Let \({{\,\mathrm{\mathcal {K}}\,}}(r)\) be the complete elliptic integral of the first kind defined on (0, 1). This paper deals with the power series of \(x\mapsto (1-x)^pF(a,b;a+b;x)\) on (0, 1), where \(F(a,b;a+b;x)\) denotes the zero-balanced hypergeometric function. This result extends the recently obtained absolutely monotonic property of \((1-x)^p{{\,\mathrm{\mathcal {K}}\,}}(\sqrt{x})\) . As a consequence, a rational approximation of zero-balanced hypergeometric function will be established, which is an arbitrary precise approximation near \(x=0\) .
