Guide Orthogonal functions, moment theory, and continued fractions: theory and applications

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Gcsarnlnelte Werke 6 Karlin ancl. A l a n y sevuer qutwcLng proce. URSS 31 , 1! N Canada 13 , Nevai, A ncw class of orthogonal polynomzals, Proc. O n a speczal s y s t e m of orthogonal polynomzals, Dissertation, Stanford Univ.. Pollaczck, S u r u n e gine'ralisation d e s polynomes d e Legen,dre, C. Paris , Pollaczek, S u r m e famille de polyn6mes orthogonaux a quatre paramztres, C. Rees, Ellzptzc orth,ogonal polynomzals, Duke Math. Lordon hlath. Sclliah A s y s t e m of orthogonal polynomials, Tech. Report 13, January, Sherman O n t h e nu7nerntzons of t h e conwerqcnts of th,e Stie-ltjes contznued fmctaons, Trans.

Szego, Orthogonal Polynomials, Amer. RI, 1st ed. O n certazn speczal sets of orthogonal polynonzzals, Proc. Haimo ctiitor, Southern Ill. Press, Edwardsville, , 3 11, reprinted in Gabor Szego. Collected Papers, volume 3, , Birkhauscr, Boston. Pures Appl. Translation of Russian paper published in ; reprinted in Oeuvres de P.

Sonin, St. Petersburg, ; reprint,ed Chelsea, N. Tcheby- chef, Tome Petersburg, ; reprinted Chelsea. Wimp, Explp2lczt fonnu1a. In this paper we give a survey of all 58 problems, emphasizing the mathematical activity generated by them since Between the years and Ramanujan submitted a total of 58 problems, several with multiple parts, to the Journal of the Indian Mathematical Society.

For t,he first five. Ebr others, significant amounts of hard analysis are necessary to effect solutioiis. Every problerri is either interesting or curious in some way. All 58 problems can be found in Rarnarnljan's Collected Papers [, pp. This was likely the practice followed by the editors of the Journal of t h e Indian Mathernatical Societ?

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The publication of the Collected Papers in brought Ramanujan's problems to a wider matherriatical audience. Several problems have become quite farnous and have attracted the attention of many mathematicians. Some problems have spawned a p1et hora of papers, nlany cont,ainirig generalizations or analogues. It Mathematics Subject Classzjication. Primary , ; Secondary llD25, llD99, 40A In referring to these problcms, we follow the numbering given in the.

Solutions of Equations: Some of Ramam? However, we have t,aken the lib- crtl- of replacing occasional archaic spelling by more contemporary spelling. After the number of the question, t,he volume and page nuniber s where the problem first appeared in the Journal of the I n d i a n Mathe- rrlntical Society. However, if a solution was pliblished after the publication of the Collected Papers in , then we record it as a separate item in the bibliography. AIany of the problenls. Normally in such a case. JIRIS 3. S h o w that it is possible t o solve the rquatsons.

Implicit assump- tions were made in Ramanujan's solution, and thus it should be emphasized that 2. For a sketch of Ramanujan's solution. Bhargava [ 2 5 ]. Another derivation of the general solution for 2. Naraniengar []. The more general system 2.


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  8. We quote the discussion of 2. In , J. Sylvester [ 1 9 6 ] ,[ 1 9 7 ] ,[, pp. Thus, the numbers XI,. Sylvester's x l ,. For a contemporary account. Kung and G. Rota []. The solutions comprise 25 values for both x and y. As pointed out in [23, pp. Ramariujaa's published solution. Then [23, pp. Ramanujan then listed the values of s as. However, observe that the third member of this set may be derived from the second member by replacing p by p2. Lastly, the sixth arises from the fifth when p is replaced by p2.

    Ramanujan missed the values. Solve completely. Rarnamijan's published solution in the. Journal of the Indian Mathematical Society is correct,. However, there are four sign errors in the solution published in his Collected Papers [, pp. The equations in 2. This polynomial factors over d m int,o a quadratic polynomial c2 T - u and t,wo cubic -.

