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Partition implications of a three-parameter q-series identity

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Abstract

A generalization of a beautiful q-series identity found in the unorganized portion of Ramanujan’s second and third notebooks is obtained. As a consequence, we derive a three-parameter identity which is a rich source of partition-theoretic information. In particular, we use this identity to obtain a generalization of a recent result of Andrews et al., which itself generalizes the famous result of Fokkink et al. This three-parameter identity also leads to several new weighted partition identities as well as a natural proof of a recent result of Garvan. This natural proof gives interesting number-theoretic information along the way. We also obtain a new result consisting of an infinite series involving a special case of Fine’s function F(a, b; t), namely \(F(0,q^n;cq^n)\). For \(c=1\), this gives Andrews’ famous identity for \(spt (n)\), whereas for \(c=-1, 0\) and q, it unravels new relations that the divisor function d(n) has with other partition-theoretic functions such as the largest parts function \(lpt (n)\).

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Notes

  1. The authors are indebted to Andrews, Private communication, June 12, (2018) for this proof.

  2. It was during the international conference on the occasion of Ramanujan’s 125th birth anniversary at University of Delhi in December 2012 that the first author learned from George E. Andrews about the connection between \(lpt (n)\) and Zagier’s identity in Eq. (7.22). Both the authors thank Andrews for the same.

References

  1. Alladi, K.: Weighted partition identities and applications. In: Berndt, B.C., Diamond, H.G., Hildebrand, A.J. (eds.) Analytic Number Theory, vol. I. Birkhäuser, Boston (1996)

    Google Scholar 

  2. Alladi, K.: Partition identities involving gaps and weights. Trans. Am. Math. Soc. 349(12), 5001–5019 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alladi, K.: Partition identities involving gaps and weights. II. Ramanujan J. 2, 21–37 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alladi, K.: A partial theta identity of Ramanujan and its number-theoretic interpretation. Ramanujan J. 20, 329–339 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alladi, K.: Partitions with non-repeating odd parts and combinatorial identities. Ann. Comb. 20, 1–20 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ando, M.: A combinatorial proof of an identity for the divisor generating function. Electron. J. Comb. 20(2), P13 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Andrews, G.E., The Theory of Partitions. Addison-Wesley Pub. Co., New York. Reissued, p. 1998. Cambridge University Press, New York (1976)

  8. Andrews, G.E.: Ramanujan’s Lost Notebook V: Euler’s partition identity. Adv. Math. 61, 156–164 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. Andrews, G.E.: Questions and conjectures in partition theory. Am. Math. Mon. 93, 708–711 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  10. Andrews, G.E.: The number of smallest parts in the partitions on \(n\). J. Reine Angew. Math. 624, 133–142 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Andrews, G.E.: The Bhargava-Adiga summation and partitions. J. Indian Math. Soc. 84(3–4), 151–160 (2017)

    Article  MathSciNet  Google Scholar 

  12. Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook. Part II. Springer, New York (2009)

    MATH  Google Scholar 

  13. Andrews, G.E., Chan, S.H., Kim, B.: The odd moments of ranks and cranks. J. Comb. Theory Ser. A 120(1), 77–91 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Andrews, G.E., Dyson, F.J., Hickerson, D.: Partitions and indefinite quadratic forms. Invent. Math. 91, 391–407 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  15. Andrews, G.E., Garvan, F.G., Liang, J.: Self-conjugate vector partitions and the parity of the spt-function. Acta Arith. 158(3), 199–218 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Banerjee, K., Dixit, A.: New representations for \(\sigma (q)\) via reciprocity theorems. In: Analytic Number Theory, Modular Forms and \(q\)-Hypergeometric Series (in honor of Krishnaswami Alladi’s \(60^{\text{th}}\) birthday). Springer Proceedings in Mathematics and Statistics, pp. 39–57 (2017)

  17. Berkovich, A., Uncu, A.K.: Variation on a theme of Nathan Fine. New weighted partition identities. J. Number Theory 176, 226–248 (2017)

  18. Berndt, B.C.: Ramanujan’s Notebooks. Part IV. Springer, New York (1991)

    Book  MATH  Google Scholar 

  19. Bhargava, S., Adiga, C.: A basic bilateral series summation formula and its applications. Integral Transforms Spec. Funct. 2, 165–184 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bressoud, D.: An easy proof of the Rogers-Ramanujan identities. J. Number Theory 16, 235 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bressoud, D., Subbarao, M.: On Uchimura’s connection between partitions and the number of divisors. Can. Math. Bull. 27, 143–145 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Corteel, S., Lovejoy, J.: Overpartitions. Trans. Am. Math. Soc. 356(4), 1623–1635 (2004)

