Padé table

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.

Henri Padé further expanded this notion in his doctoral thesis Sur la representation approchee d'une fonction par des fractions rationelles, in 1892.

Daniel Shanks and Peter Wynn published influential papers about 1955, and W. B. Gragg obtained far-reaching convergence results during the '70s.

However, it can be shown that, due to cancellation, the generated rational functions Rm, n are all the same, so that the (m, n)th entry in the Padé table is unique.

Usage of the Padé table has been extended to meromorphic functions by newer, timesaving methods such as the epsilon algorithm.

The procedure used to derive Gauss's continued fraction can be applied to a certain confluent hypergeometric series to derive the following C-fraction expansion for the exponential function, valid throughout the entire complex plane: By applying the fundamental recurrence formulas one may easily verify that the successive convergents of this C-fraction are the stairstep sequence of Padé approximants R0,0, R1,0, R1,1, ...