Constructive Approximation

, Volume 9, Issue 4, pp 509–523 | Cite as

Hypergeometric analogues of the arithmetic-geometric mean iteration

  • J. Borwein
  • P. Borwein
  • F. Garvan
Article

Abstract

The arithmetic-geometric mean iteration of Gauss and Legendre is the two-term iterationan+1=(an+bn)/2 and\(b_{n + 1} = \sqrt {a_n b_n } \) witha0≔1 andb0≔x. The common limit is2F1(1/2, 1/2; 1; 1−x2)−1 and the convergence is quadratic.

This is a rare object with very few close relatives. There are however three other hypergeometric functions for which we expect similar iterations to exist, namely:2F1(1/2−s1, 1/2+s; 1; ·) withs=1/3, 1/4, 1/6.

Our intention is to exhibit explicitly these iterations and some of their generalizations. These iterations exist because of underlying quadratic or cubic transformations of certain hypergeometric functions, and thus the problem may be approached via searching for invariances of the corresponding second-order differential equations. It may also be approached by searching for various quadratic and cubic modular equations for the modular forms that arise on inverting the ratios of the solutions of these differential equations. In either case, the problem is intrinsically computational. Indeed, the discovery of the identities and their proofs can be effected almost entirely computationally with the aid of a symbolic manipulation package, and we intend to emphasize this computational approach.

AMS classification

Primary 33A25 Secondary 33A30 11F10 39B10 

Key words and phrases

Schwartz functions Hypergeometric functions Arithmetic geometric mean iteration 

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Copyright information

© Springer-Verlag New York Inc 1993

Authors and Affiliations

  • J. Borwein
    • 1
  • P. Borwein
    • 1
  • F. Garvan
    • 1
  1. 1.Department of Mathematics, Statistics and Computing ScienceDalhousie UniversityHalifaxCanada

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