Related papers: Generalized Ramanujan-Sato Series Arising from Mod…

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 prove the Ramanujan and Sato-Tate conjectures for Bianchi modular forms of weight at least 2. More generally, we prove these conjectures for all regular algebraic cuspidal automorphic representations of $\mathrm{GL}_2(\mathbf{A}_F)$ of…

We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato--Tate density. Examples of such sequences come from…

We explain the use and set grounds about applicability of algebraic transformations of arithmetic hypergeometric series for proving Ramanujan's formulae for $1/\pi$ and their generalisations.

In the first part of the paper we characterize certain systems of first order nonlinear differential equations whose space of solutions is an $\mathfrak{sl}_2(\mathbb{C})$-module. We prove that such systems, called Ramanujan systems of…

The generalized Riordan group consists of infinite lower triangular matrices that correspond to certain operators in the space of formal power series. Each such group contains the matrix (generalized Pascal matrix), elements of which are…

Several terminating generalizations of Ramanujan's formula for $\frac{1}{\pi}$ with complete WZ proofs are given.

We use a variant of Wan's method to prove two Ramanujan-Orr type formulas for $1/\pi$. This variant needs to know in advance the formulas for $1/\pi$ that we want to prove, but avoids the need of solving a system of equations.

We give a simple unified proof for all existing rational hypergeometric Ramanujan identities for $1/\pi$, and give a complete survey (without proof) of several generalizations: rational hypergeometric identities for $1/\pi^c$, Taylor…

We consider the generalised Mathieu series \[\sum_{n=1}^\infty \frac{n^\gamma}{(n^\lambda+a^\lambda)^\mu}\qquad (\mu>0)\] when the parameters $\lambda$ ($>0$) and $\gamma$ are even integers for large complex $a$ in the sector…

By means of the extended Gould-Hsu inverse series relations, we find that the dual relation of Dougall's summation theorem for the well--poised $_7F_6$-series can be utilized to construct numerous interesting Ramanujan--like infinite series…

We state and prove a claim of Ramanujan. As a consequence, a large class of Saalchutzian hypergeometric series is summed in closed form.

This is the second part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. In this part, we give constructive results generalizing previous ones obtained by the author in…

Each of Ramanujan's series for $\frac{1}{\pi}$ is of the form $$ \sum_{n=0}^{\infty} z^n \frac{ (a_{1})_{n} (a_{2})_{n} (a_{3})_{n} }{ (b_{1})_{n} (b_{2})_{n} (b_{3})_{n} } (c_{1} n + c_2) $$ for rational parameters such that the difference…

Applying the triplicate form of the extended Gould--Hsu inverse series relations to Dougall's summation theorem for the well--poised $_7F_6$-series, we establish, from the dual series, several interesting Ramanujan--like infinite series…

In this short note, we aim to discuss some summations due to Ramanujan, their generalizations and some allied series

The notion of quasi-elliptic rings appeared as a result of an attempt to classify a wide class of commutative rings of operators found in the theory of integrable systems, such as rings of commuting differential, difference,…

We provide a general theorem for evaluating trigonometric Dirichlet series of the form $\sum_{n \geq 1} \frac{f (\pi n \tau)}{n^s}$, where $f$ is an arbitrary product of the elementary trigonometric functions, $\tau$ a real quadratic…

We present a new method for producing series for $1/\pi$ and other constants using Legendre's relation, starting from a generation function that can be factorised into two elliptic $K$'s; this way we avoid much of modular theory or creative…

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue…