This paper deals with rational approximants of the exponential function \(e^z\), a subject of interest in function theory and in some problems of applied mathematics. Given a triangular sequence of complex interpolation points \(\{z_j^{(2n)}\}_{j=0}^{2n}\), consider the associated rational functions \(r_n=p_n/q_n\), with \(p_n\), \(q_n\) polynomials of degree at most \(n\) such that as \(z\rightarrow z_j^{(2n)}\), \(j=0,\dots, 2n\),
\[ e_n(z):=p_n(z) e^{-z/2}+ q_n(z) e^{z/2}=O\bigg(\prod_{j=0}^{2n}\big(z-z_j^{(2n)}\big)\bigg) . \] The main result is the following.
Theorem1.1. Let \(\{z_j^{(2n)}\}_{j=0}^{2n}\) be a triangular family such that
\[ \rho_n:=\max_{j=0,\dots,2n}|z_j^{(2n)}| \leq c n^{1-\alpha},\qquad n\in\mathbb N, \] for some \(c>0\) and \(\alpha\in(0,1]\). Let \(p_n,q_n\) be polynomials satisfying (1). Then
i) No zeros and poles of \(r_n\) lie in the disk \(D(0,\rho_n)\) for \(n\) large. In particular, as \(z\rightarrow z_j^{(2n)}\), \(j=0,\dots, 2n\),
\[ e^z+r_n(z)=O\bigg(\prod_{j=0}^{2n}\big(z-z_j^{(2n)}\big)\bigg) . \]
ii) Assume \(q_n\) is normalised so that \(q_n(0)=1\). As \(n\to\infty\), \[ p_n(z)\rightarrow e^{z/2},\qquad q_n(z)\rightarrow e^{-z/2},\qquad r_n(z)\rightarrow -e^{z} \] locally uniformly in \(\mathbb C\).
iii) For large \(n\)
\[ e^z+r_n(z)=(-1)^n\bigg(\frac{e c_n}{4n}\bigg)^{2n+1} e^{z-1} O\bigg(\prod_{j=0}^{2n}\big(z-z_j^{(2n)}\big)\bigg) \bigg(1+ O\Big(\frac 1{n^{\alpha}}\Big)\bigg) \] locally uniformly in \(\mathbb C\), where \(c_n\) is a constant that dependes only on the interpolation points and such that
\[ c_n=1+ O\bigg(\Big(\frac{\rho_n}n\Big)^2\bigg),\qquad n\to\infty. \] Similar results for the case where \(\{\rho_n\}_n\) grows at most logarithmicly in \(n\) were given in [F. Wielonsky, J. Approximation Theory 131, No. 1, 100–148 (2004; Zbl 1069.30062)]. The proof of Theorem 1.1 follows the same scheme of this previous case: first the rational interpolants are characterised in terms of a solution of a matrix Riemann-Hilbert problem and then a steepest descent analysis of the Riemann-Hilbert problem is performed.
The second part of the paper describes the limit distributions of the zeros of the scaled polynomials \(P_n(z)=p_n(2nz)\) and \(Q_n(z)=q_n(2nz)\). Given a polynomial \(P\) of degree \(n\), let \(\nu_P=\frac 1n\sum_{p(z)=0}\delta_z\) be the normalised zero counting measure.
Theorem 1.3. As \(n\to\infty\), \[ \nu_{P_n} \longrightarrow \frac 1{i\pi}\frac{(\sqrt{s^2+1})_+}{s} ds, \qquad \nu_{Q_n} \longrightarrow \frac 1{i\pi}\frac{(\sqrt{(-s)^2+1})_+}{s} ds, \] where the convergence is in the sense of weak\(^*\)-convergence of measures. (Here the two square roots indicate the two different branches).
A final section is devoted to numerical experiments showing how the distributions of zeros and poles of the interpolants may be modified when considering different configurations of interpolation points with modulus of order \(n\).