相关论文: New series for powers of $\pi$ and related congrue…

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…

In this paper, we deduce a family of six new series for $1/\pi$; for example, $$\sum_{n=0}^\infty\frac{41673840n+4777111}{5780^n}W_n\left(\frac{1444}{1445}\right) =\frac{147758475}{\sqrt{95}\,\pi}$$ where $W_n(x)=\sum_{k=0}^n\binom…

In this paper, we evaluate some series of the form $$\sum_{k=1}^\infty\frac{ak^2+bk+c}{k(3k-1)(3k-2)m^k\binom{4k}k}.$$ For example, we prove that $$\sum_{k=1}^\infty\frac{(5k^2-4k+1)8^{k}}{k(3k-1)(3k-2)\binom{4k}k}=\frac{3}2\pi$$ and…

We evaluate some series with summands involving a single binomial coefficient $\binom{6k}{3k}$. For example, we prove that $$\sum_{k=0}^\infty\frac{(63k^2+78k+22)8^k}{(2k+1)(6k+1)(6k+5)\binom{6k}{3k}}=\frac{3\pi}2.$$ Motivated by Galois…

In this paper we collect over 150 new series identities (involving binomial coefficients) conjectured by the author in 2026. The values involved are related to $\pi$ or Riemann's zeta function or Dirichlet's $L$-function. For example, we…

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…

We obtain $q$-analogues of several series for powers of $\pi$. For example, the identity $$\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^3}=\frac{\pi^3}{32}$$ has the following $q$-analogue: \begin{equation*}…

We give $q$-analogues of the following two Ramanujan-type formulas for $1/\pi$: \begin{align*} \sum_{k=0}^\infty (6k+1)\frac{(\frac{1}{2})_k^3}{k!^3 4^k} =\frac{4}{\pi} \quad\text{and}\quad \sum_{k=0}^\infty…

Let $H_n^{(2)}$ denote the second-order harmonic number $\sum_{0<k\le n}1/k^2$ for $n=0,1,2,\ldots$. In this paper we obtain the following identity: $$\sum_{k=1}^\infty\frac{2^kH_{k-1}^{(2)}}{k\binom{2k}k}=\frac{\pi^3}{48}.$$ We explain how…

We present several supercongruences that may be viewed as $p$-adic analogues of Ramanujan-type series for $1/\pi$ and $1/\pi^2$, and prove three of these examples.

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 previous works, we presented series representations for $\pi^3$ and $\pi^5$, in which the prefactor depends only on the golden ratio appears. In this article, we derive a general relation involving trigonometric functions and an infinite…

We define bilateral series related to Ramanujan-like series for $1/\pi^2$. Then, we conjecture a property of them and give some applications.

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$…

Our main results are a WZ-proof of a new Ramanujan-like series for $1/\pi^2$ and a hypergeometric identity involving three series.

Re presenting the traditional proof of Srinivasa Ramanujan's own favorite series for the reciprocal of $\pi$ :\begin{equation}\frac{1}{\pi} = \frac{\sqrt{8}}{9801} \sum_{n=0}^{+\infty} \frac{(4n)!}{(n!)^4} \frac{1103 + 26390n}{396^{4n}} \;…

We show how to find series expansions for $\pi$ of the form $\pi=\sum_{n=0}^\infty {S(n)}\big/{\binom{mn}{pn}a^n}$, where S(n) is some polynomial in $n$ (depending on $m,p,a$). We prove that there exist such expansions for $m=8k$, $p=4k$,…

In 1914, Ramanujan presented a collection of 17 elegant and rapidly converging formulae for $\pi$. Among these, one of the most celebrated is the following series:…

It is known that $\sum_{k=0}^\infty\binom{2k}{k}/((2k+1)4^k)=\pi/2$ and $\sum_{k=0}^\infty\binom{2k}{k}/((2k+1)16^k)=\pi/3$. In this paper we obtain their p-adic analogues such as…

The author gives the full list of his conjectures on series for powers of $\pi$ and other important constants scattered in some of his public papers or his private diaries. The list contains 234 reasonable conjectural series. On the list…