A proof that Euler missed ...

  title={A proof that Euler missed ...},
  author={Alfred J. van der Poorten and R. Ap{\'e}ry},
  journal={The Mathematical Intelligencer},
176 Citations
Further Apéry-Like Series for Riemann Zeta Function
  • W. Chu
  • Mathematics
    Mathematical Notes
  • 2021
Abstract By means of two transformation formulas of classical hypergeometric series, several Apéry-like series involving harmonic numbers of higher order are derived for the Riemann zeta function,
Generating conjectures on fundamental constants with the Ramanujan Machine.
This work supports a different conceptual framework for research: computer algorithms use numerical data to unveil mathematical structures, thus trying to replace the mathematical intuition of great mathematicians and providing leads to further mathematical research.
Multilevel Evaluation of the General Dirichlet Series
  • I. Suwan
  • Computer Science
    Advances in the Theory of Nonlinear Analysis and its Application
  • 2020
An accurate method for summing the general Dirichlet series using multilevels of grid points overcomes the high cost of calculating the long range terms and results obtained are better than those of direct calculations.
Holonomic relations for modular functions and forms: First guess, then prove
One major theme of this paper concerns the expansion of modular forms and functions in terms of fractional (Puiseux) series. This theme is connected with another major theme, holonomic functions and
Infinite series identities derived from the very well-poised $$\Omega $$ Ω -sum
For the very well-poised $$\Omega $$ Ω -series, a universal iteration pattern is established that yields numerous infinite series identities including several important ones discovered by Ramanujan
Recurrence equation and integral representation of Apéry sums
  • M. Uhl
  • Mathematics
    European Journal of Mathematics
  • 2020
Various methods are used to investigate sums involving a reciprocal central binomial coefficient and a power term. In the first part, new functions are introduced for calculation of sums with a
On two congruences involving Franel numbers
  • Ji-Cai Liu
  • Mathematics
    Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
  • 2020
Via symbolic summation method, we establish the following series for $\pi^2$: \begin{align*} \sum_{k=1}^\infty \frac{H_k-2H_{2k}}{(-3)^k k} = \frac{\pi^2}{18}, \end{align*} where $H_k=\sum_{j=1}^k
A note on the number of irrational odd zeta values
Abstract We prove that, for any small $\varepsilon > 0$, the number of irrationals among the following odd zeta values: $\zeta (3),\zeta (5),\zeta (7),\ldots ,\zeta (s)$ is at least $( c_0 -
Hypergeometric rational approximations to ζ(4)
Abstract We give a new hypergeometric construction of rational approximations to ζ(4), which absorbs the earlier one from 2003 based on Bailey's 9F8 hypergeometric integrals. With the novel