Classical orthogonal polynomials: dependence of parameters

Most of the classical orthogonal polynomials (continuous, discrete and their q-analogues) can be considered as functions of several parameters c(i). A systematic study of the variation, infinitesimal and finite, of these polynomials P-n(x,c(i)) with respect to the parameters c(i) is proposed. A method to get recurrence relations for connection coefficients linking (partial derivative(r)/partial derivative c(i)(r))P-n(x,c(i)) to P-n(x,c(i)) is given and, in some situations, explicit expressions are obtained. This allows us to compute new integrals or sums of classical orthogonal polynomials using the digamma function. A basic theorem on the zeros of (partial derivative/partial derivative c(i)))P-n,(x,c(i)) is also proved. (C) 2000 Elsevier Science B.V. All rights reserved. MSC: 33C25; 42C05; 33B15.