On an Iteration Leading to a q-Analogue of the Digamma Function

Abstract

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