Skip to main content
SpringerLink
Log in

Theta Functions

  • Chapter
  • First Online:
Ramanujan's Theta Functions

Abstract

This chapter contains a detailed study of Ramanujan’s theta functions

$$\phi (q) =\sum _{ j=-\infty }^{\infty }q^{j^{2} },\quad \mbox{ and}\quad \psi (q) =\sum _{ j=0}^{\infty }q^{j(j+1)/2},$$

and the Borweins’ theta functions

$$\displaystyle\begin{array}{rcl} a(q)& =& \sum _{j=-\infty }^{\infty }\sum _{ k=-\infty }^{\infty }q^{j^{2}+jk+k^{2} }, {}\\ b(q)& =& \sum _{j=-\infty }^{\infty }\sum _{ k=-\infty }^{\infty }\omega ^{j-k}q^{j^{2}+jk+k^{2} },\quad \omega =\exp (2\pi i/3), {}\\ \mbox{ and}\quad c(q)& =& \sum _{j=-\infty }^{\infty }\sum _{ k=-\infty }^{\infty }q^{(j+\frac{1} {3} )^{2}+(j+\frac{1} {3} )(k+\frac{1} {3} )+(k+\frac{1} {3} )^{2} }. {}\\ \end{array}$$

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. C. Adiga, S. Cooper, J.H. Han, A general relation between sums of squares and sums of triangular numbers. Int. J. Number Theory 1, 175–182 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. C. Adiga, K.R. Vasuki, On sums of triangular numbers. Math. Stud. 70, 185–190 (2001)

    MathSciNet  MATH  Google Scholar 

  3. A. Alaca, Ş. Alaca, M.F. Lemire, K.S. Williams, Jacobi’s identity and representations of integers by certain quaternary quadratic forms. Int. J. Modern Math. 2, 143–176 (2007)

    MathSciNet  MATH  Google Scholar 

  4. A. Alaca, Ş. Alaca, M.F. Lemire, K.S. Williams, The number of representations of a positive integer by certain quaternary quadratic forms. Int. J. Number Theory 5, 13–40 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Alaca, Ş. Alaca, K.S. Williams, On the two-dimensional theta functions of the Borweins. Acta Arith. 124, 177–195 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Alaca, Ş. Alaca, K.S. Williams, Liouville’s sextenary quadratic forms x 2 + y 2 + z 2 + t 2 + 2u 2 + 2v 2, x 2 + y 2 + 2z 2 + 2t 2 + 2u 2 + 2v 2 and x 2 + 2y 2 + 2z 2 + 2t 2 + 2u 2 + 4v 2. Far East J. Math. Sci. 30, 547–556 (2008)

    Google Scholar 

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

    MATH  Google Scholar 

  8. P. Barrucand, S. Cooper, M. Hirschhorn, Relations between squares and triangles. Discr. Math. 248, 245–247 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. N.D. Baruah, S. Cooper, M. Hirschhorn, Sums of squares and sums of triangular numbers induced by partitions of 8. Int. J. Number Theory 4, 525–538 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Berkovich, H. Yesilyurt, Ramanujan’s identities and representation of integers by certain binary and quaternary quadratic forms. Ramanujan J. 20, 375–408 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  12. B.C. Berndt, Ramanujan’s Notebooks, Part IV (Springer, New York, 1994)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  14. B.C. Berndt, Number Theory in the Spirit of Ramanujan (American Mathematical Society, Providence, RI, 2006)

    Book  MATH  Google Scholar 

  15. B.C. Berndt, S. Bhargava, F.G. Garvan, Ramanujan’s theories of elliptic functions to alternative bases. Trans. Am. Math. Soc. 347, 4163–4244 (1995)

    MathSciNet  MATH  Google Scholar 

  16. B.C. Berndt, R.A. Rankin, Ramanujan. Letters and Commentary (American Mathematical Society, Providence, RI, 1995)

    Google Scholar 

  17. S. Bhargava, Unification of the cubic analogues of the Jacobian theta-function. J. Math. Anal. Appl. 193, 543–558 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. J.M. Borwein, P.B. Borwein, A cubic counterpart of Jacobi’s identity and the AGM. Trans. Am. Math. Soc. 323, 691–701 (1991)

    MathSciNet  MATH  Google Scholar 

  19. J.M. Borwein, P.B. Borwein, F.G. Garvan, Some cubic modular identities of Ramanujan. Trans. Am. Math. Soc. 343, 35–47 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. L. Carlitz, Note on some partition formulae. Q. J. Math. Oxf. Ser. (2) 4, 168–172 (1953)

    Google Scholar 

  21. L. Carlitz, Note on sums of four and six squares. Proc. Am. Math. Soc. 8, 120–124 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  22. A.-L. Cauchy, Second Mémoire sur les fonctions dont plusieurs valeurs sont liées entre elles par une équation linéaire. C. R. Acad. Sci. Paris 17, 567 (1843). Reprinted in Oeuvres de Cauchy [1893], Series 1, vol. 8, 50–55

    Google Scholar 

  23. A.-L. Cauchy, Mémoire sur les fonctions complémentaires. C. R. Acad. Sci. Paris 19, 1377–1384 (1844). Reprinted in Oeuvres de Cauchy [1893], Series 1, vol. 8, 378–385

    Google Scholar 

  24. H.H. Chan, S. Cooper, Powers of theta functions. Pac. J. Math. 235, 1–14 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. H.H. Chan, Z.-G. Liu, Analogues of Jacobi’s inversion formula for the incomplete elliptic integral of the first kind. Adv. Math. 174, 69–88 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. R. Chapman, Cubic identities for theta series in three variables. Ramanujan J. 8, 459–465 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. S. Cooper, On sums of an even number of squares, and an even number of triangular numbers: an elementary approach based on Ramanujan’s1 ψ 1 summation formula, in q-series with Applications to Combinatorics, Number Theory, and Physics (Urbana, IL, 2000). Contemporary Mathematics, vol. 291 (American Mathematical Society, Providence, RI, 2001), pp. 115–137

    Google Scholar 

  28. S. Cooper, Cubic theta functions. J. Comput. Appl. Math. 160, 77–94 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. S. Cooper, Cubic elliptic functions. Ramanujan J. 11, 355–397 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. S. Cooper, Theta function identities for level 15, in Ramanujan Rediscovered, ed. by N.D. Baruah, B.C. Berndt, S. Cooper, T. Huber, M. Schlosser. Ramanujan Mathematical Society Lecture Notes Series, vol. 14 (Ramanujan Mathematical Society, Mysore, 2010), pp. 79–86

    Google Scholar 

  31. S. Cooper, M. Hirschhorn, A combinatorial proof of a result from number theory. Integers 4, A9, 1–4 (2004)

    MathSciNet  MATH  Google Scholar 

  32. S. Cooper, B. Kane, D. Ye, Analogues of the Ramanujan–Mordell theorem. J. Math. Anal. Appl. 446, 568–579 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. S. Cooper, H.Y. Lam, Sums of two, four, six and eight squares and triangular numbers: an elementary approach. Indian J. Math. 44, 21–40 (2002)

    MathSciNet  MATH  Google Scholar 

  34. D.A. Cox, The arithmetic-geometric mean of Gauss. L’Enseignement Mathématique 30, 275–330 (1984)

    MathSciNet  MATH  Google Scholar 

  35. L.E. Dickson, History of the Theory of Numbers, vol. III (Carnegie Institute of Washington, 1923). Republished by AMS Chelsea, Providence, RI, 1999

    Google Scholar 

  36. L.E. Dickson, Introduction to the Theory of Numbers (University of Chicago Press, Chicago, 1929). Republished by Dover, New York, 1957

    Google Scholar 

  37. G. Eisenstein, Neue Theoreme der höheren Arithmetik. J. Reine Angew. Math. 35, 117–136 (1847). Reprinted in Mathematische Werke, vol. 1 (Chelsea, New York, 1989), pp. 483–502

    Google Scholar 

  38. N. Fine, Basic Hypergeometric Series and Applications (American Mathematical Society, Providence, RI, 1988)

    Book  MATH  Google Scholar 

  39. F.G. Garvan, Cubic modular identities of Ramanujan, hypergeometric functions and analogues of the arithmetic-geometric mean iteration, in The Rademacher Legacy to Mathematics (University Park, PA, 1992). Contemporary Mathematics, vol. 166 (American Mathematical Society, Providence, RI, 1994), pp. 245–264

    Google Scholar 

  40. J.W.L. Glaisher, On the numbers of representations of a number as a sum of 2r squares, where 2r does not exceed eighteen. Proc. Lond. Math. Soc. (2) 5, 479–490 (1907)

    Google Scholar 

  41. J.W.L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten and twelve squares. Q. J. Pure Appl. Math. 38, 1–62 (1907)

    MATH  Google Scholar 

  42. G.H. Hardy, Ramanujan. Twelve Lectures on Subjects Suggested by his Life and Work, 3rd edn. (AMS, Providence, RI, 1999)

    Google Scholar 

  43. M.D. Hirschhorn, Three classical results on representations of a number. Sém. Lothar. Combinatoire 42, 8 pp. (1999). Article B42f

    Google Scholar 

  44. M.D. Hirschhorn, Partial fractions and four classical theorems of number theory. Am. Math. Mon. 107, 260–264 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  45. M. Hirschhorn, F. Garvan, J. Borwein, Cubic analogues of the Jacobian theta functions θ(z, q). Can. J. Math. 45, 673–694 (1993)

    Google Scholar 

  46. J.G. Huard, P. Kaplan, K.S. Williams, The Chowla-Selberg formula for genera. Acta Arith. 73, 271–301 (1995)

    MathSciNet  MATH  Google Scholar 

  47. C.G.J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum. Reprinted in Gesammelte Werke, vol. I (Chelsea, New York, 1969), pp. 49–239

    Google Scholar 

  48. E. Landau, Elementary Number Theory (AMS Chelsea, Providence, RI, 1999)

    MATH  Google Scholar 

  49. R.P. Lewis, Z.-G. Liu, The Borweins’ cubic theta functions and q-elliptic functions, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics, Gainesville, FL (1999). Developments in Mathematics, vol. 4 (Kluwer Academic Publishers, Dordrecht, 2001), pp. 133–145

    Google Scholar 

  50. J. Liouville, Sur la forme x 2 + y 2 + 3(z 2 + t 2). J. Math. Pures Appl. Ser. 2 5, 147–152 (1860)

    Google Scholar 

  51. J. Liouville, Remarque nouvelle sur la forme x 2 + y 2 + 3(z 2 + t 2). J. Math. Pures Appl. Ser. 2 8, 296 (1863)

    Google Scholar 

  52. J. Liouville, Sua la forme x 2 + xy + y 2 + 2z 2 + 2zt + 2t 2. J. Math. Pures Appl. Ser. 2 8, 308–310 (1863)

    Google Scholar 

  53. J. Liouville, Extrait d’une lettre adressée à M. Besge. J. Math. Pures Appl. Ser. 2 9, 296–298 (1864)

    Google Scholar 

  54. J. Liouville, Nombre des représentations d’un entier quelconque sous la forme d’une somme de dix carrés. J. Math. Pures Appl. Ser. 2 11, 1–8 (1866)

    Google Scholar 

  55. Z.-G. Liu, The Borweins’ cubic theta function identity and some cubic modular identities of Ramanujan. Ramanujan J. 4, 43–50 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  56. Z.-G. Liu, Some Eisenstein series associated with the Borwein functions, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics, Gainesville, FL (1999). Developments in Mathematics, vol. 4 (Kluwer Academic Publishers, Dordrecht, 2001), pp. 147–169

