Let be given the orthogonal polynomials \(\{p_ i(\lambda)\}\), \(\{q_ i(\lambda)\}\). The author demonstrates four theorems concerning the expression of \(\lambda^ ip_ j(\lambda)\), \(\lambda^ iq_ j(\lambda)\), \(p_ i(\lambda)q_ j(\lambda)\) and \(q_ i(\lambda)q_ j(\lambda)\) in terms of \(p_ i(\lambda)\). The proofs are based on a former result of the author [Proc. Int. Conf. on Linear Algebra and its Applications, Vitoria, Spain, (1983), pp. 9-19] using a tridiagonal matrix for the expression of a product of two linear combinations of \(p_ i(\lambda)\) in terms of \(\{p_ i(\lambda)\}\). Various examples are given for results.