Nuttall’s theorem with analytic weights on algebraic S-contours

Given a function ff holomorphic at infinity, the nnth diagonal Padé approximant to ff, denoted by [n/n]f[n/n]f, is a rational function of type (n,n)(n,n) that has the highest order of contact with ff at infinity. Nuttall’s theorem provides an asymptotic formula for the error of approximation f−[n/n]ff−[n/n]f in the case where ff is the Cauchy integral of a smooth density with respect to the arcsine distribution on [−1,1][−1,1]. In this note, Nuttall’s theorem is extended to Cauchy integrals of analytic densities on the so-called algebraic S-contours (in the sense of Nuttall and Stahl).