The rational approximation of functions which are formally defined by a power series expansion
P. Wynn · Mathematics of Computation · 1960
P. WYNN the (v 4-l)th column and the (n 4-l)th row.Such an array is referred to as the Padé table of the function ß(x), and the condition that it may be constructed from the systems of equations ( 2.3) is that all the Hankel determinants (2.4) tlm = Cm Cm+l CiiM-1 Cm+2 Cm+n-1 Cm-f-n Cm+n-1 C»i+n Cm+2n-2 should be non-zero.Since the successive convergents of the continued fraction (2.5) {m) (m) Cm q\_x ei x 1 -I -1 -(ml (m) qr x eT xare rational functions of x, it is to be expected that there is a connection between the theory of continued fractions and the Padé 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 quotientsare the successive convergents of the continued fraction *+l " "." " (*+l>" .,, /., cn , i , k Ct+ix qx x ex x q2 x e2 x (2.8) Co 4-Ci x 4-• • • 4-ckx 4-^-^-r-^r-jwhile the quotients (2 9) Uk'°(x) i;*.i(x) k*+i,i(x) tV,.i(x) ' Uk+i.i(x)Uk+t.2(x)Vk.o(x) ' VkA(x) ' Vh+M ' Vk+1,(x)