Classical orthogonal polynomials: Dependence of parameters (Q1587396)

The authors study connection problems between classical orthogonal polynomials and their derivatives with respect to (one of) their parameter(s). They use their so-called \texttt{Navima} algorithm to derive recurrence relations for the connection coefficients linking a family of classical orthogonal polynomials (like the Laguerre and Jacobi polynomials) and a family consisting of derivatives of these polynomials with respect to a parameter. The case of the little \(q\)-Jacobi polynomials is also treated. In some cases the recurrence relation for the connection coefficients can be solved explicitly. This leads to some new integral formulas involving these classical orthogonal polynomials, some logarithms and the digamma function, the logarithmic derivative of the gamma function. In the last section of the paper the authors study the interlacing property of the zeros of the derivatives with respect to a parameter and the classical orthogonal polynomial itself, which holds if a simple condition is satisfied.