Monotonicity of CF-coefficients in Gauss-fractions
A continued fraction expansion of the hypergeometric function \[ _2F_1(a,b;c;z)=\sum_{k=0}^\infty\frac{(a)_k(b)_k}{(c)_k}\,\frac{z^k}{k!} \] has the form \[ \frac{c_1}1\,{\quad\atop +}\,\frac{c_2z}1\,{\quad\atop +}\,\frac{c_3z}1\,{\quad\atop +}\,\frac{c_4z}1\,{\quad\atop +\dots} \] where \[ c_{2m+1}=\frac{(b+m)(c-a+m)}{(c+2m-1)(c+2m)},\quad c_{2m+2}=\frac{(a+m)(c-b+m)}{(c+2m)(c+2m+1)}. \] In this paper it is proved that for real parameters \(a\), \(b\), and \(c\), with \(a\), \(b+1\), \(c\), \(c-a+1\) and \(c-b\) not non-positive integers, the coefficients \(\{c_m\}\) has the following monotonicity properties: \begin{itemize}\item[1.]If \(a=b+\frac 12\) and \((2b-c+1)(2b-c)>0\), then \(\{c_m\}\) is increasing \item[2.]If \(a=b+\frac 12\) and \((2b-c+1)(2b-c)><0\), then \(\{c_m\}\) is decreasing \item[3.]If \(a<b+\frac 12\) and \((2b-c+1)(2b-c)(2a-c)(2a-c-1)\leq 0\), then \(\{c_{2m}\}\) is increasing and \(\{c_{2m+1}\}\) is decreasing \item[4.]If \(a>b+\frac 12\) and \((2b-c+1)(2b-c)(2a-c)(2a-c-1)\leq 0\), then \(\{c_{2m}\}\) is decreasing and \(\{c_{2m+1}\}\) is increasing \noindent These results are useful for efficient numerical computation of the function with reliable a priori truncation error bounds.\end{itemize}
scientific article; zbMATH DE number 2173206