##
**The teaching of continued fractions. Vol. II. Analytical–function-theoretic continued fractions. 3rd corr. and extended edition.
(Die Lehre von den Kettenbrüchen. Band II. Analytisch–funktionentheoretische Kettenbrüche. 3. verbesserte und erweiterte Auflage.)**
*(German)*
Zbl 0077.06602

Stuttgart: B. G. Teubner Verlagsgesellschaft. vi, 316 S. (1957).

This is Volume II of the third edition of this book, the first edition of which appeared in 1913 [JFM 43.0283.04] and the second edition with a few changes in 1929 (Leipzig) [JFM 55.0262.09]. The present volume has been very materially changed and brought up to date. It deals entirely with the analytic, function-theoretic aspect of continued fraction theory, whereas Vol. I [Zbl 0056.05901] deals entirely with the elementary arithmetic portion. Although the general framework of the book is still the same as the old one, although the six chapters and a number of the individual paragraphs have the same or almost the same titles as did those of the preceding editions, and although the format is still along classical lines, the changes in the contents are far reaching. As in Vol. I, the author has again placed greatest value on ease of comprehension, and all material is presented according to standard methods. The book consists of the following topics:

Chapter I, Transformations of continued fractions: (1) recapitulation; (2) zero as numerator of a partial quotient, equivalent continued fractions; (3) continued fractions with given approximants; (4) contraction and extension; (5) equivalence of continued fractions and series; (6) equivalence of continued fractions and products; (7) the transformation of Bauer and Muir; (8) further applications, principal formula of Ramanujan.

Chapter II, Criteria for convergence and divergence: (10) general criteria of Broman, Stern and Scott-Wall; (11) convergence with positive elements; (12) convergence with real elements; (13) irrationality of certain continued fractions; (14) the convergence criterion of Pringsheim; (15) the convergence criterion of van Vleck-Jensen and Hamburger-Mall-Wall; (16) application: region of validity of the Ramanujan formula; (17) some recent criteria, the parabola theorem; (18) periodic continued fractions; (19) limit periodic continued fractions; (20) the equation \[ \frac{x_0}{x_1}=b_0+\frac{a_1|}{| b_1}+\frac{a_2|}{| b_2}+\cdots \] as a consequence of the system of recurrence equations \(x_\nu=b_\nu x_{\nu+1}+a_{\nu+1}x_{\nu+2}\).

Chapter III, Various correspondences of power series with continued fractions: (21) general \(C\)-fractions; (22) square roots; (23) regular \(C\)-fractions; (24) the continued fractions of Gauss, Heine, and related ones; (25) the associated continued fractions; (26) connection between the corresponding and associated continued fraction, some transformations of the corresponding continued fraction; (27) convergence and divergence; (28) convergence of the continued fractions of Gauss, Heine, etc.; (29) a remarkable divergence phenomenon; (30) \(J\)-fractions and their application to polynomials whose zeros have negative real parts; (31) further types of continued fractions that one can associate with power series.

Chapter IV, the continued fractions of Stieltjes: (32) the integral concept of Stieltjes; (33) the corresponding and associated continued fraction of a Stieltjes integral; (34) the theorem of Markoff; (35) the roots of the denominators of the approximants of \(G\)-, \(H\)-, and \(S\)-fractions; (36) the Grommer selection theorem; (37) convergence and analytic character of \(S\)- and \(H\)-fractions; (38) the complete convergence of \(G\)-fractions; (39) the moment problem.

Chapter V, The Padé table: (40) concept of the Padé table; (41) normal and abnormal table; (42) the exponential function; (43) the Laguerre differential equation; (44) the continued fractions of the Padé table; (45) the convergence question.

Chapter VI, The continued fractions whose elements \(a_\nu\) and \(b_\nu\) are rational functions of \(\nu\); (46) the convergence of these continued fractions; (47) connection with differential equations; (48) the continued fraction with the general term \(\frac{a_\nu|}{| b_\nu}=\frac{a+b\nu|}{| c+d\nu}\); (49) the continued fraction with the general term \(\frac{a_\nu|}{| b_\nu}=\frac{a+b\nu+c\nu^2|}{| d+e\nu}\); (50) the method of Cesàro; (51) the formula of Pincherle.

There is also a bibliography of 185 references referred to in the text, and a table of important formulas. Only those reference are given which are cited in the text.

The following improvements and additions are noted:

In Chapter I, the section 1 has been added as it contains a recapitulation of the essential recurrence relations developed in Vol. I (loc. cit.). In section 3 are derived transformations of a continued fraction into ones whose sequences of approximants are permutations of the original one, also continued fractions with prescribed approximants. Section 7 on the transformation of Bauer and Muir has been considerably altered with respect to illustrative material. Section 8 on further applications and the principal formula of Ramanujan is entirely new.

Chapter II contains many additions on new criteria for the convergence and divergence of continued fractions, since a great deal of new work in this field has been done recently. In section 10 the criterion of Scott-Wall has been added. Section 15 on the convergence criteria of Van Vleck-Jensen and Hamburger-Mall-Wall has been changed and brought up to date. Section 16 on the region of validity of the Ramanujan formula is new, as well as section 17 on the parabola theorem.

Chapter III begins with a new section on general \(C\)-fractions. In sections 20 and 24 are given more remarkable formulas proved by G. N. Watson. Other new sections are section 30 which is concerned with \(J\)-fractions and their applications to polynomials whose zeros have negative real parts, section 31 on new developments in further types of continued fractions corresponding to power series, including Schur fractions and extensions thereof, and section 38 on the convergence of \(G\)-fractions with developments of Hamburger, Hellinger, and Wall. Section 39 on the moment problem has also been changed.

