On continued fractions and infinite products
Scott, Walter T.
Doctor of Philosophy
Expansion theorem. Every power series (1 · 10) C0 + C1x + C2x2+&ldots;,+Cn xn determines uniquely a continued fraction of the form (1 · 11) a0+a1x a11+a 2xa2 1+&ldots;+anx an1+&ldots; where the exponents an are positive integers and the coefficients an are complex constants. Thus any infinite set of numbers which may be regarded as the differential coefficients of a function at the origin determines uniquely a continued fraction of the form (1 · 10). This is true, in particular, of every function analytic at the origin. The continued fraction (1 · 11) is said to correspond to the power series (1 · 10) and is called a corresponding continued fraction . The power series (1 · 10) is called the corresponding power series of the continued fraction (1 · 11). There is no loss of generality in assuming that a0 = 1. The corresponding continued fraction is then (1 · 12) 1+a1xa 11+a2 xa2 1+&cdots;+an xan1+&cdots; . The continued fractions to be discussed henceforth in this paper will be of the form (1 · 12), where all of the coefficients an are different from zero. If an = 0 for some finite index n, the continued fraction is equivalent to a terminating one and represents a rational function of x.