A large number of analytic functions are known to have continued fraction representations. Frequently a given function will be represented by several different continued fractions, each with its own convergence behavior. The purpose of this chapter is to give a wide selection of examples of continued-fraction representations of functions based on the methods of Chapter 5. Our selection is far from exhaustive. In Section 6.1 the examples consist of continued fractions of Gauss associated with various hypergeometric and confluent hypergeometric series. Section 6.2 deals with examples associated with minimal solutions of three-term recurrence relations. Further methods and examples will be discussed in Chapters 7 and 10. Additional examples can be found, among others, in the following books and articles as well as in the references contained therein: [Abramowitz and Stegun, 1964], [Andrews, 1968], [Askey and Ismail, to appear], [de Bruin, 1977], [Cody et al., 1970], [Erdelyi et al., 1953, Vols. 1, 2, 3], [Frank, 1958, 1960a, 1960b], [Gautschi, 1967, 1970, 1977], [Henrici, 1977, Vol. 2, Chapter 12], [Ince, 1919], [Khovanskii, 1963], [Luke, 1969], [Maurer, 1966], [McCabe, 1974], [Miller, 1975], [Murphy, 1971], [Murphy and O'Donohoe, 1976], [Perron, 1957a], [Phipps, 1971], [Schlömilch, 1871], [Sobhy, 1973], [Stegun and Zucker, 1970], [Tannery, 1882], [Thacher, 1967 and Tech. Rpt. 38–77], [Wall, 1948], [Widder, 1968], [Wynn, 1959].

Continued Fractions of Gauss

In this section we consider a number of examples of continued-fraction representations of functions associated with hypergeometric and confluent hypergeometric series. The method employed is described in Sections 5.2 and 5.4.