Padé Table - Generalizations

Generalizations

A formal Newton series L is of the form

$L(z) = c_0 + \sum_{n=1}^\infty c_n \prod_{k=1}^n (z - \beta_k)$

where the sequence {β_k} of points in the complex plane is known as the set of interpolation points. A sequence of rational approximants R_m,n can be formed for such a series L in a manner entirely analogous to the procedure described above, and the approximants can be arranged in a Newton-Padé table. It has been shown that some "staircase" sequences in the Newton-Padé table correspond with the successive convergents of a Thiele-type continued fraction, which is of the form

$a_0 + \cfrac{a_1(z - \beta_1)}{1 - \cfrac{a_2(z - \beta_2)}{1 - \cfrac{a_3(z - \beta_3)}{1 - \ddots}}}.$

Mathematicians have also constructed two-point Padé tables by considering two series, one in powers of z, the other in powers of 1/z, which alternately represent the function f(z) in a neighborhood of zero and in a neighborhood of infinity.