Asymptotics and inequalities for a class of infinite sums

The goal of this chapter is to establish the asymptotic behavior for large, positive x of S(a, ß). The chapter presents the transform method for R_a asymptotics and R_a asymptotics using an identity. The chapter shows that as a byproduct, a generalization of the Alzer-Brenner inequality, namely, (ß – 1)S²(1, ß) = S(1, 2ß – 1) is obtained. Finally, it presents a general criterion for inequalities of this type.