Generalized Lambert Series and Euler’s Pentagonal Number Theorem


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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  1. 1.

    Andrews, G.E.: The Theory of Partitions. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1998)

    Google Scholar 

  2. 2.

    Andrews, G.E., Merca, M.: The truncated pentagonal number theorem. J. Comb. Theory Ser. A 119, 1639–1643 (2012)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Andrews, G.E., Merca, M.: On the number of even parts in all partitions of \(n\) into distinct parts. Ann. Comb. 24, 47–54 (2020)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Berkovich, A., Garvan, F.G.: Some observations on Dyson’s new symmetries of partitions. J. Comb. Theory Ser. A 100, 61–93 (2002)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Christopher, D.: Partitions with fixed number of sizes. J. Integer Seq. 18, 15.11.5 (2015).

  6. 6.

    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Clarendon Press, Oxford (1979)

    Google Scholar 

  7. 7.

    Klazar, M.: Counting even and odd partitions. Am. Math. Mon. 110, 527–532 (2003)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Knopfmacher, A., Mays, M.E.: The sum of distinct parts in compositions and partitions. Bull. Inst. Comb. Appl. 25, 66–78 (1999)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Liu, J.-C.: Some finite generalizations of Euler’s pentagonal number theorem. Czechoslov. Math. J. 67, 525–531 (2017)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Merca, M.: A new look on the generating function for the number of divisors. J. Number Theory 149, 57–69 (2015)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Merca, M.: Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer. J. Number Theory 160, 60–75 (2016)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Merca, M.: New relations for the number of partitions with distinct even parts. J. Number Theory 176, 1–12 (2017)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Merca, M.: The Lambert series factorization theorem. Ramanujan J. 44(2), 417–435 (2017)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Merca, M.: New connections between functions from additive and multiplicative number theory. Mediterr. J. Math. 15, 36 (2018)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Merca, M., Schmidt, M.D.: A partition identity related to Stanley’s theorem. Am. Math. Mon. 125(10), 929–933 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Merca, M., Schmidt, M.D.: The partition function \(p(n)\) in terms of the classical Möbius function. Ramanujan J. 49, 87–96 (2019)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Merca, M., Schmidt, M.D.: Generating special arithmetic functions by Lambert series factorizations. Contrib. Discret. Math. 14(1), 31–45 (2019)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Merca, M., Schmidt, M.D.: Factorization theorems for generalized Lambert series and applications. Ramanujan J. 51, 391–419 (2020)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Schmidt, M.D.: New recurrence relations and matrix equations for arithmetic functions generated by Lambert series. Acta Arith. 181, 355–367 (2017)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Shanks, D.: A short proof of an identity of Euler. Proc. Am. Math. Soc. 2, 747–749 (1951)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. Published electronically (2020).

  22. 22.

    Tani, N.: Enumeration of the partitions of an integer into parts of a specified number of different sizes and especially two sizes. J. Integer Seq. 14, 11.3.6 (2011).

  23. 23.

    Warnaar, S.O.: \(q\)-hypergeometric proofs of polynomial analogues of the triple product identity, Lebesgue’s identity and Euler’s pentagonal number theorem. Ramanujan J. 8, 467–474 (2004)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Wilf, H.: Three problems in combinatorial asymptotics. J. Comb. Theory Ser. A 35, 199–207 (1983)

    MathSciNet  Article  Google Scholar 

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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).

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  • Partitions
  • pentagonal number theorem
  • Lambert series

Mathematics Subject Classification

  • 11P81
  • 05A19