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%%%% NATO Advanced Study Institute
%%%% "Special Functions 2000: Current Perspective and Future Directions"
%%%% May 29-June 9, 2000
%%%% Arizona State University, Tempe, Arizona (U.S.A.)
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%%%% Parameter derivatives of orthogonal polynomials and
%%%% connection coefficients
%%%%
%%%% by A. Ronveaux, A. Zarzo, I. Area and E. Godoy
%%%%
%%%%
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\title{Parameter derivatives of orthogonal polynomials and connection coefficients}

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\begin{center}
{A. Ronveaux}, Facult\'es Universitaires N.\,D. de la Paix,
Belgium, \\ {A. Zarzo}, Universidad Polit\'ecnica de Madrid,
Spain, \\ {I. Area}, Universidade de Vigo, Spain,
\\  {E. Godoy}, Universidade de Vigo, Spain
\end{center}

\vspace*{1cm}
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 $\displaystyle{\frac{\partial^r}{\partial c_i^r} P_n(x,c_i)}$ 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 $\displaystyle{\frac{\partial}{\partial c_i} P_n(x,c_i)}$ is also proved.

\begin{thebibliography}{99}
\bibitem{FRO}
J. Froehlich,
Parameter derivatives of the Jacoby polynomials and Gaussian hypergeometric function,
{\em Integral Transforms and Special Functions}
{\bfseries 2} (1994), 252--266.

\bibitem{KSc}
W. Koepf and D. Schmersau,
Representations of orthogonal polynomials,
{\em J.\ Comput.\ Appl.\ Math.\/} {\bfseries 90} (1998), 57--94.

\bibitem{AGRZ}
A. Ronveaux, A. Zarzo, I. Area, and E. Godoy,
Classical orthogonal polynomials: Dependence of parameters,
{\em J. Comput. Appl. Math.\/} (in print).
\end{thebibliography}



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