For integer p, |p|>1, and generic rational x and z, we establish the irrationality of the series p (x, z) = x ∞ n=1 z n p n − x. It is a symmetric (p (x, z) = p (z, x)) generalization of the q-logarithmic function (x = 1 and p = 1/q where |q| < 1), which in turn generalizes the q-harmonic series (x = z = 1). Our proof makes use of the Hankel determinants built on the Padé approximations to p (x, z).