Heine, Hilbert, Padé, Riemann, and Stieltjes: a John

In 1986 J. Nuttall published in Constructive Approximation the paper "Asymptotics of generalized Jacobi polynomials", where with his usual insight he studied the behavior of the denominators ("generalized Jacobi polynomials") and the remainders of the Pade approximants to a special class of algebraic functions with 3 branch points. 25 years later we try to look at this problem from a modern perspective. On one hand, the generalized Jacobi polynomials constitute an instance of the so-called Heine-Stieltjes polynomials, i.e. they are solutions of linear ODE with polynomial coefficients. On the other, they satisfy complex orthogonality relations, and thus are suitable for the Riemann-Hilbert asymptotic analysis. Along with the names mentioned in the title, this paper features also a special appearance by Riemann surfaces, quadratic differentials, compact sets of minimal capacity, special functions and other characters

Jacqueline Harpman's transgressive dystopian fantastic in ‘moi qui n'ai pas connu les hommes’:Between familiar territory and unknown worlds

Type I Hermite-Pad\'e polynomials for set of functions $f_0, f_1,..., f_s$ at infinity, $(Q_{n,0}f_0+Q_{n,1}f_1+Q_{n,2}f_2+...+Q_{n,s}f_s)(z)=O(\frac{1}{z^{sn+s}}), z\rightarrow \infty$ with the degree of all $Q_{n,k}<=n$. We describe an approach for finding the asymptotic zero distribution of these polynomials as $n\rightarrow \infty$ under the assumption that all $f'_js$ are semiclassical, i.e. their logarithmic derivatives are rational functions. In this situation $R_n$ and $Q_{n,k}f_k$ satisfy the same differential equation with polynomials coefficients. We discuss in more detail the case when $f'_k$s are powers of the same function $f (f_k=f^k)$; for illustration, the simplest non trivial situation of $s=2$ and $f$ having two branch points is analyzed in depth. Under these conditions, the ratio or comparative asymptotics of these polynomials is also discussed. From methodological considerations and in order to make the situation clearer, we start our exposition with the better known case of Pad\'e approximants (when $s=1$)