Generalized Lambert Series and Euler’s Pentagonal Number Theorem

Abstract

We consider the number of k’s in all the partitions of n, and provide new connections between partitions and functions from multiplicative number theory. In this context, we obtain a new generalization of Euler’s pentagonal number theorem and provide combinatorial interpretations for some special cases of this general result.

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Acknowledgements

The author wish to thank the anonymous referees for a very careful reading of the original manuscript.

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Correspondence to Mircea Merca.

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Merca, M. Generalized Lambert Series and Euler’s Pentagonal Number Theorem. Mediterr. J. Math. 18, 29 (2021). https://doi.org/10.1007/s00009-020-01663-8

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Keywords

  • Partitions
  • pentagonal number theorem
  • Lambert series

Mathematics Subject Classification

  • 11P81
  • 05A19