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

摘要

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 ν_q on the unit interval with moments 1/Σ_(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 νq 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.