Related papers: On some definite integrals connecting with certain…
In this article it is proven the existence of integration of indefinite integrals as infinite derivative's series expansion. This also opens a new way to integrate a definite integral.
A translation of Kummer`s paper On certain definite integrals and infinite series
In this note we study the convergence of recursively defined infinite series. We explore the role of the derivative of the defining function at the origin (if it exists), and develop a comparison test for such series which can be used even…
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…
We revisit an infinitely nested radical by Ramanujan. Utilizing the full strength of his method, we shall arrive at some new infinitely nested radicals.
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.
We prove a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of $\pi$, and conjecture a type of supercongruences associated to them. We support this conjecture by checking all the cases for many…
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…
This paper develops an approach to the evaluation of infinite series involving hyperbolic functions. By using the approach, we give explicit formulas for several classes of series of hyperbolic functions in terms of Riemann zeta values.…
We define bilateral series related to Ramanujan-like series for $1/\pi^2$. Then, we conjecture a property of them and give some applications.
This paper investigates Srinivasa Ramanujan's initial intuitive methodology for assigning the finite value -1/12 to the sum of the divergent infinite series of all positive integers. We systematically examine Ramanujan's initial method,…
In this paper we present experimental ways of evaluating Ramanujan`s quantities which as someone can see are related with algebraic numbers. The good thing with algebraic numbers is that can be found in a closed form, from there…
In this paper, we give explicit evaluation for some infinite series involving generalized (alternating) harmonic numbers. In addition, some formulas for generalized (alternating) harmonic numbers will also be derived.
We introduce and prove evaluations for families of multiple elliptic integrals by solving special types of ordinary and partial differential equations. As an application, we obtain new expressions of Ramanujan-type series of level 4 and…
Throughout his entire mathematical life, Ramanujan loved to evaluate definite integrals. One can find them in his problems submitted to the \emph{Journal of the Indian Mathematical Society}, notebooks, Quarterly Reports to the University of…
This article is written with the hope to draw attention to a method that uses integral transforms to find exact values for a large class of convergent series (and, in particular, series of rational terms). We apply the method to some series…
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…
In this paper, we establish the irrationality of some open problems in mathematics based on using a recursive formula that generate the complete sequence of numbers. see [1] But before getting into that we begin with some Ramanujan notable…
In this paper, we obtain analytical solutions of some definite integrals of Srinivasa Ramanujan [Mess. Math., XLIV, 75-86, 1915] in terms of Meijer's $G$-function by using Laplace transforms of $ \sin(\beta x^{2}),\cos(\beta x^{2}),…
In this article, we consider systems of linear congruences in several variables and obtain necessary and sufficient conditions as well as explicit expressions for the number of solutions subject to certain restriction conditions. These…