There have been some omissions from the 1929 edition which were deleted to make room for the new additions. Of course, not all new theorems are included. As Professor Perron states, in the case of the new convergence criteria, he included those which were the most useful and which made a comprehensive, well-rounded presentation of the subject.

Above all, he has given a simple and fine account of the theory of continued fractions, all of which is of practical interest as well as theoretical. The book is, as Vol. I, a most masterful work, and another great addition to mathematical literature.

Chapter I, Transformations of continued fractions: (1) recapitulation; (2) zero as numerator of a partial quotient, equivalent continued fractions; (3) continued fractions with given approximants; (4) contraction and extension; (5) equivalence of continued fractions and series; (6) equivalence of continued fractions and products; (7) the transformation of Bauer and Muir; (8) further applications, principal formula of Ramanujan.

Chapter II, Criteria for convergence and divergence: (10) general criteria of Broman, Stern and Scott-Wall; (11) convergence with positive elements; (12) convergence with real elements; (13) irrationality of certain continued fractions; (14) the convergence criterion of Pringsheim; (15) the convergence criterion of van Vleck-Jensen and Hamburger-Mall-Wall; (16) application: region of validity of the Ramanujan formula; (17) some recent criteria, the parabola theorem; (18) periodic continued fractions; (19) limit periodic continued fractions; (20) the equation \[ \frac{x_0}{x_1}=b_0+\frac{a_1|}{| b_1}+\frac{a_2|}{| b_2}+\cdots \] as a consequence of the system of recurrence equations \(x_\nu=b_\nu x_{\nu+1}+a_{\nu+1}x_{\nu+2}\).

Chapter III, Various correspondences of power series with continued fractions: (21) general \(C\)-fractions; (22) square roots; (23) regular \(C\)-fractions; (24) the continued fractions of Gauss, Heine, and related ones; (25) the associated continued fractions; (26) connection between the corresponding and associated continued fraction, some transformations of the corresponding continued fraction; (27) convergence and divergence; (28) convergence of the continued fractions of Gauss, Heine, etc.; (29) a remarkable divergence phenomenon; (30) \(J\)-fractions and their application to polynomials whose zeros have negative real parts; (31) further types of continued fractions that one can associate with power series.

Chapter IV, the continued fractions of Stieltjes: (32) the integral concept of Stieltjes; (33) the corresponding and associated continued fraction of a Stieltjes integral; (34) the theorem of Markoff; (35) the roots of the denominators of the approximants of \(G\)-, \(H\)-, and \(S\)-fractions; (36) the Grommer selection theorem; (37) convergence and analytic character of \(S\)- and \(H\)-fractions; (38) the complete convergence of \(G\)-fractions; (39) the moment problem.

Chapter V, The Padé table: (40) concept of the Padé table; (41) normal and abnormal table; (42) the exponential function; (43) the Laguerre differential equation; (44) the continued fractions of the Padé table; (45) the convergence question.

Chapter VI, The continued fractions whose elements \(a_\nu\) and \(b_\nu\) are rational functions of \(\nu\); (46) the convergence of these continued fractions; (47) connection with differential equations; (48) the continued fraction with the general term \(\frac{a_\nu|}{| b_\nu}=\frac{a+b\nu|}{| c+d\nu}\); (49) the continued fraction with the general term \(\frac{a_\nu|}{| b_\nu}=\frac{a+b\nu+c\nu^2|}{| d+e\nu}\); (50) the method of Cesàro; (51) the formula of Pincherle.

There is also a bibliography of 185 references referred to in the text, and a table of important formulas. Only those reference are given which are cited in the text.

The following improvements and additions are noted:

In Chapter I, the section 1 has been added as it contains a recapitulation of the essential recurrence relations developed in Vol. I (loc. cit.). In section 3 are derived transformations of a continued fraction into ones whose sequences of approximants are permutations of the original one, also continued fractions with prescribed approximants. Section 7 on the transformation of Bauer and Muir has been considerably altered with respect to illustrative material. Section 8 on further applications and the principal formula of Ramanujan is entirely new.

Chapter II contains many additions on new criteria for the convergence and divergence of continued fractions, since a great deal of new work in this field has been done recently. In section 10 the criterion of Scott-Wall has been added. Section 15 on the convergence criteria of Van Vleck-Jensen and Hamburger-Mall-Wall has been changed and brought up to date. Section 16 on the region of validity of the Ramanujan formula is new, as well as section 17 on the parabola theorem.

Chapter III begins with a new section on general \(C\)-fractions. In sections 20 and 24 are given more remarkable formulas proved by G. N. Watson. Other new sections are section 30 which is concerned with \(J\)-fractions and their applications to polynomials whose zeros have negative real parts, section 31 on new developments in further types of continued fractions corresponding to power series, including Schur fractions and extensions thereof, and section 38 on the convergence of \(G\)-fractions with developments of Hamburger, Hellinger, and Wall. Section 39 on the moment problem has also been changed.

There have been some omissions from the 1929 edition which were deleted to make room for the new additions. Of course, not all new theorems are included. As Professor Perron states, in the case of the new convergence criteria, he included those which were the most useful and which made a comprehensive, well-rounded presentation of the subject.

Above all, he has given a simple and fine account of the theory of continued fractions, all of which is of practical interest as well as theoretical. The book is, as Vol. I, a most masterful work, and another great addition to mathematical literature.

Reviewer: Evelyn Frank (Chicago)

### MSC:

30B70 | Continued fractions; complex-analytic aspects |

30-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable |

40A15 | Convergence and divergence of continued fractions |

41A21 | Padé approximation |

11J70 | Continued fractions and generalizations |