Contiguous relations and their computations for 2F1 hypergeometric series

Adel K. Ibrahim, Medhat A. Rakha*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


The hypergeometric function 2F1 [a1, a2 ; a3 ; z] plays an important role in mathematical analysis and its application. Gauss defined two hypergeometric functions to be contiguous if they have the same power-series variable, if two of the parameters are pairwise equal, and if the third pair differs by ±1. He showed that a hypergeometric function and any two other contiguous to it are linearly related. In this paper, we present an interesting formula as a linear relation of three shifted Gauss polynomials in the three parameters a1, a2 and a3. More precisely, we obtained a recurrence relation including 2F1 [a1 + α1, a2 ; a3 ; z], 2F1 [a1, a2 + α2 ; a3 ; z] and 2F1 [a1, a2 ; a3 + α3 ; z] for any arbitrary integers α1, α2 and α3.

Original languageEnglish
Pages (from-to)1918-1926
Number of pages9
JournalComputers and Mathematics with Applications
Issue number8
Publication statusPublished - Oct 2008


  • Computer algebra
  • Contiguous function relation
  • F hypergeometric function
  • Gauss hypergeometric function
  • Linear recurrence relation

ASJC Scopus subject areas

  • Modelling and Simulation
  • Computational Theory and Mathematics
  • Computational Mathematics


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