Padé Table - Connection With Continued Fractions

Connection With Continued Fractions

One of the most important forms in which an analytic continued fraction can appear is as a regular C-fraction, which is a continued fraction of the form

f(z) = b_0 + \cfrac{a_1z}{1 - \cfrac{a_2z}{1 - \cfrac{a_3z}{1 - \cfrac{a_4z}{1 - \ddots}}}}.

where the ai ≠ 0 are complex constants, and z is a complex variable.

There is an intimate connection between regular C-fractions and Padé tables with normal approximants along the main diagonal: the "stairstep" sequence of Padé approximants R0,0, R1,0, R1,1, R2,1, R2,2, … is normal if and only if that sequence coincides with the successive convergents of a regular C-fraction. In other words, if the Padé table is normal along the main diagonal, it can be used to construct a regular C-fraction, and if a regular C-fraction representation for the function f(z) exists, then the main diagonal of the Padé table representing f(z) is normal.

Read more about this topic:  Padé Table

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