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J van der Poorten Macquarie University It is well known that one can obtain explicit continued fraction expansions of for various interesting values of but the details of appropriate constructions are not widely known We provide a reminder of those ID: 23178

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CONTINUED FRACTION EXPANSIONS OF VALUES OF THE EXPONENTIAL FUNCTION AND RELATED FUN WITH CONTINUED FRACTIONS A. J. van der Poorten Macquarie University It is well known that one can obtain explicit continued fraction expansions of for various interesting values of ; but the details of appropriate constructions are not widely known. We provide a reminder of those methods and do that in a way that allows us to mention a number of techniques generally useful in dealing with continued fractions. Moreover, we choose to consider some expansions in aussian integers, allowing us to display some new results and to indicate some generalisations of classical results. 1. Introduction We have fun with continued fractions. That ‘fun’ is intended to demystify a variety of simple facts often disguised in the literature, or proved by turgid methods. In particular we provide some brief notes sketching an explanation for some of the well known continued fraction expansions of for special values of . Our remarks are motivated by suggestions [5] of Jerome Minkus on (aussian integer continued fraction expansions of certain complex values of the exponential function. With his permission we mention and give our proof of his new results and conjectures. Once again we show the power and congeniality of the matrix methods inspired, for this author, by observations of Stark [+,] and reintroduced in [-] and [.]. These methods had of course been used earlier, for example by Walters [++]; indeed, precisely in the context of the exponential function. We make liberal use of Walters’ remarks below. Incidentally, the earliest mention of these matrix methods for continued fractions which Ihave looked at is 0olden [1]; scholarly colleagues also remind me to mention 2rame [3]. Iam occasionally asked who ‘invented’ the matrix approach to continued fractions. Ifeel that the correct answer is that the approach was invented by those who invented matrices. It is after all obvious to anyone who has learned matrix notation that the traditional recurrence formulas for the convergents insist on a matrix formulation. Work supported in part by grants from the Australian Research Council and by a research agreement with Digital %quipment Corporation. 1991Mathematicssubjectclassification : 11A55, 11Y65, 11J,- .ypeset by -.
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Alf van der Poorten 2. First principles Suppose we are given a sequence of arbitrary nonsingular 3 3 matrices, say with complex elements, or even polynomials over 4, ,... with the property that the two limits lim /C and lim /D are equal to one another. Suppose the common limit is . Then we will say that the sequence yields an expansion for We claim that a sequence of 3 3 matrices yielding an expansion for yields a formal continued fraction expansion of . Indeed, there is no loss of generality in arranging that the matrices be unimodular, since multiplying each matrix of the sequence by some complex 4 , does not change , and then those matrices have a decomposition as a product of elementary unimodular matrices corresponding to a continued fraction expansion. We remind the reader that5 6 continued fraction is an expression of the shape which one denotes in a space8saving 9at notation by ,a ,a ,a ,...... :verything one needs now follows from the correspondence whereby +, +, +, for 4,,+,3, ...... entails 4[ ,a ,......,a ] for 4,,+,3, ...... and conversely, up to the ambiguities in choosing the and given just their quotients /q . If the all are positive integers for + it su;ces that and be chosen relatively prime. Our remark is now no more than the observation that every unimodular matrix has decompositions corresponding, in the sense just indicated, to a continued fraction expansion; and this is not intended to be more than the remark that a unimodular matrix can be written as a product of elementary row transformations.
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Fun with continued fractions 3. The exponential function It is not too di;cult to verify directly that the continued fraction expansion of corresponds, in the sense just described, to the sequence of matrices (+= =0 (3 7+=7 (3 7+= (3 7+= (3 7+= Indeed, 4( += +1 2( +1) shows that the formal power series lim /C and lim /D coincide. One sees readily that =4 = and =4 = and one confirms more laboriously that (3 7 +=? 4 +7 3( 7+= .(3 7+= ... and (3 7 +=? 4 3( 7+= .(3 7+= ... 2or further details see Walters [++]. Since 73 3 7+ 7+ 3 ++ +, +, ++ +, we immediately obtain the expansion 4[+ ,... Aoting that (3= +, ,+ +, +, +, we have the familiar expansion +4[+ ,... ]4[ =1 Here the overline indicates quasi8periodicity with the variable sequentially taking the values +, 3, ... with each repetition of the quasi8period. 4. Digression Fun with matrices :veryone knows about the transpose ac bd of a matrix ab cd
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Alf van der Poorten but, correctly, less well known is the false transpose db ca Writing 01 10 , we see that JAJ is appropriately called the double transpose of and then 4( JAJ readily yields such properties of the false transpose as ( AB We already know that continued fraction expansions correspond to matrix products +, +, +, In the sequel it will be convenient to define +, ++ and ++ ,+ and to note that JL +, Then, observing that , we see that continued fraction expansions also cor8 respond to so8called RL –sequences in the sense that +, +, +, JL 5. Some expansions in Gaussian integers We notice the decomposition (3 7+=73 (3 7+= (3 7+= (3 7+= ++ ,+ +, ia +7 ,+ +7 ++ whence alternatively it also equals +73 +7 ++ ,+ +, ia ++ ,+ Of course one need do no more than check a claim of this kind, yet it warrants explanation. The first decomposition is a more or less natural row decomposition; Cynical remark to be omitted by publications that believe that mathematics cannot be fun: 4owever, it seems that 215 of undergraduate students cannot distinguish the false transpose and the transpose; if students are explicitly warned to avoid this blunder this number increases to 355. 6ne also must not tell students that the double transpose has the congenial property that the double transpose of a product is the product of the double transposes without any bothersome reversal of order.
