Several q-series related to Ramanujan’s theta functions

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.

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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.

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Correspondence to Dazhao Tang.

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The first author was supported by the National Natural Science Foundation of China (No. 11501061) and the Fundamental Research Funds for the Central Universities (No. 2018CDXYST0024). The second author was supported by the National Natural Science Foundation of China (No. 11571143) and Jiangsu National Funds for Distinguished Young Scientists (No. BK20180044).

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Tang, D., Xia, E.X.W. Several q-series related to Ramanujan’s theta functions. Ramanujan J 53, 705–724 (2020). https://doi.org/10.1007/s11139-019-00187-4

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Keywords

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

Mathematics Subject Classification

  • 33D15
  • 11F33
  • 30B10