Orthogonal polynomials in the spherical ensemble with two insertions

We consider asymptotics of planar orthogonal polynomials $P_{n,N}$ (where $\mathrm{d}\mathrm{e}\mathrm{g}{P}_{n,N}=n$) with respect to the weight $$\frac{\mathrm{\mid }z-w{\mathrm{\mid }}^{2N{Q}_1}}{(1+\mathrm{\mid }z{\mathrm{\mid }}^2{)}^{N(1+Q_0+Q_1)+1}},(Q_0,Q_1>0)$$ in the whole complex plane. With $n,N\to \mathrm{\infty }$ and $N-n$ fixed, we obtain the strong asymptotics of the polynomials, asymptotics for the weighted $L^2$ norm and the limiting zero counting measure. These results apply to the pre-critical phase of the underlying two-dimensional Coulomb gas system, when the support of the equilibrium measure is simply connected. Our method relies on specifying the mother body of the two-dimensional potential problem. It relies too on the fact that the planar orthogonality can be rewritten as a non-Hermitian contour orthogonality. This allows us to perform the Deift-Zhou steepest descent analysis of the associated $2\times 2$ Riemann-Hilbert problem.