    Each of t,hese two cubic: polynomials has two sign errors. The formulation of Question indicates t,hat the three examples can be deduced from the solutiorls of 2. However, in his published solution, Ramanujan did not do this but established each ident,it,y ad hoc. In his so1ut ionto Question Rao [] derived each of the examples, a - c , from t,he general solut,ion of 2.

    On pages in his second notebook []. Rarnanujm offtm a more extensive version of Question [23, pp. By taking successive square roots in 2. The 2" different sequences of nested radicals correspond t o the eight root,s of thc octic polynomial arising from 2. Of course, one must determine the values of a for which the eight infinite sequences convergc.

    In [23, pp. For example, it was indicated in [23, p. In our discussion of Quest,iorl For a part,icular value of a , one can numerically check which infinite sequence of nested radicals corresponds to a given root. In general, for the t,wo infinite sequences of nested radicals arising from the two roots of the quadratic polynomial.

    On pages in his second not,ebook. Rarnanujan made thcse general identifications [23, p. In [23, p. There is also a very brief discussion of the system 2. Cajori's book [53, pp. Question is obviously an analogue of Question The first complete solu- tion to this problem was given by XI.

    Rao [] in He also solved Question The second solution was given in by G. Watson []. A considerably shorter solution was found by A. Salam [I in Question can also be found in Ramanujan's third notebook [, vol. Solve i n posztive rational numbers. Since the published solution by J.


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    Swaminarayan arid R. Vythynathaswarny is short and elegant and was not published in Rarnanujan's Collected Papers, we reproduce it here. The values of i and ii are 3 and 4, respectively. Rarnanujan's solutions in volume 4 [, p. A note by T. Vijajaraghavan at the end of Appendix 1 in Rarnanujan's Collected Papers [, p. This note was considerably amplified in a letter from Vijayaraghavan t o B. Herschfeld [96] also proved Vijayaragliavan's criterion.

    Orthogonal Functions

    In a 1at erquestion s u b i n i t t d t,o the Jour- nal of the Indian Mathenmtzcul Society, Vijayaragllavan [] claimed a stronger theorem for t,lle convergence of t,,, which he said was best possible. However, the problenl is flawed, and no correction or solution was evident,ly publishrd. Problem in P61ya and Szego's book [, pp. Tllron []. Convergence criteria for certain iiifiriit,e nested radicals of pth roots liaw been given by J. Borwein and G. The values of i and ii appear as examples for a gcncral tlicwrcm of Rarrianujan on ncsted radicals in Section 4 of Chapter 12 in his second notebook [21, p.

    Show thof. Part i appears on the last page of Rarnanujan's second notebook [, p. Of course, one can establish each of the nine identities in the four preceding problems by taklng the appropriate powcr of each side of each equality. However, such a proof provides no insight whatsoever into such an equality. Both the left and right sides of each of the equalities are units in sornt. Although Ramanujan never used the term unzt, and probably did not formally know what a unit was, he evidently rralized their funcldinental properties.

    He then recognized that taking certain powers of units often led to elegant identities. Berndt, Chan. As another example, for any real number a ,. In both the original formulation and the Collected Papers [, p. The inversion likcly was not irnrnediat,ely discovered, as the first solution t,o the corrected problems was not given until by S.

    Srinivasan []. In Kothantlararnan []. On page in his second notebook [], Ramanujan stated two general radical identities involving cube roots. For example,. See [23, pp. Many papers have been written on simplifying radicals. In particular. Zippel [] and T. Osler []. Landau []. Ranganatha Aiyar, R. Karve, G. Kamtekar, L. Datta, and L. Subramanyam is clever and short, and so we give it. Nore recently, another solution was given by V. Thiruvenkatachar and K. It would seem desirable to have a briefer, more elegant solution, but perhaps this is not possible.

    This problem gives a two-parameter family of solutions to Euler's diophantine ccluation. Equation 5. The equality 5. Hooley employed 5. First, he gave t,wo families of solut,iorls which include 5. According to Dickson , p. Euler [77]. Rarnanlijan, in his t,hird notebook , vol. Both Hardy [92, p. There arc. Further references to gerlcral solutions of 5. However, C. Sdndor's paper also cont,ains a summary of further especially recent progress on the problem of finding all integral solutions t,o 5. The solutions in both volumes 13 and 14 of the Journal of the Indian Mathr- nmtacal Soczefy are by S.

