A new non-negative distribution with both finite and infinite support

The Tukey-$\lambda$ distribution has interesting properties including (i) for some parameters values it has finite support, and for others infinite support, and (ii) it can mimic several other distributions such that parameter estimation for the Tukey distribution is a method for identifying an appropriate class of distribution to model a set of data. The Tukey-$\lambda$ is, however, symmetric. Here we define a new class of {\em non-negative} distribution with similar properties to the Tukey-$\lambda$ distribution. As with the Tukey-$\lambda$ distribution, our distribution is defined in terms of its quantile function, which in this case is given by the polylogarithm function. We show the support of the distribution to be the Riemann zeta function (when finite), and we provide a closed form for the expectation, provide simple means to calculate the CDF and PDF, and show that it has relationships to the uniform, exponential, inverse beta and extreme-value distributions.