Generalized Gauss-Rys orthogonal polynomials

Let $(P_n(x;z;\lambda))_{n\geq 0}$ be the sequence of monic orthogonal polynomials with respect to the symmetric linear functional $\mathbf{s}$ defined by $$\langle\mathbf{s},p\rangle=\int_{-1}^1 p(x)(1-x^2)^{(\lambda-1/2)} e^{-zx^2}dx,\qquad\lambda>-1/2, \quad z>0.$$ In this contribution, several properties of the polynomials $P_n(x;z;\lambda)$ are studied taking into account the relation between the parameters of the three-term recurrence relation that they satisfy. Asymptotic expansions of these coefficients are given. Discrete Painlev\'e and Painlev\'e equations associated with such coefficients appear naturally. An electrostatic interpretation of the zeros of such polynomials as well as the dynamics of the zeros in terms of the parameters $z$ and $\lambda$ are given.