Continued Fractions

Abstract

In assessing the content of Ramanujan’s first letter, dated January 16, 1913, to him, Hardy [9, p. 9] remarked: “but (1.10)–(1.12) defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class. They must be true because, if they were not true, no one would have had the imagination to invent them.” These comments were directed at three continued fraction representations. Indeed, Ramanujan’s contributions to the continued fraction expansions of analytic functions are one of his most spectacular achievements. The three formulas that challenged Hardy’s acumen are not found in Chapter 12, but this chapter, which is almost entirely devoted to the study of continued fractions, contains many other beautiful and penetrating formulas. Unfortunately, Ramanujan left us no clues as to how he discovered these elegant continued fraction formulas. Especially enigmatic are the several representations for products and quotients of gamma functions. Three of the principal formulas involving gamma functions are Entries 34, 39, and 40. Entries 20 and 22, giving Gauss’s and Euler’s continued fractions, respectively, for a quotient of two hypergeometric functions, also play prominent roles. Several other formulas are dependent on these five entries, and it may be helpful to schematically indicate these connections among entries.