Interpolation on Gauss hypergeometric functions with an application

Abstract

We use some standard numerical techniques to approximate the hypergeometric function

$${}_2{F}_1\left[a,b;c;x\right]=1+\frac{ab}{c}x+\frac{a\left(a+1\right)b\left(b+1\right)}{c\left(c+1\right)}\frac{x^2}{2!}+\cdots$$

for a range of parameter triples $\left(a,b,c\right)$ 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 $\left(a,b,c\right)$. Finally, an application to continued fractions of Gauss is discussed followed by concluding remarks consisting of recent works on related problems.