On interpolation series related to the Abel-Goncharov problem, with applications to arithmetic-geometric mean relationship and Hellinger integrals

In this paper a convergence class is characterized for special series associated with Gelfond's interpolation problem (a generalization of the Abel-Goncharov problem) when the interpolation nodes are equidistantly distributed within the interval $[0,1]$. As a result, an expansion is derived of the arithmetic-geometric mean difference in terms of certain central moments. Another result concerns an expansion of the Hellinger integral.