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Fun with continued fractions the fun way to obtain the second from it is to note that the conjugate of the given matrix is its false transpose. Aext we observe that +7 ++ 73 ib bb +73 +7 ++ +, 7+ + 2inally let 43 in (+= and take its matrices in threes to obtain =0 RL ki 1+ (3 +2) R, which corresponds to Theorem 1. (Minkus [5]= (D= 4[+ hi , +7 i, i, (D 73= i, 3] =0 The experienced reader will recognise, however, that hindsight assisted the quoted calculation; it did5 it is a formula demonstrated by less friendly methods in [5]. Might one have discovered (D= spontaneouslyE To that end consider a continued fraction expansion of coth3 4( 7+= += Our remarks show that coth3 corresponds to ++ =0 (3 7+=73 (3 7+= (3 7+= (3 7+= and because ++ (3 7+=73 (3 7+= (3 7+= (3 7+= 43 (3 7+=7 (3 7+= zz 43 (3 7+= /z +, ++ this immediately yields a formula known to Fambert (see [1]=, that (1= 7+ 4[ (3 7+= /z =0 Taking this as our starting point, we seek to recover an expansion for by mul8 tiplying the continued fraction expansion (1= by the matrix ++ ,+ 3, +, ++
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Alf van der Poorten We do that by following suggestions of Ganey [H], and to that end, recalling (1=, interpret our task as being to transform the matrix product 3, 11 so that it once again corresponds to a continued fraction expansion, to wit to an RL –sequence consisting just of ((aussian= integer powers of and of .We do that by applying appropriate transition formulae to move the oIending matrix through the RL 8sequence until it disappears in the ... on the right. This will be less complex once we see that we may write +, 11 ,+ RL 11 The trick is to notice that multiplying a continued fraction [ a,b,c,d,e,... ]by yields [ ax,b/x,cx,d/x,ex,... ]. We have multiplied by and have neglected , taking advantage of the fact that the 4 sign can only mean that the two sequences of matrices yield the same continued fraction expansion. We set +7 ,+ ; so its conjugate is ,+ We apply the transition formulJ 1+ 1+ 1+ LA 1+ RA RL LR LR LA RL RA and their conjugates to obtain sequentially the transductions RL RL 4+3 9+9 11 6+5 RL ... ... and then RL LR 1+ 4+4 RL 8+10 1+9 LR 7+5 ... ... 7y the way, these require only a few calculations, since they are mostly transposes and or false transposes one of the other.
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Fun with continued fractions telling us that ,+ RL 11 ,+ RL RLR RL 1+9 LR 2+ RL 1+ RL 1+6 which corresponds to the continued fraction expansion i, 37 i,i, ,i, i, ,i, +7 i, i,i, i, i, +76 i,... The trouble is that this is not obviously the same result as (D=. 6. Digression Fun with continued fractions The curious calculation 4, 7 / +7( += / α/ += 4+ 7 + += +4 +7 / 4,7+ / says that ...,a, b, ]4 4[ ...,a, ,b, ]4[ ...,a ,b Its converse, A, ,B, K] 4 [ 7+ K] is just a special case. 6lmost identical trickery shows that also ...,a, b, ]4 4[ ...,a, ,b, ]4[ ...,a 7+ ,b 7+ with converse A, ,B, K] 4 [ 7+ K] The imaginary analogue of the first formula is just (5= [ a,b, ]4 ia , ib , i ]4 ia ib ,i ]4[ i, i,b i, and its converse conjugate is, A,i,B, K] 4 [ i,B i, K]
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Alf van der Poorten 7. Related results With some care the reader will succeed in showing that such transductions do provide a systematic fairly uncontrived method for obtaining the expansion (D=. Specifically, i, 37 i,i, ,i, i, ,i, +7 i, i,i, i, i, +76 i,... plainly commences [+ +7 i,... ]. That’s good because we’re claiming to be rediscovering the expansion [+ hi , +7 i, i, (D 73= i, 3] =0 Aext 37 i,i, ,i,... ]4[ i,i,... and i, i, +7 i, i,... ]4[3 i, +7 i,i, +7 i, i,... ]4[3 i, i,... 