III and IV. Using an approach based on residue-class…
This paper gives a short but reasonably comprehensive review of Ramanujan's {_1\psi_1} summation and its generalisations. It covers the history of Ramanujan's summation, simple applications to sums of squares and orthogonal polynomials,…
Inspired by a Zudilin-Zhao's supercongruences pattern related to Ramanujan-like series for $1/\pi^k$, we conjecture a kind of $p$-adic expansions.
We extend a holomorphic projection argument of our earlier work to prove a novel divisibility result for non-holomorphic congruences of Hurwitz class numbers. This result allows us to establish Ramanujan-type congruences for Hurwitz class…
Quadratic forms over Z that represent all positive integers are called universal. Starting with Ramanujan, 54 universal quaternary quadratic forms without cross product terms were discovered. The form that is the sum of four squares was…
In 1987 Jonathan and Peter Borwein, inspired by the works of Ramanujan, derived many efficient algorithms for computing $\pi$. We will see that by using only a formula of Gauss's and elementary algebra we are able to prove the correctness…
We prove the last remaining case of the original 13 Ramanujan-type supercongruence conjectures due to Van Hamme from 1997. The proof utilizes classical congruences and a WZ pair due to Guillera. Additionally, we mention some future…
We present a proof of Ramanujan's congruences $$p(5n+4) \equiv 0 \pmod 5 \text{ and } \tau(5n+5) \equiv 0 \pmod 5.$$ The proof only requires a limiting case of Jacobi's triple product, a result that Ramanujan knew well, and a technique…
The Ramanujan zeta function was in $1916$ proposed by an Indian mathematician Srinivasa Ramanujan. As an analogue of the Riemann hypothesis, an English mathematician Godfrey Harold Hardy proposed in $1940$ that the real part of all complex…
We apply the method established in our previous work to derive a Chudnovsky-Ramanujan type formula for the Legendre family of elliptic curves. As a result, we prove two identities for $1/\pi$ in terms of hypergeometric functions.
We shall make use of the method of partial fractions to generalize some of Ramanujan's infinite series identities, including Ramanujan's famous formula for $\zeta(2n+1)$, and we shall also give a generalization of the transformation formula…
During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Based on numerical computations, Van Hamme recently conjectured $p$-adic analogues to such formulae. Using a combination…
A comprehensive study of the generalized Lambert series $\displaystyle\sum_{n=1}^{\infty}\frac{n^{N-2h}\exp{(-an^{N}x)}}{1-\exp{(-n^{N}x)}}, 0<a\leq 1,\ x>0$, $N\in\mathbb{N}$ and $h\in\mathbb{Z}$, is undertaken. Two of the general…
In the former part of this paper, we give functional equations for Barnes multiple zeta-functions and consider some relevant results. In particular, we show that Ramanujan's classical formula for the Riemann zeta values can be derived from…
"Divergent" Ramanujan-type series for $1/\pi$ and $1/\pi^2$ provide us with new nice examples of supercongruences of the same kind as those related to the convergent cases. In this paper we manage to prove three of the supercongruences by…
We obtained a new formula for $\pi$.
One of the celebrated formulas of Ramanujan is about odd zeta values, which has been studied by many mathematicians over the years. A notable extension was given by Grosswald in 1972. Following Ramanujan's idea, we rediscovered a…