New properties of the Lerch's transcendent

A new representation of the Lerch's transcendent Φ(z,s,a)Φ(z,s,a), valid for positive integer s=n=1,2,…s=n=1,2,… and for z and a   belonging to certain regions of the complex plane, is presented. It allows to write an equation relating Φ(z,n,a)Φ(z,n,a) and Φ(1/z,n,1−a)Φ(1/z,n,1−a), which in turn provides an expansion of Φ(z,n,a)Φ(z,n,a) as a power series of 1/z1/z, convergent for |z|>1|z|>1.