A new presentation of orthogonal polynomials with applications to their computation

In this paper a new presentation of orthogonal polynomials is given. It is based on the introduction of two auxiliary sequences of arbitrary monic polynomials and it leads to a very simple derivation of the usual determinantal formulae for orthogonal polynomials and of their recurrence relations either in the definite or in the indefinite case. New expressions for the coefficients of these recurrence relations are obtained and they are compared to the usual ones from the point of view of their numerical stability. The qd-algorithm is also recovered very easily.