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Rogers–Ramanujan type identities via Abel’s lemma on summation by parts

  • Wenchang ChuEmail author
Article
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Abstract

The Abel’s lemma on summation by parts is employed to review identities of Rogers–Ramanujan type. Twenty examples are illustrated including several new RR identities.

Keywords

Rogers–Ramanujan identities Abel’s lemma on summation by parts Basic hypergeometric series 

Mathematics Subject Classification

Primary 33D15 Secondary 05A30 

Notes

Acknowledgements

The author expresses his sincere gratitude to three anonymous referees for their careful reading, critical comments and valuable suggestions, that have improved the manuscript during the revision.

References

  1. 1.
    Agarwal, A.K., Bressoud, D.M.: Lattice paths and multiple basic hypergeometric series. Pac. J. Math. 136, 209–228 (1989)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andrews, G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1976)zbMATHGoogle Scholar
  3. 3.
    Andrews, G.E.: Combinatorics and Ramanujans “Lost” Notebook “Sureys in Combinatorics”. Lecture Note Series, vol. 103. London Mathematical Society, pp. 1–23 (1985)Google Scholar
  4. 4.
    Andrews, G.E.: q-series: Their Development and Application in Analysis Number Theory, Combinatorics, Physics, and Computer Algebra, CBMS Regional Conference Series in Mathematics, vol. 66. American Mathematical Society, Providence, RI, (1986)Google Scholar
  5. 5.
    Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook (Part I). Springer, New York (2005)zbMATHGoogle Scholar
  6. 6.
    Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook, vol. Part II. Springer, New York (2005)zbMATHGoogle Scholar
  7. 7.
    Bailey, W.N.: Some identities in combinatory analysis. Proc. Lond. Math. Soc. (2) 49, 421–435 (1947)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bailey, W.N.: Identities of the Rogers-Ramanujan type. Proc. Lond. Math. Soc. (2) 50, 1–10 (1948)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London and New York (1982)zbMATHGoogle Scholar
  10. 10.
    Chu, W.: Gould-Hsu-Carlitz inversions and Rogers-Ramanujan identities. Acta Math. Sin. 33(1), 7–12 (1990). MR 91d:05010 & Zbl 727:11037MathSciNetzbMATHGoogle Scholar
  11. 11.
    Chu, W.: Abel’s Lemma on summation by parts and Ramanujan’s \(_1\psi _1\)-series identity. Aequationes Math. 72(1/2), 172–176 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chu, W.: Bailey’s very well-poised \({_6\psi _6}\)-series identity. J. Comb. Theory Ser. A 113(6), 966–979 (2006)CrossRefGoogle Scholar
  13. 13.
    Chu, W.: Abel’s lemma on summation by parts and basic hypergeometric series. Adv. Appl. Math. 39(4), 490–514 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chu, W., Chen, X.J.: Carlitz inversions and identities of Rogers-Ramanujan type. Rocky Mountain J. Math. 44(4), 1125–1143 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chu, W., Wang, C.Y.: The Multisection method for triple products and identities of Rogers-Ramanujan type. J. Math. Anal. Appl. 339(2), 774–784 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Chu, W., Wang, C.Y.: Identities of Rogers-Ramanujan type via \(\pm 1\). J. Comput. Anal. Appl. 12(1A), 83–91 (2010)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Chu, W., Wang, X.Y.: Telescopic creation for identities of Rogers-Ramanujan type. J. Differ. Equ. Appl. 18(2), 167–183 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Chu, W., Zhang, W.L.: Abel’s method on summation by parts and nonterminating \(q\)-series identities. Collect. Math. 60(2), 193–211 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Chu, W., Zhang, W.L.: Bilateral Bailey lemma and Rogers-Ramanujan identities. Adv. Appl. Math. 42(3), 358–391 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Chu, W., Zhang, W.L.: The \(q\)-binomial theorem and Rogers-Ramanujan identities. Utilitas Math. 84, 173–188 (2011)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Chu, W., Zhang, W.L.: Four classes of Rogers-Ramanujan identities with quintuple products. Hiroshima Math. J. 41(1), 27–40 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gessel, I., Stanton, D.: Applications of \(q\)-Lagrange inversion to basic hypergeometric series. Trans. Am. Math. Soc. 277(1), 173–201 (1983)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Jackson, F.H.: Examples of a generalization of Euler’s transformation for power series. Messenger Math. 57, 169–187 (1928)Google Scholar
  24. 24.
    Lepowsky, J., Milne, S.C.: Lie algebraic approaches to classical partition identities. Adv. Math. 29(1), 15–59 (1978)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mohammed, M.: The \(q\)-Markov-WZ method. Ann. Comb. 9(2), 205–221 (2005)MathSciNetCrossRefGoogle Scholar
  26. 26.
    McLaughlin, J., Sills, A.V.: Ramanujan-Slater type identities related to moduli 18 and 24. J. Math. Anal. Appl. 344, 765–777 (2008)MathSciNetCrossRefGoogle Scholar
  27. 27.
    McLaughlin, J., Sills, A.V., Zimmer, P.: Rogers–Ramanujan–Slater type identities. J. Electron. Comb. 15, #DS15 (2008)Google Scholar
  28. 28.
    McLaughlin, J., Sills, A.V., Zimmer, P.: Rogers-Ramanujan computer searches. J. Symb. Comput. 44, 1068–1078 (2009)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Ramanujan, S.: Proof of certain identities in combinatory analysis. Proc. Cambridge Philos. Soc. 19, 214–216 (1919)zbMATHGoogle Scholar
  30. 30.
    Rogers, L.J.: Second memoir on the expansion of certain infinite products. Proc. Lond. Math. Soc. 25, 318–343 (1894)MathSciNetGoogle Scholar
  31. 31.
    Rogers, L.J.: On two theorems of combinatory analysis and some allied identities. Proc. Lond. Math. Soc. 16, 315–336 (1917)CrossRefGoogle Scholar
  32. 32.
    Sills, A.V.: Finite Rogers–Ramanujan type identities. Electron. J. Comb. 10 (2003), Research Paper#13Google Scholar
  33. 33.
    Slater, L.J.: Further identities of the Rogers-Ramanujan type. Proc. Lond. Math. Soc. (Ser. 2) 54, 147–167 (1952)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhoukou Normal UniversityZhoukouPeople’s Republic of China
  2. 2.Dipartimento di Matematica e Fisica “Ennio De Giorgi”Università del SalentoLecceItaly

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