Definitions, recursion relations, and computation

In this chapter we do not aspire to summarize the companion volume of Jones and Thron [1980] which is devoted to the general theory of continued fractions. Here, we set out to present a working knowledge of the basic concepts of continued fractions, so that we may give a self-contained account of how continued-fraction theory supplements our understanding of Padé approximation. The discovery of continued fractions in the West seems to have been made by Bombelli [1572]; Jones and Thron [1980] and Brezinski [1990] give historical surveys. In Section 4.7, we quote the basic convergence theorems for general continued fractions, and refer to the companion volume for the proofs. We are primarily concerned with continued fractions associated with power series, for which the continued fractions happen to be Padé approximants. Indeed, in the next chapter we will see that S-fractions are associated with Stieltjes series and that real J-fractions are associated with Hamburger series. The convergents of these fractions form simple sequences in the Padé table.

There is no doubt that part of Padé approximation theory grew out of continued-fraction theory. We choose to regard the Padé table as the fundamental set of rational approximants, and the convergents of various continued fractions derived from power series as particular subsequences of the Padé table. We suggest that which continued-fraction representation is the most useful is often seen most clearly by considering first which sequence from the Padé table has the desired asymptotic behavior or rate of convergence.