    In volume The second family contains as spcclal cases all examples in the first column above, and the second example in the second column. In volurne Alitra established the general rational solution of 5. Solve In tnfegers. Clearly, 5. Observe t,ha. As in the previous problem, the solut,ions in both volumes 13 and 14 of the Journal of the Indian Mathematical Society are by S. Venkata R a n a Ayl-ar [15] found other met hods for examining Quest ions In an earlier paper [16]. Kao foimd further solntioils to 5. Some very special solut,ions to 5.

    Hirschhorn [97]. In contrast to Question , Qucst,ion has at,tractcd corisiderable att,mtioii in t,he literature. Carrnichael [55] raised the problem of finding t,he general integral solution of 5. Bradley [44] asked if t,he om--paramctcr family. Carrnicliael, Bradley, and hlitra. In response, hlahler [] con5tr1lcted the family of solutions 5. Lchmer [] showed how to construct infinite sequmces of solutions to 5. Supplenienting the work of Lehmer, H.

    Godwin [86]and V. Podsypariin [] found further families of solutions. Finding a complete description of all zntegral solutions to 5. It 1s natural to generalize Question by asking which positive integers n are the sum of three cubes. For early references, consult Dickson's Hzstory [73, pp.

    It is conjectured that, if C is the set of all integers representable as a sum of three cubes. Vaughan [] who showed that. Breinrler [45]. Up to the present time. Tsuruoka, and H. Sekigawa [ I l l ]. Fznd other values. The equation. Journal of the Indian Mathematical Society. Sanjana and T. Trivedi in volume 5 offers a systern- atic derivation of the five given solutions but does not show that these are the only solutions as the authors make clear.

    Unaware of Rarnanujan's problem. Ljunggren [] proposed the same problem in Nagell's name is att,ached to 5. However, since his paper was writt,en in Norwegian in a relatively obscure Norwegian journal [].

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    Skolem, S. Chowla, and D. Lewis [] used Skolem's p-adic nlethod to prove Ramanujan's conjecture in , Nagell repub- lished his proof in English in a more prominent journal [] and pointed out that he had solved the problem much earlier than had Skolem. Chowla, and Lewis. Shapiro and D. Slotnik [],in t,heir work on error correcting codes. Significantly improving the methods in [], Chowla, hI. Dunton, and Lewis [66] gave in another proof of Ranxmujan's conjecture. In , using the arithmetic of certain cubic fields, L. Hossain []:W. Johnson []. Turnwald [], and P.

    Buiidschuh [52] are some of the aut,hors who have also found proofs. In his book []. Instruct,ive surveys of all known proofs with a p1et hoi-aof references have been writt,eri by E. Coheii [68] and A. Rarnasamy []. We confine further remarks to the generalued Rnmnnujan-Nagell equation. Browkiri and A. Furthermore: if any solut,ion exist,s. They also pointed out that, in fact, in , they [48] had completely solved a diophantine equation, easily shown to be equivalent t,o 5. Apiiry [8]showed t,hat. More recently. Cohn [70] has shown that for 46 values of D 5 , 5.

    Hasse [94] and F. Beukers [37] have written excellent surveys on 5. Beukers' thesis [37] and his two papers [38] and [39] contain substantial new results as well. In [38]. Beukers studied 5. Le has written several papers on 5. In [I and [], Le considered the case of an odd prime p and showed that if m a x D , p is sufficiently large which is made precise , then N D ,p 5 3.

    Find other values. Rao wrote that the question was originally posed by H. Brocard [46] in , and then again in [47]. The problem was also nlentioned on t,he Norwegian radio program, Verdt aa vit,e Worth knowing. Dickson [73, p. G6rardin [83] remarked that, if further solutions of.

    Overholt [I proved that 5. To state the weak form of Szpiro's conlecturc. For a furtlier discussion of Szpiro's conjecture. Lang [, pp. The more general q u a t i o n. Dabrowski [72] in By a short. He also showed that. The problem of finding solutions to 5. Guy's book [89, pp. Krislmaswami Aiyangar [2] later posed a problem giving analogues.

