Interpolation on Gauss hypergeometric functions with an application. (English) Zbl 06864400

Arora, Hina Manoj; Sahoo, Swadesh Kumar

Involve 11, No. 4, 625-641 (2018).

Summary: We use some standard numerical techniques to approximate the hypergeometric function \[ {}_2F_1[a,b;c;x]=1+\frac{ab}{c}x+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{x^2}{2!}+\cdots \] for a range of parameter triples \((a,b,c)\) on the interval \(0<x<1\). Some of the familiar hypergeometric functional identities and asymptotic behavior of the hypergeometric function at \(x=1\) play crucial roles in deriving the formula for such approximations. We also focus on error analysis of the numerical approximations leading to monotone properties of quotients of gamma functions in parameter triples \((a,b,c)\). Finally, an application to continued fractions of Gauss is discussed followed by concluding remarks consisting of recent works on related problems.

MSC:

65D05	Numerical interpolation
33B15	Gamma, beta and polygamma functions
33B20	Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
33C05	Classical hypergeometric functions, \({}_2F_1\)
33F05	Numerical approximation and evaluation of special functions

Keywords:

interpolation; hypergeometric function; gamma function; error estimate

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