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RAMANUJAN SERIES WITH A SHIFT

Part of: Other special functions Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  23 October 2018

JESÚS GUILLERA Open the ORCID record for JESÚS GUILLERA [Opens in a new window]
JESÚS GUILLERA*
Affiliation:
Department of Mathematics, University of Zaragoza, 50009 Zaragoza, Spain email jguillera@gmail.com
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Abstract

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 $x_{0}=1/2$.

Keywords

Ramanujan series for 1/𝜋hypergeometric serieslattice sumsDirichlet L-valuesmodular forms

MSC classification

Primary: 33C20: Generalized hypergeometric series, $_pF_q$ 33C05: Classical hypergeometric functions, $_2F_1$
Secondary: 11F03: Modular and automorphic functions 33C75: Elliptic integrals as hypergeometric functions 33E05: Elliptic functions and integrals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 107 , Issue 3 , December 2019 , pp. 367 - 380
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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