dc.creator Scott, Walter T. 2007-08-21T01:07:54Z 2007-08-21T01:07:54Z 1938 https://hdl.handle.net/1911/18441 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. application/pdf eng Mathematics On continued fractions and infinite products Thesis Text Mathematics Natural Sciences Rice University Doctoral Doctor of Philosophy Scott, Walter T.. "On continued fractions and infinite products." (1938) Diss., Rice University. https://hdl.handle.net/1911/18441.
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