Related papers: Reciprocal Hyperbolic Series of Ramanujan Type
Inspired by the recent pioneering work, dubbed "The Ramanujan Machine" by Raayoni et al. (arXiv:1907.00205), we (automatically) [rigorously] prove some of their conjectures regarding the exact values of some specific infinite continued…
The theory of Mellin transform is an incredibly useful tool in evaluating some of the well known results for the zeta function. Ramanujan in his quarterly reports \cite{1} gave a theorem for Mellin transform which is now known as…
We use Zeilberger's algorithm for proving some identities of Ramanujan-type via $_2F_1$ evaluations.
Let $k,N \in \mathbb{N}$ with $N$ square-free and $k>1$. We prove an orthogonal relation and use this to compute the Fourier coefficients of the Eisenstein part of any $f(z) \in M_{2k}(\Gamma_0(N))$ in terms of sum of divisors function. In…
We establish new transformation formulas involving theta functions and certain bilateral basic hypergeometric series. From these formulas, we construct companion $q$-series for a class of $q$-series such that the asymptotic expansion of…
The classical transformation of Jacobi's theta function admits a simple proof by producing an integral representation that yields this invariance apparent. This idea seems to have first appeared in the work of S. Ramanujan. Several examples…
Hyperbolic beta integrals are analogues of Euler's beta integral in which the role of Euler's gamma function is taken over by Ruijsenaars' hyperbolic gamma function. They may be viewed as $(q,\widetilde{q})$-bibasic analogues of the beta…
We find new hypergeometric identities which, in a certain aspect, are stron-ger than others of the same style found by the author in a previous paper. The identities in Section \ref{section-pi} are related to some Ramanujan-type series for…
In 1997, van Hamme developed $p-$adic analogs, for primes $p$, of several series which relate hypergeometric series to values of the gamma function, originally studied by Ramanujan. These analogs relate truncated sums of hypergeometric…
We establish some identities of Euler related sums. By using these identities, we discuss the closed form representations of sums of harmonic numbers and reciprocal parametric binomial coefficients through parametric harmonic numbers,…
Ramanujan (1916) expressed quotients of certain q-series as polynomials of Eisenstein series of degree 2, 4, 6 and derived the famous Ramanujan's differential equations. We continue this research with the variants of Eisenstein-type series…
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…
Results are presented for some infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-, two- and three-dimensional series. The sums of these series can be evaluated…
The Ramanujan Machine project detects new expressions related to constants of interest, such as $\zeta$ function values, $\gamma$ and algebraic numbers (to name a few). In particular the project lists a number of conjectures involving even…
In the paper, the authors establish Maclaurin's series expansions and series identities for positive integer powers of the inverse sine function, for positive integer powers of the inverse hyperbolic sine function, for the composite of…
We derive product and series representations of the gamma function using Newton interpolation series. Using these identities, a new formula for the coefficients in the Taylor series of the reciprocal gamma function is found. We also find…
In the present work, we extend current research in a nearly-forgotten but newly revived topic, initiated by P. A. MacMahon, on a generalized notion which relates the divisor sums to the theory of integer partitions and two infinite families…
Explicit evaluations of the Tornheim-like double series in the form \[ \sum_{n,m=1}^\infty \frac{H_{n+m+s}}{nm\left( n+m+s \right)},\ s\in \mathbb{N\cup } \left\{ 0 \right\} \] and their extensions are given. Furthermore, series of the type…
We study Whittaker coefficients for maximal parabolic Eisenstein series on metaplectic covers of split reductive groups. By the theory of Eisenstein series these coefficients have meromorphic continuation and functional equation. However…
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…