    In a subsequent paper [3], he established theorems generalizing the results in his problem. Cheii [60] has also established extensions of ii and iii. Part iii appeared on thc M'illiam L o w d Putnam exam in [85]. JIhIS 8. The first is by S. Chowla [63] and usc5 two thcorerns of E.

    Landau The second, by G. Esterinam [76]. J liin! Question is concerned with the approximation of irrational nunlbers by rational riumbers. The proposed equalities should be compared with Hurwitz's theorem [, p. To sec how Question rclates t o Hurwitz's theorem. A partial solution was given by A. Krishnaswarny Aiyangar [I]. Vijayaraghavan and G. Express A. Question appears in Rarnanujan's second notebook [, p. In Berndt's book [23, pp. The two published solutions in the Journal of the Indian Mathematical Society are more complicated.

    In fact, Question is a special case of Gauss' theory of composition of binary quadratic forms [69, p. Suppose that Q l x,y and Q2 x,g are integral, positive definite quadratic forms of discriminant d. Then if Q:j is a form of discriminant d,. Show that. Wilkinson observed in volume 7, Question gives the value of Weber's class invariant f m [, p. Recall [24, p. Weber [] calculated a total of class invariants, or the monic irreducible polynomials satisfied by them, for the primary purpose of generating Hilbert class fields.

    Without knowledge of Weber's work, Ramanujan calculated a total of class invariants, or the monic irreducible polynomials satisfied by thern. Not sur- prisingly, many of these had also been calculated by Weber. After arriving in England, he learned of Weber's work, and so when he wrote his fa- mous paper on modular equations, class invariants, and approximat,ions to. Since Ramanujan did not supply any proofs in his paper [], [, pp. After Watson's work, a total of 18 of Ra- manujan's class invariants remained to be verified up to recent times.

    Using four distinct methods, Berndt, H. Chan, and L. Zhang complet,ed the verification of Ramanujan's class invariarits in two papers [28], [30]. This work can also be found in Chapter 34 of Berndt's book [24]. Not all of Watson's verifications of Ramanujan's class invariants are rigorous. Zhang [], [] has given rigorous derivations of the invariants calculated by Watson by means of his "empirical method. Show that the roots of the equations. Thus, Ramanujan's problem can be reduced to solving two quintic polynomials. It is doubtful that Ramanujan had actually solved these two quintic polynomials.

    It is unclear why Ramanujan introduced these two linear factors. The class equation for G7y is not found in Weber's book []. This result can be deduced from equivalent results due to R. Russell [] and later by Watson []. See also Berndt's book [24, pp. Watson [] furthermore pointed out that 5. Young in a paper devoted to the general problem of explicitly finding solutions to solv- able quintic polynomials and to working out many examples, the first of which is 5. Young remarks that 5. Greenhill, who had also studied 5. A few years later, A. Cayley [58] considerably simpli- fied Young's calculations.

    Since the solutions of 5. The work of Ramanujan and Watson is summarized in a paper by S. Chowla [65]. Dummit [75] and V. Galkin and 0. Kozyrev [82] have also examined 5. For example, the Galois groups of the Hilbert class fields over Q are dihedral groups of order Unaware of the work of Russell, Watson, and others, Galkin and Kozyrev rederived the class equation 5.

    Prove that 5. The second and third solutions by N. Durai Rajan and M. Bhimasena Rao, respectively, are correct. Equality 5. Part ii is given as Example iv in Section 7 of Chapter 17 in Ramanujan's second notebook [22, p. After Ramanujan, set. In particular [22, p. In view of 5. Part i is found in the second notebook, where it is an example in the aforementioned Section 23 [22, p. In Berndt's book [22, p. Part i? We shorten Ramanujan's formulation of Question Examine the correctness of the following results:.

    The identities 5. As the word- ing of the problem intimates. Ramanujan had not proved these identities when he submitted them. Rogers [] in Ramanujan had evidently stated them in one of his initial letters to Hardy, for Hardy [, p. Hardy in- formed several mathematicians about 5. The "unproved" identities became well-known, and P. MacMahon stated them without proof and devoted an entire chapter to them in volume 2 of his treatise Combznatory Analyszs [].

