Skip to main content

On continued fractions associated with polynomial type pade approximants, with an application

  • 363 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1406)

Abstract

The diagonal Padé approximants of power series that satisfy a certain reciprocal property involve, essentially, one polynomial. The continued fractions, whose convergents are the sequence of diagonal approximants, are consequentially of a simplified form. An interesting example is a continued fraction given by Stieltjes, and this is seen to have an application in current approximation theory.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   29.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   39.99
Price excludes VAT (Canada)
  • Compact, lightweight 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bowman, K O and Shenton, L R, ‘Asymptotic series and Stieltjes continued fractions for a gamma function ratio’, J. Comp. App. Maths., V.4, No.2, 1978, pp 105–111.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Burgoyne, F D, ‘Practical LP polynomial approximation’, Maths. of Comp., V.21, 1967, pp 113–115.

    MathSciNet  MATH  Google Scholar 

  3. Frenzen, C L, Private Communication.

    Google Scholar 

  4. Frenzen, C L, ‘Error bounds for asymptotic expansions of the ratio of two gamma functions’ Preprint of paper submitted to SIAM J. Math. Anal.

    Google Scholar 

  5. Magnus, A, Private Communication.

    Google Scholar 

  6. McCabe, J H, ‘On an asymptotic series and corresponding continued fraction for a gamma function ratio’, J. Comp. App. Maths., V.9, 1983, pp 125–130.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Németh, G and Zimani, M, ‘Polynomial type Padé approximants’, Maths. of Comp. V.38, No.158, 1982, pp 553–565.

    CrossRef  MATH  Google Scholar 

  8. Perron, O, ‘Die Lehre von den Kettenbruchen’, Chelsea, New York 1929.

    Google Scholar 

  9. Phillips, G M, ‘Error estimates for best polynomial approximations’, contributed to Approximation Theory’ edited by A Talbot, Academic Press 1970.

    Google Scholar 

  10. Phillips, G M and Sahney, B N, ‘An error estimate for least squares approximation’, B.I.T., V.15, 1975, pp 426–430.

    MATH  Google Scholar 

  11. Stieltjes, T J, ‘Correspondence d’Hermite et de Stieltjes’, Tomes 1 and 2, Gauthier-Villars, 1905.

    Google Scholar 

  12. Timan, A F, ‘Theory of approximation of functions of a real variable’, Pergamon Press, 1963 (Translated from Russian by J Berry).

    Google Scholar 

  13. Tricomi, F C and Erdelyi, A, ‘The asymptotic expansion of a ratio of gamma functions’, Pacific J. Math., V.1, 1951, pp 133–142.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Wrench, J W, ‘Concerning two series for the gamma function’, Maths. of Comp., V.22, 1968, pp 617–626.

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Additional information

DEDICATED TO WOLF THRON ON THE OCCASION OF HIS 70TH BIRTHDAY

Rights and permissions

Reprints and Permissions

Copyright information

© 1989 Springer-Verlag

About this paper

Cite this paper

McCabe, J.H. (1989). On continued fractions associated with polynomial type pade approximants, with an application. In: Jacobsen, L. (eds) Analytic Theory of Continued Fractions III. Lecture Notes in Mathematics, vol 1406. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096166

Download citation

  • DOI: https://doi.org/10.1007/BFb0096166

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51830-3

  • Online ISBN: 978-3-540-46820-2

  • eBook Packages: Springer Book Archive