  23. Chern, S.: On a conjecture of George Beck. Int. J. Number Theory 14(3), 647–651 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Corteel, S., Lovejoy, J.: Overpartitions. Trans. Am. Math. Soc. 356(4), 1623–1635 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dixit A., Yee A.J.: Generalized higher order spt-functions. Ramanujan J. 31(1–2) , 191–212 (2013) (Special issue in honor of Mourad Ismail and Dennis Stanton)

  26. Fine, N.J.: Basic Hypergeometric Series and Applications. Amer. Math. Soc, Mathematical Surveys and Monographs (1989)

    Google Scholar 

  27. Fokkink, R., Fokkink, W., Wang, Z.B.: A relation between partitions and the number of divisors. Am. Math. Mon. 102, 345–347 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. Garvan, F.G.: Weighted partition identities and divisor sums, Talk, AMS special Session on ‘Partition Theory and Related Topics’, Joint Mathematics Meeting, Atlanta, January 6 (2017)

  29. Garvan, F.G.: Weighted partition identities and divisor sums, Ch. 12. In: Nashed, M.Z., Li, X. (eds.) Frontiers in Orthogonal Polynomials and \(q\)-Series, Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes, vol. 1, pp. 239–249. World Scientific (2018)

  30. Gasper, G., Rahman, M.: Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, vol. 96. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  31. Guo, V.J.W., Zeng, J.: Basic and bibasic identities related to divisor functions. J. Math. Anal. Appl. 431, 1197–1209 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Guo, V.J.W., Zhang, C.: Some further q-series identities related to divisor functions. Ramanujan J. 25(3), 295–306 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ismail, M.E.H., Stanton, D.: Some combinatorial and analytical identities. Ann. Comb. 16(4), 755–771 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Ismail, M.E.H., Stanton, D.: Some combinatorial and analytical identities. Ann. Comb. 16(4), 755–771 (2012)

  35. Kluyver, J.C.: Vraagstuk XXXVII (Solution by S.C. van Veen), Wiskundige Opgaven, 92–93 (1919)

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

    Article  MathSciNet  MATH  Google Scholar 

  37. Patkowski, A.E.: Divisors, partitions and some new \(q\)-series identities. Colloq. Math. 117(2), 289–294 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ramanujan, S.: Notebooks of Srinivasa Ramanujan, vol. II, p. 393. Tata Institute of Fundamental Research, Bombay (1957)

    MATH  Google Scholar 

  39. Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi (1988)

    MATH  Google Scholar 

  40. Ramanujan, S.: Notebooks of Srinivasa Ramanujan, vol. II. Tata Institute of Fundamental Research, Mumbai (2012)

    MATH  Google Scholar 

  41. Uchimura, K.: An identity for the divisor generating function arising from sorting theory. J. Comb. Theory Ser. A 31, 131–135 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  42. van Hamme, L.: Problem \(6407\). Am. Math. Mon. 89(9), 703–704 (1982)

    MathSciNet  Google Scholar 

  43. Wang, L., Yee, A.J.: Some Hecke-Rogers type identities (preprint)

  44. Xu, A.: On a general \(q\)-identity. Electron. J. Comb. 21(2), 2.28 (2014)

  45. Xu, A., Cen, Z.: Combinatorial identities from contour integrals of rational functions. Ramanujan J. 40(1), 103–114 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  46. Yan, Q., Fu, J.: A new generalization of Dilcher’s formula. J. Nanjing Univ. Posts Telecommun. (Nat. Sci.) 32(1), 115–117 (2012)

  47. Zagier, D.: Vassiliev invariants and a strange identity related to the Dedekind eta-function. Topology 40, 945–960 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  48. Zudilin, W.: On the irrationality of generalized \(q\)-logarithm. Res. Number Theory 2(15), 1–10 (2016)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank Christian Krattenthaler who also informed them about the short proof of (2.1) which is the same as that of Andrews, Private communication, June 12, (2018). They also thank Bruce C. Berndt for his suggestions which improved the exposition of the paper. The authors sincerely thank the referee for the careful reading of the paper and for important suggestions which improved the exposition. The first author’s research is supported by the SERB-DST grant ECR/2015/000070, whereas the second author is a SERB National Post Doctoral Fellow (NPDF) supported by the fellowship PDF/2017/000370. Both sincerely thank SERB-DST for the support.

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  1. Discipline of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, Gujarat, 382355, India

    Atul Dixit & Bibekananda Maji

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  1. Atul Dixit
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Dixit, A., Maji, B. Partition implications of a three-parameter q-series identity. Ramanujan J 52, 323–358 (2020). https://doi.org/10.1007/s11139-019-00177-6

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