    Ramanujan was perusing old volumes of the Proceedzngs of the London Mathematzcal Soczety and accidently came across Rogers' paper []. Shortly thereafter, Rainanujan [l7O], [, pp. There now exist many proofs, which have been classified and discussed by G. Andrews in a very informative paper [6].

    The Rogers-Ramanujan identities are recorded as Entries 38 i. It is ironic that they are. Askey [22, pp. Entry 7 is a limiting case of ViTatson'stransformation for 8P7, and, in view of the complexity of Entry 7. The Rogers-Ramanujan identities have interesting combinatorial interpreta- tions.

    The first. The second implies that: The number of partitions of a positive integer n into distinct parts. For further history and information on the Rogers-Ramanujan identities, see Andrews' paper [6] and books [4, pp. Using this notation and the notation 5. The deduction of 6. Ramanujan evidently never had a rigorous proof of 6.

    A direct proof depending on Cauchy's theorem will be found in Mr Hardy's note which follows this paper. The solution of Question by N. Durai Rajan in volume 7 employs a partial fraction decomposition of the integrand. Askey [9]. For references to extensions and further related work, see the aforemen- tioned papers by Askey. A nice discussion of 6. The evaluation 6. See Berndt's book [22, p. This is a very beautiful result which has been greatly generalized by Berndt and R. Evans [34] in the following theorem. The proof by N. Durai Rajan and "Zero" in volume 10 is longer than the proof by Berndt and Evans of the more general result.

    Question can be found on page of Ramanujan's third notebook [22, pp.

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    The arithmetical nature of C is unknown, and this is a long outstanding, famous problem. The function. In fact, ii is a special case of Entry 16 of Chapter 9 in the second notebook, which we can write in the form. For i , observe that an integration by parts gives. Prudnikov, Yu. Marichev [, p. The solution by K. Sanjana in volume 3 proceeds similarly. Part i is Example 2 in Section 13 of Chapter 10 in Ramanujan's second notebook. For values of other integrals akin to that on the right side of 6. Ramanujan's solution involving double integrals and Fourier transforms in vol- ume 5 is very short and clever.

    His idea is examined in greater generality in his paper [, Sect. The identity 6. It is also found in Chapter 13 of Ramanujan's second notebook [21, p. In both the letter and notebooks, 6. If n is any positive odd integer, show that. The proofs of 6. Wilkinson in volume 8 employ contour integration, but his proof of 6.

    In volume 16, Chowla showed that a much shorter proof of 6. Related results can be found in Section 22 of Chapter 14 in Ramanujan's second notebook [21, pp. In particular [21, p. A complete discussion of these results can be found in Ramanujan's paper [], [, pp.

    Wilkinson's solution in volume 8 uses contour integration. Show, without using calculus, that. Question is the first question submitted by Ramanujan t o the Journal of the Indian Mathematical Society. In fact, the first two questions were communicated by P. The equality 7. Ramanujan was not the first to pose 7. The problem can also be found in G. Chrystal's book [67, p. Show that Euler's constant, namely [the limit ofl. However, the problem appears as Entry 16 in Chapter 8 in Ramanujan's second notebook, and a proof can be found in [20, p.

    The original formulation of ii is incorrect. Ramanujan recorded further identities for tan-' sums in Chapter 2 of his second notebook [20, pp. The sign of the right side of ii is incorrect in the original formulation. Part i can be found in Ramanujan's third notebook [23, pp. A companion result is given on the same page [23, p.

    The original formulation contains an obvious misprint. In volume 9, K. Madhava, M. Kewalramani, N. Durairajan, and S. Venkatachala Aiyar offered the generalization. Question coincides with Entry ll iii in Chapter 13 of Ramanujan's second notebook [21, p. Although no solutions were published, 7. Its home is in the theory of elliptic functions, as Ramanujan himself indicated when he proved 7. However, Ramanujan was not the first to establish 7.

    The first mathematician known to us to have proved 7. Schlomilch [], [I in Although not explicitly stated, 7.

