Related papers: Ramanujan-Nagell type equations and perfect number…
We show with some examples how to prove some Ramanujan-type series for $1/\pi$ in an elementary way by using terminating identities.
We study the properties of a general continued fraction of Ramanujan. In some certain cases we evaluate it completely.
An algebraic transformation of the DeTemple-Wang half-integer approximation to the harmonic series produces the general formula and error estimate for the Ramanujan expansion for the nth harmonic number.
In this paper, we utilize operational methods to obtain closed-form solutions for certain classes of integrals in the spirit of Ramanujan's Master Theorem and provide several analogs to it. Although the use of operational calculus makes the…
We define bilateral series related to Ramanujan-like series for $1/\pi^2$. Then, we conjecture a property of them and give some applications.
The Ramanujan--Mordell Theorem for sums of an even number of squares is extended to other quadratic forms and quadratic polynomials.
Explicit formulas involving a generalized Ramanujan sum are derived. An analogue of the prime number theorem is obtained and equivalences of the Riemann hypothesis are shown. Finally, explicit formulas of Bartz are generalized.
We analyze the double series of Bessel functions given by Ramanujan. Using a very simple lemma we establish the uniform convergence of these series. By this we address to the Gauss circle problem.
We completely solve the Diophantine equation $x^2+2^k11^\ell19^m=y^n$ in integers $x,y\geq 1;~ k,\ell, m\geq 0~$ and $n\geq 3$ with $\gcd(x,y)=1$, except the case $2\mid k, 2\nmid \ell m$ and $5\mid n$. We use this result to recover some…
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…
In this paper we want to prove some formulas listed by S. Ramanujan in his paper "Modular equations and approximations to $\pi$" \cite{24} with an elementary method.
We give new nested radical equations of similar kind to Ramanujan's questions to the Indian Mathematical Society 100 years ago. While many have since considered these from the perspectives of the Notebooks of Ramanujan and from the theory…
During the course of verifying the results of Ramanujan on hypergeometric series, Berndt in his notebooks, Part II mentioned corrected forms of two of the Ramanujan's results. The aim of this short research note is to point out that one of…
We define sequence patterns of length $n$ and level $\ell$ to be equivalence classes of sequences that have $n$ elements from the set of $\ell$ integer symbols $\{1,2,\ldots,\ell\}$ with no restriction on repetition, where the equivalence…
We define a length function for a perfect crystal. As an application, we derive a variant of the Rogers-Ramanujan identities which involves (a $q$-analog of) the Fibonacci numbers.
We use an integral method to establish a number of Rogers-Ramanujan type identities involving double and triple sums. The key step for proving such identities is to find some infinite products whose integrals over suitable contours are…
We prove two new series of Ramanujan type for $1/\pi^2$.
We prove q-analogues of two Ramanujan-type series for $1/\pi$ from $q$-analogues of ordinary WZ pairs.
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…
In this paper, we study how close the terms of a finite arithmetic progression can get to a perfect square. The answer depends on the initial term, the common difference and the number of terms in the arithmetic progression.