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Several q-series related to Ramanujan’s theta functions

  • Dazhao TangEmail author
  • Ernest. X. W. Xia
Article
  • 52 Downloads

Abstract

Quite recently, the first author investigated vanishing coefficients of the arithmetic progressions in several q-series expansions. In this paper, we further study the signs of coefficients in two q-series expansions and establish some interlinked identities for several q-series expansions by means of Ramanujan’s theta functions. We obtain the 5-dissections of these two q-series and give combinatorial interpretations for these dissections. Moreover, we obtain four q-series identities involving the aforementioned q-series, two of which were proved by Kim and Toh via modular forms.

Keywords

Ramanujan’s theta functions q-series expansions Jacobi’s triple product identity Interlinked identities 

Mathematics Subject Classification

33D15 11F33 30B10 

Notes

Acknowledgements

The authors are indebted to Shishuo Fu for his helpful comments on a preliminary version of this paper. The authors also would like to thank the anonymous referee for his/her careful reading and helpful comments on an earlier version of the paper.

References

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center for Applied MathematicsTianjin UniversityTianjinPeople’s Republic of China
  2. 2.Department of MathematicsJiangsu UniversityZhenjiangPeople’s Republic of China

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