Related papers: Series for $1/\pi$ arising from Cauchy product

By applying the partial derivative operator to several summation formulas for hypergeometric series, we prove several double series for $\pi$ in this paper. Similarly, we also establish several $q$-analogues of them.

In this article using the theory of Eisenstein series, we give rise to the complete evaluation of two Gauss hypergeometric functions. Moreover we evaluate the modulus of each of these functions and the values of the functions in terms of…

In this paper, we shall prove two conjectures of Z.-W. Sun concerning Ap\'ery-like series. One of the series is alternating whereas the other one is not. Our main strategy is to convert the series (resp.~the alternating series) to…

Using the theory of Calabi-Yau differential equations we obtain all the parameters of Ramanujan-Sato-like series for $1/\pi^2$ as $q$-functions valid in the complex plane. Then we use these q-functions together with a conjecture to find new…

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…

We confirm a conjectural supercongruence involving Catalan numbers, which is one of the 100 selected open conjectures on congruences of Sun. The proof makes use of hypergeometric series identities and symbolic summation method.

First we give general formulas for proving real or complex Ramanujan series for $1/\pi$. Then, as an example, we apply them for providing complete proofs of the fastest series for $1/\pi$ due to Ramanujan using Russell and Weber modular…

We make a summary of the different types of proofs adding some new ideas. In addition we conjecture some relations which could be necessary in "modular type proofs" (not still found) of the Ramanujan-like series for 1/\pi^2.

In our previous publication we have shown a method for calculating series of even powers of $\pi$ based on the product representation of the $sinc$ function. We refer the readers to [1] for more details. In this work we apply the method to…

We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and higher-order harmonic numbers. When the rates of convergence are certain rational numbers, modularity allows us to…

We use the correlation functions of vertex operators to give a proof of Cauchy's formula \begin{align*} \prod^K_{i=1}\prod^N_{j=1}(1-x_iy_j)=\sum_{\mu\subseteq [K\times N]}(-1)^{|\mu|}s_{\mu}\{x\}s_{\mu'}\{y\}. \end{align*} As an…

We provide a double-series formula for $\pi$ obtained using the Fourier series expansion of $1/\cos(x/4)$ and applying the Parseval-Plancherel identity. We show that such a formula involves the Grothendieck-Krivine constant, and that the…

We prove two new series of Ramanujan type for $1/\pi^2$.

In a well-known 1914 paper, Ramanujan gave a number of rapidly converging series for $1/\pi$ which are derived using modular functions of higher level. D. V. and G. V. Chudnovsky in their 1988 paper derived an analogous series representing…

Using generalized binomial coefficient identities and some results of John Dougall, we derive some families of series involving the cubes of Catalan numbers. We also establish a family of series containing fourth powers of Catalan numbers.…

The celebrated Cauchy identity expresses the product of terms $(1 - x_i y_j)^{-1}$ for $(i,j)$ indexing entries of a rectangular $m\times n$-matrix as a sum over partitions $\lambda$ of products of Schur polynomials:…

Recently, the authors with Lea Beneish established a recipe for constructing Ramanujan-Sato series for $1/\pi$, and used this to construct 11 explicit examples of Ramanujan-Sato series arising from modular forms for arithmetic triangle…

With the help of the partial derivative operator and several summation formulas for hypergeometric series, we find three double series for $\pi$. In terms of the operator just stated and several summation formulas for basic hypergeometric…

In terms of Sear's transformation formula for $_4\phi_3$-series, we give new proofs of a summation formula for ${_4\phi_3}$-series due to Andrews [2] and another summation formula for${_4\phi_3}$-series conjectured in the same paper.…

Infinite series are evaluated through the manipulation of a series for $\cos(2t \sin^{-1}x)$ resulting from Clausen's Product. Hypergeometric series equal to an expression involving $\frac{1} {\pi}$ are determined. Techniques to evaluate…