TITLE: Padé multipoint approximants of the exponential function
QUESTION [5 upvotes]: One says that a pair of polynomials $(P_m,Q_n)$ over $\mathbb C[z]$, with 
$\text{deg }P_m=m$, $\text{deg }Q_n=n$, is a "multipoint Padé approximant of the exponential function" if $P_m(z)e^z-Q_n(z)$ vanishes for $z=0,\ldots, n+m$. Is there an explicit formula for the $P_m$ and $Q_n$?

REPLY [4 votes]: Yes, indeed there is. Explicit expressions for multipoint Padé approximants to the exponential (and power) function at points 
$z=0,\ldots,m+n$, were given in

A. Zhedanov, Explicit multipoint rational interpolation Padé table
  for exponential and power functions, Workshop on Group Theory and
  Numerical Analysis, CRM, Montréal, May 26 - 31, 2003, CRM
  Proceedings and Lecture Notes 39, 2004, 285--298.

With your notations, the formulas read as follows,
\begin{align*}
Q _ { n } ( z  ) & = ( - 1 ) ^ { m } ( 1 - 1 / e ) ^ { - m } ( 1 + n ) _ { m  }~ {}_{2}F _ { 1 } \left( \begin{array} { c } { - n , - z } \\ { - m - n } \end{array} ; 1 - e \right),\\
P_ {m } ( z  ) & = ( - 1 ) ^ { m } ( 1 - 1 / e ) ^ { - m } ( 1 + n ) _ { m  }~ {}_{2}F _ { 1 } \left( \begin{array} { c } { - m , - z } \\ { - m - n } \end{array} ; 1 - 1/e \right),
\end{align*}
where $P_{m}(z)$ is a monic polynomial, and here is a plot of $e^z-Q_n/P_n(z)$ for $n=2,3,4$,