2ortunately [+ i,i, i, ]4[3 i, i, H7 i, 4[3 i, ,i, H7 i, ]4[3 i, i, i, 4[3 i, +7 i, , Moreover i, +76 i, ]4[ +7 i, i, 4[ ,i, i, ]4[ i, i, and we’ve pretty well completed the second quasi8period. Of course, if one were set on obtaining (D= one would avoid tedious applications of this ‘fun with continued fractions’ by contriving more appropriate transition formulJ, but that would miss the point. Our instancing Ganey’s remark that linear fractional transformations of a continued fraction may be eIected by one or more finite state transductions of a correspond8 ing RL –sequence should readily yield a proof of a generalisation, as suggested by Minkus, of a theorem of Hurwitz [D] according to which if has a continued fraction expansion of the shape ,a ,...,a ,f ,...,f =] =0 with polynomials ... taking integer values at nonnegative integers then each linear fractional transformation ( a c =of with ab cd (F = has a continued fraction expansion of the same form. The generalisation replaces by the ring of integers of (or indeed, by any order of= an algebraic number field. Our hint of the argument relies upon observing that a unimodular transformation of changes only the entries preceding its quasi8period, and our example illustration that multiplication by matrices 01 or 10 corresponds to finite state transductions of an RL –sequence should easily allow a verification that it inter alia preserves the property of present interest. 6s an example, we mention the continued fraction expansion
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Fun with continued fractions Theorem 2. (6= 73 +3 7+6 71] =0 We might obtain this from (D= by multiplying the corresponding matrix products by . However, it happens that first principles are more convenient and we note that 73 ia aa aia 73 ia 73 ia ,+ +, ih +3 ,+ 1, with 3 7+4 ; and also 4 +, +3 ,+ +, ih ,+ by false transposition. Moreover 1, bib 73 ib 73 ib +, 73= ,+ So, taking the matrices in threes yields =0 (3 7+=73 7+ 7+ (3 7+= =0 ki 2((6 +3)+2) (3 +2) [+ k, 3(6 7D= 73] =0 We now employ ‘fun with continued fractions’ to obtain (6=. 6n expansion equiva8 lent to (6= is conjectured in [5]. 8. Concluding remarks It is useful to recall the remarks of Walters [++] and to apply those methods to re8 trieving complex analogues suggested by Minkus [5] of classical results of :uler and Stieltjes; see Merron [6]. Our methods also provide an opportunity to mention ideas of Ganey [H] and to demonstrate the utility of a number of simple transformations of continued fractions that we have labeled ‘fun with ... ’. It may not be evident from our remarks — which appear to cite just isolated wonders — just why /q and /q , and i/q and i/q , should have continued fraction expansions of Hurwitz type whilst other powers of seemingly do not (see, for example, [+]=. 6 review of our examples reveals, however, that taking the matrices (3 7+= (3 7+= (3 7+= (3 7+= three at a time if 4 3 (or just one at a time if 4 += gives . times a unimodular matrix (respectively, a unimodular matrix=, whilst, as [+] shows for the case 4 D, apparently no collecting the matrices for other values of yields products of unimodular matrices multiplied just by a constant.
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1- Alf van der Poorten References [+] J. F. Oavison, ‘6n algorithm for the continued fraction of l/m ’, Congressus Numerantium XXII Proc. Eighth Manitoba Conference on Numerical Mathematics and Computing (+H-.=, +6H–+-H [3] J. S. 2rame, ‘The solution of equations by continued fractions’, Amer. Math. Monthly 60 (+H5D=, 3HD–D,5 [D] 6. Hurwitz, Uber die 0ettenbruche, deren Teilnenner arithmetische Gei8 hen bilden’, ierteljahrsschrift der naturforschenden Gesellschaft in Z urich Jahrg. 41 (+.-6= [1] 0jell 0olden, ‘Continued fractions and linear substitutions’, Arch. Math. Naturvid. 50 (+H1H=, +1+–+H6 [5] Jerome Minkus, ‘(aussian integer continued fraction expansions for and ’, manuscript, +HH+ [6] Oskar Merron, Die Lehre von den Kettenbr uchen (Chelsea reprint of +H3H edition= D3, +D6–+D. [-] 6. J. van der Moorten, ‘6n introduction to continued fractions’, in Dio- phantine Analysis , FMS Fecture Aotes in Math. 10 , ed. J. H. Foxton and 6. J. van der Moorten, (Cambridge University Mress +H.6=, HH–+D.. [.] 6. J. van der Moorten, ‘2ractions of the period of the continued fraction expansion of quadratic integers’, Bull. Austral. Math. Soc. 44 (+HH+=, +55 +6H [H] (. A. Ganey, ‘On continued fractions and finite automata’, