Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Introduction Read chapter Read first chapter

2012 | OriginalPaper | Chapter

9. Circular Summation

Authors : George E. Andrews, Bruce C. Berndt

Published in: Ramanujan's Lost Notebook

Publisher: Springer New York

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

On page 54 in his lost notebook, Ramanujan derives identities for the sum of the nth powers of n general theta functions. He states a beautiful general theorem, and then provides five particular examples.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Ramanujan’s Forty Identities for the Rogers–Ramanujan Functions
next chapter Highly Composite Numbers
Literature
[4]
S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan’s theta function, Proc. Amer. Math. Soc. 128 (2000), 1333–1338. MathSciNetMATHCrossRef
[15]
G.E. Andrews and B.C. Berndt, Ramanujan’s Lost Notebook, Part I, Springer, New York, 2005.
[16]
G.E. Andrews and B.C. Berndt, Ramanujan’s Lost Notebook, Part II, Springer, New York, 2009. MATH
[52]
A. Berkovich and H. Yesilyurt, Ramanujan’s identities and representation of integers by certain binary and quaternary quadratic forms, Ramanujan J. 20 (2009), 375–408. MathSciNetMATHCrossRef
[53]
A. Berkovich, F.G. Garvan, and H. Yesilyurt, manuscript in preparation.
[55]
[56]
B.C. Berndt, On a certain theta-function in a letter of Ramanujan from Fitzroy House, Ganita 43 (1992), 33–43. MathSciNetMATH
[57]
[58]
[79]
J.M. Borwein and P.B. Borwein, A cubic counterpart of Jacobi’s identity and the AGM, Trans. Amer. Math. Soc. 323 (1991), 691–701. MathSciNetMATH
[101]
H.H. Chan, Z.-G. Liu, and S.T. Ng, Circular summation of theta functions in Ramanujan’s lost notebook, J. Math. Anal. Appl. 316 (2006), 628–641. MathSciNetMATHCrossRef
[108]
[115]
[116]
K.S. Chua, The root lattice \(A_{n}^{*}\) and Ramanujan’s circular summation of theta functions, Proc. Amer. Math. Soc. 130 (2002), 1–8. MathSciNetMATHCrossRef
[119]
T. Dai and X. Ma, A bivariate circular summation formula for cubic theta functions and its implications, J. Ramanujan Math. Soc. 26 (2011), 237–259. MathSciNetMATH
[234]
T. Murayama, On an identity of theta functions obtained from weight enumerators of linear codes, Proc. Japan Acad., Ser. A 74 (1998), 98–100. MathSciNetMATHCrossRef
[257]
K. Ono, On the circular summation of the eleventh powers of Ramanujan’s theta functions, J. Number Thy. 76 (1999), 62–65. MATHCrossRef
[282]
S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957. MATH
[283]
S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988. MATH
[286]
[319]
[320]
[322]
S.H. Son, Ramanujan’s symmetric theta functions in his lost notebook, in Special Functions and Orthogonal Polynomials, D. Dominici and R.S. Maier, eds., Contemp. Math. 471, American Mathematical Society, Providence, RI, 2008, pp. 187–202. CrossRef
[339]
E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1966.
[345]
P. Xu, An elementary proof of Ramanujan’s circular summation formula and its generalizations, Ramanujan J. 27 (2012), 409–417. MathSciNetMATHCrossRef
[350]
[352]
J.-M. Zhu, An alternate circular summation formula of theta functions and its applications, Applic. Anal. Discrete Math. 6 (2012), 114–125. CrossRef
Metadata
Title
Circular Summation
Authors
George E. Andrews
Bruce C. Berndt
Copyright Year
2012
Publisher
Springer New York
Book
Ramanujan's Lost Notebook

Print ISBN: 978-1-4614-3809-0

Electronic ISBN: 978-1-4614-3810-6

Copyright Year: 2012

DOI
https://doi.org/10.1007/978-1-4614-3810-6_9

Premium Partner

    Image Credits