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Two-dimensional series evaluations via the elliptic functions of Ramanujan and Jacobi

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Abstract

We evaluate in closed form, for the first time, certain classes of double series, which are remindful of lattice sums. Elliptic functions, singular moduli, class invariants, and the Rogers–Ramanujan continued fraction play central roles in our evaluations.

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References

  1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Dover, New York (1965)

    Google Scholar 

  2. Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook, Part I. Springer, New York (2005)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  4. Berndt, B.C.: Ramanujan’s Notebooks, Part V. Springer, New York (1998)

    Book  MATH  Google Scholar 

  5. Borwein, D., Borwein, J.M., Taylor, K.F.: Convergence of lattice sums and Madelung’s constant. J. Math. Phys. 26(11), 2999–3009 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cox, D.: Primes of the Form x 2+ny 2. Wiley, New York (1989)

    Google Scholar 

  7. Crandall, R.E.: New representations for the Madelung constant. Exp. Math. 8(4), 367–379 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Forrester, P.J., Glasser, M.L.: Some new lattice sums including an exact result for the electrostatic potential within the NaCl lattice. J. Phys. A: Math. Gen. 15, 911–914 (1982)

    Article  MathSciNet  Google Scholar 

  9. Gradshteyn, I.S., Ryzhik, I.M. (eds.): Table of Integrals, Series, and Products, 5th edn. Academic Press, San Diego (1994)

    MATH  Google Scholar 

  10. Rogers, M.: Hypergeometric formulas for lattice sums and Mahler measures. Int. Math. Res. Not. 17, 4027–4058 (2011)

    Google Scholar 

  11. Rogers, M., Zudilin, W.: From L-series of elliptic curves to Mahler measures. Compos. Math., to appear. arXiv:1012.3036 [math.NT] (2010)

  12. Rogers, M., Zudilin, W.: On the Mahler measure of 1+X+X −1+Y+Y −1. Preprint arXiv:1102.1153 [math.NT] (2011)

  13. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1966)

    Google Scholar 

  14. Zucker, I.J.: Madelung constants and lattice sums for hexagonal crystals. J. Phys. A 24(4), 873–879 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zucker, I.J., McPhedran, R.: Problem 11294. Am. Math. Mon. 114, 452 (2007) (With unpublished solutions by the proposers, R. Chapman, and A. Stadler)

    Google Scholar 

  16. Zucker, I.J., McPhedran, R.: Dirichlet L-series with real and complex characters and their application to solving double sums. Proc. R. Soc. A 464(2094), 1405–1422 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, IL, 61801, USA

    Bruce C. Berndt

  2. 2942 Ave. del Conquistador, Tucson, AZ, 85749-9304, USA

    George Lamb

  3. Département de Mathématiques et Statistique, Université de Montréal, CP6128, Succ. Cenrte-Ville, Montréal, QC, H3T 1J4, Canada

    Mathew Rogers

Authors
  1. Bruce C. Berndt
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  2. George Lamb
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  3. Mathew Rogers
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Corresponding author

Correspondence to Bruce C. Berndt.

Additional information

The research of Bruce C. Berndt was supported by National Security Agency grant H98230-11-1-0200.

The research of Mathew Rogers was supported by National Science Foundation grant DMS-0803107.

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Berndt, B.C., Lamb, G. & Rogers, M. Two-dimensional series evaluations via the elliptic functions of Ramanujan and Jacobi. Ramanujan J 29, 185–198 (2012). https://doi.org/10.1007/s11139-011-9351-9

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  • DOI: https://doi.org/10.1007/s11139-011-9351-9

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