In this paper we consider properties and power expressions of the functions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$f:(-1,1)\rightarrow \mathbb{R }$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$f_L:(-1,1)\rightarrow \mathbb{R }$$</EquationSource> </InlineEquation>, defined by <Equation ID="Equa1"> <EquationSource Format="TEX">$$\begin{aligned} f(x;\gamma )=\frac{1}{\pi }\int \limits _{-1}^1 \frac{(1+xt)^\gamma }{\sqrt{1-t^2}}\,\mathrm{d}t \quad...</equationsource></equation></equationsource></inlineequation></equationsource></inlineequation>

In this work, simple exact results are presented for summations in two-particle potential with long-range interactions. Polygamma function is used to evaluate summations. Results are found when a periodic media is considered. Periodic boundary conditions are applied by symmetric repetitions of a...