The concept of correspondence of continued fractions with formal Laurent series (fLs) was introduced in Chapter 5. Some general theory of correspondence was developed there, and two types of corresponding continued fractions were discussed, the C-fractions and the P-fractions. Some applications of correspondence to obtain continued-fraction representations of analytic functions were given in Chapters 5 and 6. The present chapter deals with special properties of the three most important types of corresponding continued fractions: regular C-fractions (Section 7.1), associated continued fractions (Section 7.2) and general T-fractions (Section 7.3). J-fractions are discussed in connection with associated continued fractions, since they are essentially of the same type. Also g-fractions are considered with regular C-fractions.

A great deal of this chapter is devoted to algorithms. For each type of continued fraction an algorithm is given to compute the coefficients of the continued fraction in terms of the corresponding fLs. The quotientdifference algorithm (Section 7.1.2) and the FG algorithms (Section 7.3.2) can also be used to compute zeros and poles of analytic functions. Among the power series to which the quotient-difference algorithm can be applied are all normal series. Subsumed among these are the Pólya frequency series discussed in Section 7.1.2. Some examples of analytic functions represented by general T-fractions are given in Section 7.3.3; these consist of ratios of confluent hypergeometric functions and include as special cases the error function and Fresnel integrals. The close connection between J-fractions and the general theory of orthogonal polynomials is described briefly in Section 7.2.2.