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Multi-dimensional q-summations and multi-colored partitions

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Abstract

Motivated by Alladi’s recent multi-dimensional generalization of Sylvester’s classical identity, we provide a simple combinatorial proof of an overpartition analogue, which contains extra parameters tracking the numbers of overlined parts of different colors. This new identity encompasses a handful of classical results as special cases, such as Cauchy’s identity, and the product expressions of three classical theta functions studied by Gauss, Jacobi and Ramanujan.

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Acknowledgements

We would like to acknowledge our gratitude to Ae Ja Yee for her helpful suggestions, which strengthen our original version of Theorem 1.2. We also want to thank the referee for the careful reading and useful comments.

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Authors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, USA

    Shane Chern

  2. College of Mathematics and Statistics, Chongqing University, Huxi Campus LD506, Chongqing, 401331, People’s Republic of China

    Shishuo Fu

  3. College of Mathematics and Statistics, Chongqing University, Huxi Campus LD208, Chongqing, 401331, People’s Republic of China

    Dazhao Tang

Authors
  1. Shane Chern
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  2. Shishuo Fu
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  3. Dazhao Tang
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Correspondence to Dazhao Tang.

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Shishuo Fu and Dazhao Tang were supported by National Natural Science Foundation of China (No. 11501061).

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Chern, S., Fu, S. & Tang, D. Multi-dimensional q-summations and multi-colored partitions. Ramanujan J 51, 297–306 (2020). https://doi.org/10.1007/s11139-018-0079-7

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  • DOI: https://doi.org/10.1007/s11139-018-0079-7

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