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 ci. A systematic study of the variation, infinitesimal and finite, of these polynomials Pn(x,ci) with respect to the parameters ci is proposed. A method to get recurrence relations for connection coefficients linking (∂r/∂cir)Pn(x,ci) to Pn(x,ci) 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 (∂/∂ci)Pn(x,ci) is also proved.