Related papers: Proofs of conjectures on Ramanujan-type series of …
Using the Wolfram NumberTheory package and the Recognize command, together with numerical estimates involving the elliptic lambda and elliptic alpha functions, Bagis and Glasser, in 2013, introduced a conjectural Ramanujan-type series…
Using a modular equation of level $3$ and degree $23$ due to Chan and Liaw, we prove the fastest convergent rational Ramanujan-type series for $1/\pi$ of level $3$.
Our main results are a WZ-proof of a new Ramanujan-like series for $1/\pi^2$ and a hypergeometric identity involving three series.
About 40 years ago Jonathan and Peter Borwein discovered the series identity $$ \sum_{n=0}^\infty \frac{(-1)^n(6n)!}{(3n)!(n!)^3} \frac{(A+nB)}{C^{n+1/2}} = \frac{1}{12\pi} $$ where \begin{align*} A&=1657145277365+212175710912\sqrt{61},\cr…
In this paper we prove theorems related to the Ramanujan-type series for $1/\pi$ (type $_3F_2$) and to the Ramanujan-like series, discovered by the author, for $1/\pi^2$ (type $_5F_4$). Our developments for the cases $_3 F_2$ and $_5 F_4$…
Via symbolic computation we deduce 97 new type series for powers of $\pi$ related to Ramanujan-type series. Here are three typical examples: $$\sum_{k=0}^\infty \frac{P(k) \binom{2k}k\binom{3k}k…
We find new hypergeometric identities which, in a certain aspect, are stron-ger than others of the same style found by the author in a previous paper. The identities in Section \ref{section-pi} are related to some Ramanujan-type series for…
Through an application of a remarkable result due to Mishev in 2018 concerning the inverses for a class of transformations of sequences of complex numbers, we obtain a very simple proof for a famous series for $\frac{1}{\pi}$ due to…
In this work, we start an investigation of asymmetric Rogers--Ramanujan type identities. The first object is the following unexpected relation $$\sum_{n\ge 0} \frac{(-1)^n q^{3\binom{n}{2}+4n}(q;q^3)_n}{(q^9;q^9)_n} =…
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…
We prove a polynomial continued fraction identity for the constant $-\pi/4$, conjectured by the Ramanujan Machine project. The proof proceeds by explicitly solving the underlying second-order linear difference equation. We derive a…
Bhoria, Eyyunni and Maji recently obtained a four-parameter $q$-series identity which gives as special cases not only all five entries of Ramanujan on pages 354 and 355 of his second notebook but also allows them to obtain an analytical…
Motivated by Ramanujan-type series and Zeilberger-type series, in this paper we investigate two new types of series for powers of $\pi$. For example, we prove that…
We show with some examples how to prove some Ramanujan-type series for $1/\pi$ in an elementary way by using terminating identities.
We consider integral and series transformations, which are associated with Ramanujan's identities, involving various arithmetic functions and a ratio of products of Riemann's zeta functions of different arguments. Reciprocal inversion…
We revisit old conjectures of Fermat and Euler regarding representation of integers by binary quadratic form x^2+5y^2. Making use of Ramanujan's_1\psi_1 summation formula we establish a new Lambert series identity for…
Recently, Rosengren utilized an integral method to prove a number of conjectural identities found by Kanade and Russell. Using this integral method, we give new proofs to some double sum identities of Rogers-Ramanujan type. These identities…
Ramanujan wrote the following identity \begin{align*} \sqrt{2 \left(1 - \frac{1}{3^2}\right) \left(1 - \frac{1}{7^2}\right) \left(1 - \frac{1}{11^2}\right) \left(1 - \frac{1}{19^2}\right)} \ = \ \left(1 + \frac{1}{7}\right) \left(1 +…
The two Rogers-Ramanujan $q$-series \[ \sum_{n=0}^{\infty}\frac{q^{n(n+\sigma)}}{(1-q)\cdots (1-q^n)}, \] where $\sigma=0,1$, play many roles in mathematics and physics. By the Rogers-Ramanujan identities, they are essentially modular…
"Divergent" Ramanujan-type series for $1/\pi$ and $1/\pi^2$ provide us with new nice examples of supercongruences of the same kind as those related to the convergent cases. In this paper we manage to prove three of the supercongruences by…