On the irrationality of generalized q-logarithm

For integer p, |p|>1, and generic rational x and z, we establish the irrationality of the series [formula could not be replicated].It is a symmetric (ℓ_p(x,z)=ℓ_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 ℓ_p(x,z).