# Asymptotics of the Gauss hypergeometric function with large parameters, I

Richard B. Paris

Research output: Contribution to journalArticle

### Abstract

We obtain asymptotic expansions for the Gauss hypergeometric function F(a+ε1λ,b+ε2λ;c+ε3λ;z) as |λ| →∞ when the εj are ﬁnite by an application of the method of steepest descents, thereby extending previous results corresponding to εj = 0, ±1. By means of connection formulas satisﬁed by F it is possible to arrange the above hypergeometric function into three basic groups. In Part I we consider the cases (i) ε1 > 0, ε2 = 0, ε3 = 1 and (ii) ε1 > 0, ε2 = −1, ε3 = 0; the third case ε1, ε2 >0, ε3 =1 is deferred to Part II. The resulting expansions are of Poincar´e type and hold in restricted domains of the complex z-plane. Numerical results illustrating the accuracy of the different expansions are given.
Original language English 183-203 21 Journal of Classical Analysis 2 2 https://doi.org/dx.doi.org/10.7153/jca-02-15 Published - 2013

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Gauss Hypergeometric Function
Steepest Descent
Hypergeometric Functions
Poincaré
Asymptotic Expansion
Numerical Results

### Cite this

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title = "Asymptotics of the Gauss hypergeometric function with large parameters, I",
abstract = "We obtain asymptotic expansions for the Gauss hypergeometric function F(a+ε1λ,b+ε2λ;c+ε3λ;z) as |λ| →∞ when the εj are ﬁnite by an application of the method of steepest descents, thereby extending previous results corresponding to εj = 0, ±1. By means of connection formulas satisﬁed by F it is possible to arrange the above hypergeometric function into three basic groups. In Part I we consider the cases (i) ε1 > 0, ε2 = 0, ε3 = 1 and (ii) ε1 > 0, ε2 = −1, ε3 = 0; the third case ε1, ε2 >0, ε3 =1 is deferred to Part II. The resulting expansions are of Poincar´e type and hold in restricted domains of the complex z-plane. Numerical results illustrating the accuracy of the different expansions are given.",
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Asymptotics of the Gauss hypergeometric function with large parameters, I. / Paris, Richard B.

In: Journal of Classical Analysis, Vol. 2, No. 2, 2013, p. 183-203.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Asymptotics of the Gauss hypergeometric function with large parameters, I

AU - Paris, Richard B.

PY - 2013

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N2 - We obtain asymptotic expansions for the Gauss hypergeometric function F(a+ε1λ,b+ε2λ;c+ε3λ;z) as |λ| →∞ when the εj are ﬁnite by an application of the method of steepest descents, thereby extending previous results corresponding to εj = 0, ±1. By means of connection formulas satisﬁed by F it is possible to arrange the above hypergeometric function into three basic groups. In Part I we consider the cases (i) ε1 > 0, ε2 = 0, ε3 = 1 and (ii) ε1 > 0, ε2 = −1, ε3 = 0; the third case ε1, ε2 >0, ε3 =1 is deferred to Part II. The resulting expansions are of Poincar´e type and hold in restricted domains of the complex z-plane. Numerical results illustrating the accuracy of the different expansions are given.

AB - We obtain asymptotic expansions for the Gauss hypergeometric function F(a+ε1λ,b+ε2λ;c+ε3λ;z) as |λ| →∞ when the εj are ﬁnite by an application of the method of steepest descents, thereby extending previous results corresponding to εj = 0, ±1. By means of connection formulas satisﬁed by F it is possible to arrange the above hypergeometric function into three basic groups. In Part I we consider the cases (i) ε1 > 0, ε2 = 0, ε3 = 1 and (ii) ε1 > 0, ε2 = −1, ε3 = 0; the third case ε1, ε2 >0, ε3 =1 is deferred to Part II. The resulting expansions are of Poincar´e type and hold in restricted domains of the complex z-plane. Numerical results illustrating the accuracy of the different expansions are given.

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