Related papers: A method for proving Ramanujan series for $1/\pi$
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…
Faulhaber's formula expresses the sum of the first $n$ positive integers, each raised to an integer power $p\geq 0$, as a polynomial in $n$ of degree $p+1$. Ramanujan expressed this sum for $p\in\{\frac12,\frac32,\frac52,\frac72\}$ as the…
In his notebooks, Ramanujan presented without proof many remarkable formulae for the solutions to generalized modular equations. Much later, proofs of the formulae were provided by making use of highly nontrivial identities for theta series…
Using an infinite family of generalizations of the Chudnovsky brothers' series recently obtained via the analytic continuation of the Borwein brothers' formula for Ramanujan-type series of level 1, we apply the Gauss-Salamin-Brent iteration…
In this paper, we propose a new primality test, and then we employ this test to find a formula for {\pi} that computes the number of primes within any interval. We finally propose a new formula that computes the nth prime number as well as…
A sharp asymptotic formula for the sum of reciprocals of $\pi(n)$ is derived, where $\pi(x)$ is the number of primes not exceeding $x$. This result improves the previous results of De Koninck--Ivi\'c and L. Panaitopol.
We find, in the form of a continued fraction, the generating function for the number of (132)-avoiding permutations that have a given number of (123) patterns, and show how to extend this to permutations that have exactly one (132) pattern.…
We present a new proof of the Rogers-Ramanujan identities. Surprisingly, all its ingredients are available already in Rogers seminal paper from 1894, where he gave a considerably more complicated proof.
Towards the end of his life Ramanujan wrote a manuscript on properties of the partition and tau functions, some parts of which remained unpublished until very recently. Nevertheless, this manuscript gave rise to a lot of subsequent work. In…
In 1939, H. S. Zuckerman provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the overpartition function $\overline{p}(n)$. Computing $\overline{p}(n)$ by this method requires…
Ramanujan wrote the following identity \begin{align*} \sqrt{2 \left(1 - \frac{1}{3^2}\right) \left(1 - \frac{1}{7^2}\right) \left(1 - \frac{1}{11^2}\right) \left(1 - \frac{1}{19^2}\right)} \ = \ \left(1 + \frac{1}{7}\right) \left(1 +…
We apply Heine's method---the key idea Heine used in 1846 to derive his famous transformation formula for $_2\phi_1$ series---to multiple basic series over the root system of type $A$. In the classical case, this leads to a bibasic…
We prove a number of new Rogers-Ramanujan type identities involving double, triple and quadruple sums. They were discovered after an extensive search using Maple. The main idea of proofs is to reduce them to some known identities in the…
The constant $\pi$ has fascinated scholars throughout the centuries, inspiring numerous formulas for its evaluation, such as infinite sums and continued fractions. Despite their individual significance, many of the underlying connections…
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.
In 1730 James Stirling, building on the work of Abraham de Moivre, published what is known as Stirling's approximation of $n!$. He gave a good formula which is asymptotic to $n!$. Since then hundreds of papers have given alternative proofs…
An algebraic transformation of the DeTemple-Wang half-integer approximation to the harmonic series produces the general formula and error estimate for the Ramanujan expansion for the nth harmonic number.
In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula…
We deduce new q-series identities by applying inverse relations to certain identities for basic hypergeometric series. The identities obtained themselves do not belong to the hierarchy of basic hypergeometric series. We extend two of our…
This is an item on Ramanujan Graphs for a planned encyclopedia on Ramanujan. The notion of Ramanujan graphs is explained, as well as the reason to name these graphs after Ramanujan.