Strong asymptotics for Laguerre polynomials with varying weights

Supplementing and extending classical and recent results strong asymptotics for the Laguerre polynomials Ln(αn) are established, as n → ∞, when the parameter αn depends on the degree n suitably. A case of particular interest is the one for which αn grows faster than n. Rescaling the argument z appropriately the resulting asymptotic forms are described by elementary functions, thereby extending the classical formulae of Plancherel-Rotach type for Laguerre polynomials Ln(α)(z), α being fixed.