Exact and simple formulas for the linearization coefficients of products of orthogonal polynomials and physical application

We obtain exact, simple and very compact expressions for the linearization coefficients of products orthogonal polynomials; both conventional Clebsch-Gordan-type modified version. The are general depending only on three-term recursion relation linearizing polynomials. These more appropriate useful doing numerical calculations when compared to other exact formulas found in mathematics literature, some which apply special class polynomials while others may involve evaluation intractable integrals. As an application physics, we present a remarkable phenomenon where nonlinear coupling physical system with pure continuous spectrum generates mixed discrete energies.