We consider the asymptotic behaviour of the Gauss hypergeometric function when several of the parameters {it a, b, c} are large. We indicate which cases are of interest for orthogonal polynomials (Jacobi, but also Meixner, Krawtchouk, etc.), which results are already available and which cases need more attention. We also consider a few examples of ${}_3${it F}${}_2$ functions of unit argument, to explain which difficulties arise in these cases, when standardintegrals or differential equations are not available.

Classical hypergeometric functions, ${}_2F_1$ (msc 33C05), Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) (msc 33C45), Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (msc 41A60), Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) (msc 30C15), Approximation by polynomials (msc 41A10)
CWI
Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Temme, N.M. (2002). Large parameter cases of the Gauss hypergeometric function. Modelling, Analysis and Simulation [MAS]. CWI.