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, in nitesimal and nite, of these polynomials Pn(x; ci) with respect to the parameters ci is proposed. A method to get recurrence relations for connection coe cients linking (@=@c i )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. c © 2000 Elsevier Science B.V. All rights reserved. MSC: 33C25; 42C05; 33B15

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