We use the multiplicative renormalization method to characterize a class of probability measures on R determined by five parameters. This class of probability measures contains the arcsine and the Wigner semicircle distributions (the vacuum distributions of the field operators of interacting Fock spaces related to the Anderson model), as well as new nonsymmetric distributions. The corresponding orthogonal polynomials and Jacobi-Szegö parameters are derived from the orthogonal-polynomial generating functions. These orthogonal polynomials can be expressed in terms of the Chebyshev polynomials of the second kind.

We discover a family of probability measures μa, 0 < a ≤ 1, [Formula: see text] which contains the arcsine distribution (a = 1) and semi-circle distribution (a = 1/2). We show that the multiplicative renormalization method can be used to produce orthogonal polynomials, called Chebyshev polynomials with one parameter a, which reduce to Chebyshev polynomials of the first and second kinds when a = 1 and 1/2 respectively. Moreover, we derive the associated Jacobi–Szegö parameters. This one-parameter family of probability measures coincides with the vacuum distribution of the field operator of the interacting Fock spaces related to the Anderson model.

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