相关论文: Explaining Ramanujan's "Unclosed" Comment in His L…
Ramanujan's last letter to Hardy explored the asymptotic properties of modular forms, as well as those of certain interesting $q$-series which he called \emph{mock theta functions}. For his mock theta function $f(q)$, he claimed that as $q$…
Let $c_q(n)$ be the Ramanujan sums. Many results concerning Ramanujan-Fourier series $f(n)=\sum_{q=1}^\infty a_q c_q (n)$ are obtained by many mathematicians. In this paper we study series of the form $f(q)=\sum_{n=1}^\infty a_n c_q (n)$,…
In some recent papers, the authors considered regular continued fractions of the form \[ [a_{0};\underbrace{a,...,a}_{m}, \underbrace{a^{2},...,a^{2}}_{m}, \underbrace{a^{3},...,a^{3}}_{m}, ... ], \] where $a_{0} \geq 0$, $a \geq 2$ and $m…
We prove an interesting symmetric $q$-series identity which generalizes a result due to Ramanujan. A proof that is analytic in nature is offered, and a bijective-type proof is also outlined.
Ramanujan's 1920 last letter to Hardy contains seventeen examples of mock theta functions which he organized into three "orders." The most famous of these is the third-order function $f(q)$ which has received the most attention of any…
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 his lost notebook, Ramanujan listed 5 identities related to the false theta function $$f(q)=\sum_{n=0}^\infty (-1)^nq^{n(n+1)/2}.$$ A new combinatorial interpretation and proof of one of these identities is given. The methods of the…
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…
Let $\mathbb{A}=\mathbb{F}_{q}[T]$ be the polynomial ring over finite field $\mathbb{F}_{q}$, and $\mathbb{A}_{+}$ be the set of monic polynomials in $\mathbb{A}$. In this paper, we show that a large class of arithmetic functions in…
We studied Ramanujan series $\sum_{q=1}^{\infty}G(q)c_q(a)$, where $c_q(a)$ is the well-known Ramanujan sum and the complex numbers $G(q)$, as $q\in$N, are the Ramanujan coefficients; of course, we mean, implicitly, that the series…
Ramanujan's $q$-continued fractions are a central part of Ramanujan's development of basic hypergeometric series. They appear in Chapter 16 of Part III and Chapter 32 of Part V of {\em Ramanujan's Notebooks} edited by Berndt, and in Volume…
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 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…
Ramanujan, in his famous first letter to Hardy, claimed a very precise estimate for the number of integers that can be written as a sum of two squares. Far less well-known is that he also made further claims of a similar nature for the…
We use the method of generating functions to find the limit of a $q$-continued fraction, with 4 parameters, as a ratio of certain $q$-series. We then use this result to give new proofs of several known continued fraction identities,…
This note is a complement to an article which was published, six years ago, in The Ramanujan Journal (vol. 45.3, 2018). Here, the goal is to fully describe a singular transcendental continued fraction in Q((T^-1)), tied to a particular…
In this short note, we aim to discuss some summations due to Ramanujan, their generalizations and some allied series
In this short research note, we aim to establish an interesting extension of a summation due to Ramanujan.The result is derived with the help of an extension of Gauss's summation theorem available in the literature.
We define a series $\mathcal{F}_{M,N}$ as a certain generalization of $q$-hypergeometric function. We study its duality and the system of $q$-difference nonlinear equations which admits particular solutions in terms of $\mathcal{F}_{1,M}$.
We continue work started in [1] concerning integer sequences q(n), n in N, defined by q(n) = q(n-q(n-1)) + f(n), with q(1) = 1. Here, f(n), with f(1) = 0, is a given sequence. We define F as the set of semi-infinite sequence f such that the…