Laguerre-Sobolev orthogonal polynomials: Asymptotics for coherent pairs of type II
A Laguerre-Sobolev orthogonal polynomial \(S_n\) of degree \(n\) corresponding to a coherent pair of type II satisfies \((S_n, q)_S=0\) for polynomials \(q\) of degree \(\leq n-1\), where \[ (p, q)_S=\int_0^{+\infty} p q d\mu_0+ \int_0^{+\infty} p' q' d\mu_1 , \] \(\mu_0\) is standard Laguerre measure on \((0, +\infty)\) and \(\mu_1\) is its rational modification plus eventually a single mass point on the negative semiaxis. The authors establish strong asymptotics for the sequence \(S_n(x)\) for \(x \notin (0,+\infty)\), bounds on \((0,+\infty)\), along with Mehler-Heine (for \(S_n(x/n)\) on \(\mathbb C\)) and Plancherel-Rotach (for \(S_n(nx)\) on \(\mathbb C \setminus [0, 4]\)) formulas. As a corollary, asymptotic behavior of the zeros of \(S_n\) is obtained, illustrated by several numerical experiments. As an auxiliary result, several properties of polynomials orthogonal with respect to \(\mu_1\) are studied.
scientific article; zbMATH DE number 1924732