In complex analysis, a Padé table is an array, possibly of infinite extent, of the rational Padé approximants
Rm, n to a given complex formal power series. Certain sequences of approximants lying within a Padé table can often be shown to correspond with successive convergents of a continued fraction representation of a holomorphic or meromorphic function.
History Although earlier mathematicians had obtained sporadic results involving sequences of rational approximations to transcendental functions, Frobenius (in 1881) was apparently the first to organize the approximants in the form of a table. Henri Padé further expanded this notion in his doctoral thesis Sur la representation approchee d'une fonction par des fractions rationelles, in 1892. Over the ensuing 16 years Padé published 28 additional papers exploring the properties of his table, and relating the table to analytic continued fractions. Modern interest in Padé tables was revived by H. S. Wall and Oskar Perron, who were primarily interested in the connections between the tables and certain classes of continued fractions. Daniel Shanks and Peter Wynn published influential papers about 1955, and W. B. Gragg obtained far-reaching convergence results during the '70s. More recently, the widespread use of electronic computers has stimulated a great deal of additional interest in the subject.
Notation A function f(z) is represented by a formal power series:
f ( z ) = c 0 + c 1 z + c 2 z 2 + ⋯ = ∑ l = 0 ∞ c l z l , {\displaystyle f(z)=c_{0}+c_{1}z+c_{2}z^{2}+\cdots =\sum _{l=0}^{\infty }c_{l}z^{l},}
where c0 ≠ 0, by convention. The (m, n)th entry Rm, n in the Padé table for f(z) is then given by
R m , n ( z ) = P m ( z ) Q n ( z ) = a 0 + a 1 z + a 2 z 2 + ⋯ + a m z m b 0 + b 1 z + b 2 z 2 + ⋯ + b n z n {\displaystyle R_{m,n}(z)={\frac {P_{m}(z)}{Q_{n}(z)}}={\frac {a_{0}+a_{1}z+a_{2}z^{2}+\cdots +a_{m}z^{m}}{b_{0}+b_{1}z+b_{2}z^{2}+\cdots +b_{n}z^{n}}}}
where Pm(z) and Qn(z) are polynomials of degrees not more than m and n, respectively. The coefficients {ai} and {bi} can always be found by considering the expression
f ( z ) ≈ ∑ l = 0 m + n c l z l =: f a p p r o x ( z ) {\displaystyle f(z)\approx \sum _{l=0}^{m+n}c_{l}z^{l}=:f_{\mathrm {approx} }(z)}
Q n ( z ) f a p p r o x ( z ) = P m ( z ) {\displaystyle Q_{n}(z)f_{\mathrm {approx} }(z)=P_{m}(z)}
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