    Hurwitz [loll, [I in his thesis in Others who discovered 7. Krishnamachari [], S. Malurkar [],M. Rao and M. Ayyar [],H. Sandham [],and C. Ling []. The discovery of 7. More precisely, 7. The discussion in Berndt's book , p. Schlijmilch and several others cited above, in fact, proved a more general for- mula than 7. Equality 7. There is a further generalization. Observe, by 7. The first proof of 7. Ramanujan also discovered 7.

    Apparently the first proof of 7. Glaisher [84] in Many proofs of 7. Analogues of 7. See Berndt's book , p. Most of the proofs of 7. However, Berndt [19] has shown that all of the identities discussed above, and others as well, can be derived from one general modular transformation formula for a large class of functions generalizing the logarithm of the Dedekind eta-function. If n is a multiple of 4, excluding 0, show that. Although not stated, n must be a positive integer.

    In fact, 7. Other proofs of 7. Narasimha Murthy Rao []. Question has a beautiful generalization. Proofs have also been given by Nanjundiah [] and Berndt [19, p. Such results were also found by Malurkar [], Nanjundiah [], and Berndt [19]. These formulas, 7. The evaluations i and ii are Examples v and vi , respectively, in Section 32 of Chapter 9 in Ramanujan's second notebook [20, p.

    These and several other evaluations of this sort were derived in [20, pp. In turn, these corollaries may be deduced from Whipple's quadratic transformation for a well poised generalized hypergeometric function 3F2. The evaluation 7. The origina11 formulation of i is incorrect, as pointed out by M.

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    Rao in his solution. Ramanlian made the same mistake when he recorded it as Example ii in Section 8 of Chapter 9 in his second notebook. Part ii in Question is given as Example i in that same section [20, p. For other examples of this sort, see Catalan's paper [56]. We have slightly reformulated the next question. Sum the series. The sum of the series is.

    The two published proofs are elementary.

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    The first, by K. Rama Aiyar, uses Euler's elementary identity. In both the original formulation and Ramanujan's Collected Papers [, p. Question gives the values of the Rogers-Ramanujan continued fraction. As the name suggests, R q was first studied by L. Rogers in [].

    In his first letter t o Hardy, Ramanujan stated both i and ii [, p. In both letters, Ramanujan wrote that R eP"fi "can be exactly found if n be any positive rational quantity" [, p. We have quoted from the first letter; the statement in the second letter is similar but is omitted from the excerpts of Ramanujan's letters in the Collected Papers. In both his first notebook [, p. On page of [], Ramanujan planned to list fourteen values, but only three are actually given.

    Since the lost notebook was written in the last year of his life, his illness and subsequent death obviously prevented him from determining the values he intended to compute. In several papers []-[], K. Ramanathan derived some of Ramanujan's values for R q. All of the values in the first notebook were systematically computed by Berndt and Chan in [26], while all the values, including the eleven omitted values, were established by Berndt, Chan, and Zhang in [29].

    More precisely, they used modular equations to derive some general formulas for R e - " A in terms of class invariants. Thus, if the requisite class invariants are known, R e-"fi can be determined exactly. A brief expository account of this work can be found in Prove that. Madhava offered an informative discussion of this problem in volume 8. His solution employs Prym's identity []for the incomplete gamma function. Question is a special case of the first part of Entry 43 in Chapter 12 in Ramanujan's second notebook [21, p. In turn, Entry 43 is a corollary of Entry 42 [21, p.

    Lastly, we remark that Question is closely related to a result communicated by Ramanujan in his first letter to Hardy [, p. However, the first rigorous proof is due to Jacobi []. If we set. Thus f and g are self-reciprocal with respect to Fourier sine transforms. The first solution t o Question was given by E. Phillips In his proof of i , Phillips used 8. Stieltjes [].

    Lange [I independently found a similar solution. It is curious that immediately following Phillips' paper is the obituary of M. Hill, the first mathematician whom Ramanujan wrote from India [36, pp. Hill did not fully understand Ra- manujan's work, and there is no mention of Ramanujan in the obituary. Bailey [17]generalized 8. He gave a simpler proof of this result in a later paper [18]. View via Publisher. Open Access. Save to Library. Create Alert. Share This Paper.

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    Choque Rivero , Sergey M. References Publications referenced by this paper. Recherches sur les fractions continues T.