Explicit expressions for one family of the four parametric linear Euler sums

In this paper, we investigate one in the family of four parametric linear Euler sums as given by Alzer and Choi and later extended by Sofo and Choi. We give explicit expressions for a variety of special cases involving a free parameter. Our analysis incorporates the properties of the polylogarithm function, commonly referred to as the Fermi–Dirac integral. A reciprocity property is utilized to highlight an explicit representation of a particular skew harmonic linear Euler sum. A number of examples are also given which highlight the theorems. This work generalizes some results in the published literature.