The rational approximation of functions which are formally defined by a power series expansion

Abstract

WYNNthe (v 4-l)th column and the (n 4-l)th row.Such an array is referred to as the Pad table of the function (x), and the condition that it may be constructed from the systems of equations (2.3) is that all the Hankel determinants (2.4)should be non-zero.Since the successive convergents of the continued fractionare rational functions of x, it is to be expected that there is a connection between the theory of continued fractions and the Pad table.In fact, if (2.5) is the continued fraction expansion, which may be derived by a number of methods one of which willexplicitly Ix; described in a later section, of the power series (2.6) 22 c"+rX then [2, p. 447] the quotients Uo.k(x) t/p,M-l(x) Ul.k+l(x) V0.k(4 ' V'o.k+l(x) ' Vr.k+1(x) (2.7) Ui.k+t(x) U2.k+t(x) Vr.k+2(x) ' V2.k+i(x)'are the successive convergents of the continued fraction *+l " <*+!>"." <*+i> " (*+l>" .<*+!>,,