Skip to main content
SpringerLink
Log in

q-Extensions of Gauss’ Fifteen Contiguous Relations for 2 F 1-Series

  • Conference paper
Information Computing and Applications (ICICA 2010)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 105))

Included in the following conference series:

  • 1542 Accesses

Abstract

Contiguous relation is a fundamental concept within the theories of hypergeometric series and orthogonal polynomials. Gauss’ fifteen contiguous relations imply that any three 2 F 1-series whose corresponding parameters differ by integers are linearly related. As q-extensions of hypergeometric series, basic hypergeometric series are used widely in Statistics and Physics. By the method of comparing coefficients, we establish fifteen interesting three-term relations for 2 φ 1-series. Their limiting cases recover Gauss’ fifteen contiguous relations for 2 F 1-series.

Project supported by National Nature Science Foundation of China (No.60533060), Educational Commission of Hebei Province of China (No.2009448) and Natural Science Foundation of Hebei Province of China (No.A2010000908).

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  2. Bailey, W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935)

    MATH  Google Scholar 

  3. Chu, W., Wei, C.: Legedre inversions and balanced hypergeometric series identities. Discrete Math. 308, 541–549 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chu, W., Wei, C.: Set partitions with restrictions. Discrete Math. 308, 3163–3168 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gupta, D.P., Masson, D.R.: Use of the Gauss Contiguous Relations in Computing the Hypergeometric Functions F(n + 1/2,n + 1/2;m;z ). Interdisciplinary Information Sciences 2, 63–74 (1996)

    Article  MathSciNet  Google Scholar 

  6. Ibrahim, A.K., Rakha, M.A.: Contiguous relations and their computations for 2 F 1 hypergeometric series. Computers & Mathematics with Applications 56, 1918–1926 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Rakha, M.A., Ibrahim, A.K.: On the contiguous relations of hypergeometric series. J. Comput. Appl. Math. 192, 396–410 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Rakha, M.A., Ibrahim, A.K., Rathie, A.K.: On the computations of contiguous relations for 2 F 1hypergeometric series. Commun. Korean Math. Soc. 24, 291–302 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Slater, L.J.: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966)

    MATH  Google Scholar 

  10. Takayama, N.: Gröbner Basis and the Problem of Contiguous Relations, Japan. J. Appl. Math. 6, 147–160 (1989)

    MathSciNet  Google Scholar 

  11. Vidunas, R.: Contiguous relations of hypergeometric series. J. Comput. Appl. Math. 153, 507–519 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Wei, C., Gu, Q.: q-Generalizations of a faimily of harmonic number identities Adv. Appl. Math. 45, 24–27 (2010)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Information Technology, Hainan Medical College, Haikou, 571101, China

    Chuanan Wei

  2. College of Sciences, Hebei Polytechnic University, Tangshan, 063009, China

    Dianxuan Gong

Authors
  1. Chuanan Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dianxuan Gong
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. College of Computer Science, South-Central University for Nationalities, 708 Minyuan Road, 430074, Wuhan, China

    Rongbo Zhu

  2. Victoria University, 8001, Melbourne, VIC, Australia

    Yanchun Zhang

  3. College of sciences, He’Bei polytechnic University, 063000, Tangshan, Hebei, China

    Baoxiang Liu  & Chunfeng Liu  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Wei, C., Gong, D. (2010). q-Extensions of Gauss’ Fifteen Contiguous Relations for 2 F 1-Series. In: Zhu, R., Zhang, Y., Liu, B., Liu, C. (eds) Information Computing and Applications. ICICA 2010. Communications in Computer and Information Science, vol 105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16336-4_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-16336-4_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-16335-7

  • Online ISBN: 978-3-642-16336-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation