Uniform convergent expansions of the Gauss hypergeometric function in terms of elementary functions. Issue 12 (2nd December 2018)

• Record Type: Journal Article
• Authors: Ferreira, Chelo; López, José L.; Pérez Sinusía, Ester
• Is Part Of: Integral transforms and special functions. Volume 29:Issue 12(2018)
• Page Start: 942
• Page End: 954
• Publication Date: 2018-12-02
• Subjects: Hypergeometric function -- convergent expansions -- uniform expansions; 33C05 -- 41A58 -- 41A80; Integral transforms -- Periodicals; Transcendental functions -- Periodicals; Transformations (Mathematics) -- Periodicals; Calculus, Integral -- Periodicals; 515
• DOI: 10.1080/10652469.2018.1525369
• Languages: English
• ISSNs: 1065-2469

Abstract

ABSTRACT: We consider the hypergeometric function 2 F 1 ( a, b ; c ; z ) forz ∈ C ∖ [ 1, ∞ ) . Forℜ a ≥ 0, we derive a convergent expansion of 2 F 1 ( a, b ; c ; z ) in terms of the function( 1 − z ) − a and of rational functions of z that is uniformly valid for z in any compact inC ∖ [ 1, ∞ ) . Whena ∈ N, the expansion also contains a logarithmic term of the formlog ⁡ ( 1 − z ) . Forℜ a ≤ 0, we derive a convergent expansion of( 1 − z ) a 2 F 1 ( a, b ; c ; z ) in terms of the function( 1 − z ) − a and of rational functions of z that is uniformly valid for z in any compact inC ∖ [ 1, ∞ ) in the exterior of the circle| z − 1 | = r for arbitrary r >0. The expansions are accompanied by realistic error bounds. Some numerical experiments show the accuracy of the approximation.