Means and divided differences
Let \(f\) be \(C^{2n}\) on the positive half line with nonzero \(2n\)th derivative and let \(L_ n\) be the limit of the \((2n- 1)\)st divided difference when \(n\) variables tend to \(a\) and \(n\) to \(b\). This paper examines which mean values of \(a\) and \(b\), substituted into \((1/(2n- 1)!) f^{(2n- 1)}\), give \(L_ n\). In much of the paper \(f(x)= x^ p\). Among the results offered are the following. The mean is arithmetic for \(p= 2n\), geometric for \(p= -1\), \({1\over 2}\)th power mean for \(p= (2n- 1)/2\) and for no \(p\) harmonic. The limit of \(L_ n\) as \(n\to \infty\) is the geometric mean for each \(x^ p\) but the arithmetic mean for \(e^ x\).
scientific article; zbMATH DE number 755534