# Manual Analytic Theory of Continued Fractions II

Hubert Stanley Wall presents a unified theory correlating certain parts and applications of the subject within a larger analytic structure. Prerequisites include a first course in function theory and knowledge of the elementary properties of linear transformations in the complex plane.

Some background in number theory, real analysis, and complex analysis may also prove helpful. The two-part treatment begins with an exploration of convergence theory, addressing continued fractions as products of linear fractional transformations, convergence theorems, and the theory of positive definite continued fractions, as well as other topics. The second part, focusing on function theory, covers the theory of equations, matrix theory of continued fractions, bounded analytic functions, and many additional subjects. This book deals with the analytic theory of continued fractions, that is, with continued fractions in relation to analysis: the theory of equations, orthogonal polynomials, power series, infinite matrices and quadratic forms in infinitely many variables, definite integrals, the moment problem, analytic functions, and the summation of divergent series.

In contrast with the analytic theory of continued fractions, there is an extensive arithmetic theory which is not touched upon here.

## Books Continued Fractions: Analytic Theory and Applications (Encyclopedia of Mathematics and Its

The celebrated memoir of T. Here is to be found the development of fundamental function theory and integral theory necessary for a complete treatment of an important class of continued fractions. For several years, Stieltjes had been interested in the problem of summation of divergent power series. In — he published a considerable number of examples of continued fraction expansions for series of this kind, all arising as formal power series expansions of definite integrals. The integrals are of the form. In the memoir of , Stieltjes developed a general theory of these continued fractions, covering questions of convergence and connection with definite integrals and divergent power series.

In order to complete the theory, he had to extend the customary notion of integral, and to develop a general convergence continuation theorem for sequences of analytic functions. In , E. Van Vleck [ ] undertook to extend the Stieltjes theory to continued fractions of the form b in which the pk are arbitrary positive numbers and the bk arbitrary real numbers. He was able to connect in certain cases these continued fractions with definite integrals of the type found by Stieltjes, but with the range of integration taken over the entire real axis.

A complete extension of the Stieltjes theory to these continued fractions was first obtained by Hamburger [26] in , following the pattern laid down by Stieltjes. In the interim, Hilbert and his pupils developed their famous theory of infinite matrices and quadratic forms in infinitely many variables, in which the ideas of Stieltjes are in the background.

Several other mathematicians reached the same goal by different methods at about the same time Carleman [6], R. Nevanlinna [62], M. Riesz [79]. Another kind of investigation had been going on in the meantime. Around Pringsheim [ 73, 75] and Van Vleck [, ] considered the question of convergence of continued fractions with complex elements.

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This is an extension of an older theorem of von Koch []. These results and the proofs which were employed bear little relationship to one another or to the Stieltjes theory. During the years —, in which this book has been written, it has been our desire to develop a unified theory extending the various results indicated in the preceding sketch and tying them together within a larger analytic structure. Second, we developed the theory of positive definite continued fractions, which extends the Stieltjes theory to a class of continued fractions b with complex pk and bk , and also contains and extends the other results we have described, including the parabola theorems just mentioned Hellinger and Wall [35], Wall and Wetzel [, ], Dennis and Wall[9].

The aforementioned larger analytic structure is obtained here by regarding the continued fraction as generated by an infinite sequence of linear fractional transformations in a single variable, and also as arising from a single linear transformation in infinitely many variables. There is often an interplay of these two ideas.

Otherwise the continued fraction diverges. Thus the value of a continued fraction is the limit of the images of a fixed point under a certain sequence of linear fractional transformations. In order to investigate the continued fraction, we proceed as follows. Thus, we determine a nest of circles such that the p th approximant of the continued fraction is in the p th circle. There are two possible cases. Either the circles Kp have one, and only one, point L in common limit-point case , or else the circles Kp have a circular region in common limit-circle case.

In the first case the continued fraction converges to the value L; in the second case the question of convergence remains undecided. Criteria for determining which of the two cases holds may be found if we first obtain an explicit formula for the radius rp of Kp. The above-described program can be carried out under various hypotheses upon the coefficients ap and bp of e.

We have sought to define a class of continued fractions for which this can be done, which is sufficiently general to include all particular classes which have been studied in the literature. The class of positive definite continued fractions , for which the quadratic forms. The continued fraction is positive definite if, and only if, numbers g 0, g 1, g 2, … can be found satisfying the above inequalities.

This parabola has its focus at the origin and its vertex upon the negative half of the real axis, and it depends upon the index p. The class of positive definite continued fractions is first introduced in Chapter IV. This could have been done earlier, but we have followed, instead, the plan of first investigating by appropriately simple methods certain special positive definite continued fractions.

After covering some preliminary ideas in Chapter I, and some necessary conditions for convergence and a treatment of periodic continued fractions in Chapter II, we take up, in Chapter III, the aforementioned special positive definite continued fractions. These can be taken in the form c , with the positive definiteness condition.

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The treatment of these cases in Chapter III is based upon a system of fundamental inequalities [86]. The moment problem for an infinite interval The continued fraction of Gauss Stieltjes summability In writing this book, I have tried to keep in mind the student of rather modest mathematical preparation, presupposing only a first course in function theory. Thus, I have included such things as a proof of Schwarz's inequality, theorems on uniformly bounded families of analytic functions, properties of Stieltjes integrals, and an introduction to the matrix calculus.

I have presupposed a knowledge of the elementary properties of linear fractional transformations in the complex plane. It has not been my intention to write a complete treatise on the subject of continued fractions, covering all the literature, but rather to present a unified theory correlating certain parts and applications of the subject within a larger analytic structure. I have not touched upon the arithmetic theory, and have, for the most part, refrained from developing formulas of a more general character than are actually used in the proofs.