Combinatorial probability interpretation of certain modified orthogonal polynomials

A probabilistic interpretation of a modified Gegenbauer polynomial is furnished 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, two limiting forms of the compound urn model indicated, which lead to analogous interpretations of a modified Hermite and a modified Chebyshev polynomial of the first kind.