Canonical products of small order and related Pick functions

Abstract

We show that various functions related to the logarithms of the canonical products P ρ ( z ) = ∏ n = 1 ∞ ( 1 + z / n ρ ) , ρ > 1 and Q ( z ) = ∏ n = 0 ∞ ( 1 + zq n ) , 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.