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RAMANUJAN SERIES WITH A SHIFT
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 107 / Issue 3 / December 2019
- Published online by Cambridge University Press:
- 23 October 2018, pp. 367-380
- Print publication:
- December 2019
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- Article
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We consider an extension of the Ramanujan series with a variable
$x$. If we let 
$x=x_{0}$, we call the resulting series ‘Ramanujan series with the shift 
$x_{0}$’. Then we relate these shifted series to some 
$q$-series and solve the case of level 
$4$ with the shift 
$x_{0}=1/2$. Finally, we indicate a possible way towards proving some patterns observed by the author corresponding to the levels 
$\ell =1,2,3$ and the shift 
$x_{0}=1/2$.
RAMANUJAN SERIES UPSIDE-DOWN
- Part of
-
- Journal:
- Journal of the Australian Mathematical Society / Volume 97 / Issue 1 / August 2014
- Published online by Cambridge University Press:
- 07 July 2014, pp. 78-106
- Print publication:
- August 2014
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- Article
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We prove that there is a correspondence between Ramanujan-type formulas for
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1/\pi $ and formulas for Dirichlet 
$L$-values. Our method also allows us to reduce certain values of the Epstein zeta function to rapidly converging hypergeometric functions. The Epstein zeta functions were previously studied by Glasser and Zucker.
