We show that the q-Digamma function ψq for 0 < q < 1 appears in an
iteration studied by Berg and Durán. This is connected with the determination of the probability measure νq on the unit interval with moments 1/n+1 k=1(1 − q)/(1 −
qk), which are q-analogues of the reciprocals of the harmonic numbers. The Mellin transform of the measure νq can be expressed in terms of the q-Digamma function.
It is shown that νq has a continuous density on ]0, 1], which is piecewise C∞ with kinks at the powers of q. Furthermore, (1 − q)e−xνq (e−x ) is a standard p-function from the theory of regenerative phenomen.