On the even extension of an M fraction

Abstract

One result of the surge of interest in Padé approximations during the last two decades has been the study of two-point Padé approximations. In particular, rational functions which are derived from power series expansions about the origin and the point at infinity have found several applications and the theory associated with them has developed accordingly.

These particular two-point Padé approximations are convergents of continued fractions of the form

$$c_0 + c_1 z + c_2 z^2 + \cdots + \frac{{^c k^{z^k } }}{{1 + d_1 z}} + \frac{{^n 2^z }}{{1 + d_2 z }} + \frac{{^n 3^z }}{{1 + d_3 z}} + \cdots , k \geqslant 0,$$

\]now generally known as M fractions or, alternatively, general T fractions. The coefficients of these continued fractions can be obtained by a variety of methods, including the well known q-d algorithm.

The purpose of this short talk is to discuss an even extension of the above continued fractions, that is a continued fraction whose even order convergents are the successive convergents of the above fraction. The extension is a continued fraction of a form not frequently met in the literature, but is of a simpler type than the M fraction. The same q-d algorithm, with two slight modifications, can be used to provide the coefficients of the even extension, and this will be described.

Finally, an example for which the extension will provide error bounds, whereas the M fraction will not, is considered.