Asymptotics of Diagonal Hermite–Padé Approximants toez Original Research Article

Letmbe a fixed positive integer. We consider Hermite–Padé approximants to the exponential functionR(z)=∑p=0m Ap(z) epz=O(z^mn+n−1),where the degree of the polynomialsAp, 0⩽p⩽m, is less thann. Asn→∞, exact asymptotics for theApʹs and the remainder termR, along with an upper bound on the zeros of the polynomialsAp, are given. These asymptotics show that shifted Hermite–Padé approximants asymptotically minimize exponential polynomials of the above form on a disk {|z|⩽ρ}, providedρdoes not exceedπ/m. These results generalize some of those obtained by Borwein (Const. Approx.2(1986), 291–302) on quadratic Hermite–Padé approximants.