Extensions of Ramanujan's two formulas for $1/\pi$

  title={Extensions of Ramanujan's two formulas for \$1/\pi\$},
  author={Chuanan Wei and Dianxuan Gong},
  journal={arXiv: Combinatorics},

Ramanujan-type formulae for $1/\pi$: The art of translation

We outline an elementary method for proving numerical hypergeometric identities, in particular, Ramanujan-type identities for $1/\pi$. The principal idea is using algebraic transformations of

$q$-analogues of several $\pi $-formulas

  • Chuanan Wei
  • Physics, Mathematics
    Proceedings of the American Mathematical Society
  • 2020
According to the $q$-series method, a short proof for Hou and Sun's identity, which is the $q$-analogue of a known $\pi$-formula, is offered. Furthermore, $q$-analogues of several other

Extensions of the classical theorems for very well-poised hypergeometric functions

The well-known classical summation theorems due to Dougall and certain transformation formulas due to Whipple and Bailey for very well-poised hypergeometric functions are extended by introducing two

Pell's equation and series expansions for irrational numbers

Solutions of Pell's equation and hypergeometric series identities are used to study series expansions for $\sqrt{p}$ where $p$ are arbitrary prime numbers. Numerous fast convergent series expansions

Series expansions for $1/\pi^m$ and $\pi^m$

By means of the telescoping method, we establish two sum- mation formulas on sine function. As the special cases of them, several interesting series expansions for $1/\pi^m$ and $\pi^m$.



Gauss summation and Ramanujan type series for $1/{\pi}$

Using some properties of the gamma function and the well-known Gauss summation formula for the classical hypergeometric series, we prove a four-parameter series expansion formula, which can produce

Ramanujan-type formulae for 1/π: a second wind?

In 1914 S. Ramanujan recorded a list of 17 series for 1=…. We survey the methods of proofs of Ramanujan’s formulae and indicate recently discovered generalisations, some of which are not yet proven.

Hypergeometric identities for 10 extended Ramanujan-type series

We prove, by the WZ-method, some hypergeometric identities which relate ten extended Ramanujan type series to simpler hypergeometric series. The identities we are going to prove are valid for all the

Ramanujan's Class Invariant λn and a New Class of Series for 1/π

On page 212 of his lost notebook, Ramanujan defined a new class invariant λn and constructed a table of values for λn. The paper constructs a new class of series for 1/π associated with λn. The new

About a New Kind of Ramanujan-Type Series

A new kind of Ramanujan-type formula for 1/π is proposed and it is conjectured that it is related to the theory of modular functions.

New 5F4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/π

Abstract New relations are established between families of three-variable Mahler measures. Those identities are then expressed as transformations for the 5F4 hypergeometric function. We use these

Eisenstein series and Ramanujan-type series for 1/π

Using certain representations for Eisenstein series, we uniformly derive several Ramanujan-type series for 1/π.