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 $Q_n$?
1 Answer
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 $n=2,3,4$,