    Google Scholar 

  57. L. Lorenz, Bidrag til tallenes theori. Tidsskrift for Mathematik (3) 1, 97–114 (1871)

    Google Scholar 

  58. J. Lützen, Joseph Liouville 1809–1882. Master of Pure and Applied Mathematics (Springer, New York, 1990)

    Google Scholar 

  59. L.J. Mordell, On the representation of numbers as the sum of 2r squares. Q. J. Pure Appl. Math. Oxf. 48, 93–104 (1917)

    Google Scholar 

  60. P.S. Nazimoff, Applications of the Theory of Elliptic Functions to the Theory of Numbers, Moscow, 1884. Translated from Russian into English by A.E. Ross, Chicago, 1928

    MATH  Google Scholar 

  61. H. Petersson, Modulfunktionen und quadratische Formen. Ergebnisse der Mathematik und ihrer Grenzgebiete (Springer, Berlin, 1982)

    Google Scholar 

  62. K. Petr, O počtu tříd forem kvadratických záporného diskriminantu. Rozpravy Ceské Akademie Císare Frantiska Josefa I 10, 1–22 (1901). JFM 32.0212.02

    Google Scholar 

  63. S. Ramanujan, On certain arithmetical functions. Trans. Camb. Philos. Soc. 22, 159–184 (1916). Reprinted in [255, pp. 136–162]

    Google Scholar 

  64. S. Ramanujan, On certain trigonometrical sums. Trans. Camb. Philos. Soc. 22, 259–276 (1918). Reprinted in [255, pp. 179–199]

    Google Scholar 

  65. S. Ramanujan, Notebooks, 2 vols. (Tata Institute of Fundamental Research, Bombay, 1957)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  67. D. Schultz, Cubic theta functions. Adv. Math. 248, 618–697 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  68. D. Schultz, Identities involving theta functions and analogues of theta functions, PhD Thesis, University of Illinois, 2013

    Google Scholar 

  69. J.-P. Serre, A course in Arithmetic (Springer, New York, 1973)

    Book  MATH  Google Scholar 

  70. H.J.S. Smith, Report on the theory of numbers. Part VI. Report of the British Association for 1865, pp. 322–375. Reprinted in Collected Mathematical Papers, vol. 1, 1894. Reprinted by Chelsea, New York, 1979

    Google Scholar 

  71. Z.-H. Sun, On the number of representations of n by ax(x − 1)∕2 + by(y − 1)∕2. J. Number Theory 129, 971–989 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  72. Z.-H. Sun, K.S. Williams, On the number of representations of n by ax 2 + bxy + cy 2. Acta Arith. 122, 101–171 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  73. Z.-H. Sun, K.S. Williams, Ramanujan identities and Euler products for a type of Dirichlet series. Acta Arith. 122, 349–393 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  74. P.C. Toh, Representations of certain binary quadratic forms as Lambert series. Acta Arith. 143, 227–237 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  75. A. Walfisz, Zur additiven Zahlentheorie. VI. Trav. Inst. Math. Tbilissi [Trudy Tbiliss. Mat. Inst.] 5, 197–254 (1938)

    Google Scholar 

  76. A. Walfisz, Zur additiven Zahlentheorie. IX. Trav. Inst. Math. Tbilissi [Trudy Tbiliss. Mat. Inst.] 9, 75–96 (1941)

    Google Scholar 

  77. K.S. Williams, Number Theory in the Spirit of Liouville. London Mathematical Society Student Texts, vol. 76 (Cambridge University Press, Cambridge, 2011)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Natural and Mathematical Science, Massey University, Auckland, New Zealand

    Shaun Cooper

Authors
  1. Shaun Cooper
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Cooper, S. (2017). Theta Functions. In: Ramanujan's Theta Functions. Springer, Cham. https://doi.org/10.1007/978-3-319-56172-1_4

Download citation

Publish with us

Policies and ethics

Search

Navigation