Interpolation of Chebyshev polynomials and interacting Fock spaces

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We discover a family of probability measures μa, 0 < a ≤ 1, dμa(x) = a√1-x2/π[a2 + ( 1 - 2a) x2]/dx |x| <1, 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. © World Scientific Publishing Company.

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Infinite Dimensional Analysis, Quantum Probability and Related Topics

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