Generalizing classical and recent results for Jacobi polynomials P (� n,�n) n (z), strong asymptotics are established as n → ∞ where, in contrast to the standard situation, the real parametersn, �n depend on the degree n and satisfy certain limiting conditions. By rescaling the argument z suitably, extensions of the well-known Darboux formulae are derived. 1. Introduction and summary In this paper, we deal with generalizations of asymptotic forms for Jacobi polynomials P (�,�) n . As usual, we assume these functions to be defined via the orthogonality relation Z 1 −1 P (�,�) n (x)P (�,�) m (x)(1 − x) � (1 + x) � dx