The q-Stirling numbers, continued fractions and the q-Charlier and q-Laguerre polynomials

We give a simple proof of the continued fraction expansions of the ordinary generating functions of the q-Stirling numbers of both kinds. By generalizing the method of Touchard (1956) and Milne (1978), we obtain the explicit formulas and measure of one set of the polynomials whose moments are the q-Stirling numbers.