Some identities involving orthogonal polynomials

Formulas of Christoffel and Uvarov for changing the weight of real orthogonal polynomials by polynomial multiplication and division are extended to polynomials orthogonal on the unit circle. Elementary methods are used to give an alternative proof of a result of Askey and Hahn.

Next, some by-products that come from our proofs, not the theorems themselves, are explored. These by-products have to do with certain non-vanishing determinants having orthogonal polynomials as entries.

Finally, an elementary approach is given regarding two problems in orthogonal polynomials: (i) that of filling out certain sequences of orthogonal polynomials by finding intermediate ones, and (ii) that of finding certain frequencies involved in digital signal processing.