Multiplication of Generalized Polynomials, with Applications to Classical Orthogonal Polynomials

A simple scheme is presented for computing the product of two polynomials in generalized form, i.e. expressed relative to a given orthogonal polynomial basis. If the polynomials have degrees m and $r\,(r\leqq m)$, then the method requires r multiplications of vectors by a tridiagonal matrix of order $m + r$. No conversions to standard power form are involved. As particular cases, some explicit formulae are easily derived for products of pairs of classical orthogonal polynomials.