Stieltjes transforms of a class of probability measures

Let the probability measures μN, N=2,3,… be defined by μN({λk,N})=1/N, μN(A)=0 for λk,N∉A, where λk,N, k=1,2,…,N are the zeros of the orthogonal polynomial PN+1(x), which is obtained recursively by P0(x)=0, P1(x)=1, anPn+1(x)+an−1Pn−1(x)+bnPn(x)=xPn(x). Conditions on an and bn are found such that the sequence μN, N=2,3,… converges weakly to the Dirac measure at the point zero. This is achieved through the convergence of the sequence of Stieltjes transforms ∫−∞∞dμN(t)/(λ−t) to the function 1/λ. Typical examples are the Al-Salam and Carlitz polynomials, the Wall polynomials, the Lommel polynomials and the Tricomi–Carlitz polynomials.