Interpolation on Gauss hypergeometric functions with an application

TL;DR

This paper develops numerical approximation methods for the Gauss hypergeometric function over a parameter range, analyzes errors, and explores applications to continued fractions, advancing computational techniques for special functions.

Contribution

It introduces new numerical approximation formulas for the hypergeometric function, analyzes their errors, and applies these methods to continued fractions of Gauss, with insights into gamma function properties.

Findings

Derived approximation formulas for hypergeometric functions

Established error bounds and monotonicity properties

Applied methods to continued fractions of Gauss

Abstract

In this paper, 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 quotient 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.