Combinatorial probability interpretation of certain modified orthogonal polynomials

A probabilistic interpretation of a modified Gegenbauer polynomial is supplied by its expression in terms of a combinatorial probability defined on a compound urn model. Also, a combinatorial interpretation of its coefficients is provided. In particular, probabilistic interpretations of a modified Chebyshev polynomial of the second kind and a modified Legendre polynomial together with combinatorial interpretations of their coefficients are deduced. Further, probabilistic interpretations of a modified Hermite and a modified Chebyshev polynomial of the first kind are supplied by their expressions in terms of combinatorial probability functions defined on two limiting forms of the compound urn model. Finally, combinatorial interpretations of their coefficients are obtained.