Related papers: Chudnovsky's formula for $1/\pi$ revisited
In this note we give a short proof of Cherednik's generalization of Macdonald-Mehta identities for the root system $A_{n-1}$ using the representation theory of quantum groups. These identities, suggested and proved by Cherednik, give an…
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 his 1685 paper "Observationes cyclometricae" published in Acta Eruditorum, Adam Adamandy Kocha\'nski presented an approximate ruler-and-compass construction for rectification of the circle. It is not generally known that the first part…
Through an application of a remarkable result due to Mishev in 2018 concerning the inverses for a class of transformations of sequences of complex numbers, we obtain a very simple proof for a famous series for $\frac{1}{\pi}$ due to…
We prove that there is a correspondence between Ramanujan-type formulas for 1/\pi, and formulas for Dirichlet L-values. The same method also allows us to resolve certain values of the Epstein zeta function in terms of rapidly converging…
In this paper, we study the Carlitz's degenerate Bernoulli numbers and polynomials and give some formulae and identities related to those numbers and polynomials.
Using mostly elementary results and functions from probability, we prove Wallis's formula for pi: pi/2 = prod_n (2n * 2n) / ((2n-1) * (2n+1)). The proof involves normalization constants and the Gamma function, Standard normal, and the…
We use properties of modular forms to prove the following extension of the Ramanujan-Mordell formula, \begin{align*} z^{k-j}z_p^{j}=&\frac{p_{\chi}^{k-j}-1}{p_{\chi}^{k}-1}F_p(k,j;\tau)+…
An identity by Ramanujan is expressed using polar coordinates, so that its proof reduces to the verification of an elementary trigonometric identity. This approach produces a few variations on Ramanujan's original identity.
We develop an approach to establish $1/\pi$-series from bimodular forms. Utilizing this approach, we obtain new families of $2$-variable $1/\pi$-series associated to Zagier's sporadic Apery-like sequences.
In this Note, we start off with the primary representation of e and from there present an elementary short proof for the Wallis formula for $\pi$.
Motivated by work of Chan, Chan, and Liu, we obtain a new general theorem which produces Ramanujan-Sato series for $1/\pi$. We then use it to construct explicit examples related to non-compact arithmetic triangle groups, as classified by…
In 1914, Ramanujan gave a list of 17 identities expressing $1/\pi$ as linear combinations of values of hypergeometric functions at certain rational numbers. Since then, identities of similar nature have been discovered by many authors.…
In this paper we prove some transformation formulae for congruences modulo a prime and deduce some congruences for Domb numbers and Almkvist-Zudilin numbers. We also pose some conjectures on congruences modulo prime powers.
In this paper, we give a short proof of a relation generalizing many identities for Bernoulli numbers.
Via the contour integral method, we establish a reduction formula from a double series to a single series with parameters, which not only implies Uncu and Zudilin's two results and Cao and Wang's two results, but also is related to…
We study the quotient of hypergeometric functions \begin{equation*} \mu_{a}^*(r)=\frac{\pi}{2\sin{(\pi a)}}\frac{F(a,1-a;1;1-r^3)}{F(a,1-a;1;r^3)} \quad (r\in(0,1)) \end{equation*} in the theory of Ramanujan's generalized modular equation…
There have been many works on proving the integrals in the table of integrals compiled by Gradshteyn and Ryzhik, and in this paper we prove some doubly logarithmic integral identities in the Gradshteyn and Ryzhik table.
In 2007 Lam and Pylyavskyy found a combinatorial formula for the dual stable Grothendieck polynomials, which are the dual basis of the stable Grothendieck polynomials with respect to the Hall inner product. In 2016 Galashin, Grinberg, and…
The original proof of the Sharkovsky theorem is presented in full detail. The proof should be accessible to readers with basic Real Analysis background. Although nowadays there are several alternative proofs of this classical result, we…