Little q-Legendre polynomials and irrationality of certain Lambert series

We show how one can obtain rational approximants for $q$-extensions of the harmonic series and the logarithm (and many other similar quantities) by Pad\'e approximation using little $q$-Legendre polynomials and we show that properties of these orthogonal polynomials indeed prove the irrationality, with an upper bound of the measure of irrationality which is as sharp as the upper bound given by Bundschuh and V\"a\"an\"anen for the harmonic series and a better upper bound than the one given by Matala-aho and V\"a\"an\"anen for the logarithm.