Canonical products of small order and related Pick functions

We show that various functions related to the logarithms of the canonical products Pρ(z)=∏n=1∞(1+z/nρ), ρ>1 and Q(z)=∏n=0∞(1+zqn), q∈(0,1) are Pick functions. As a consequence we find an integral expansion of a function involving the logarithm of Jacksons q-gamma function.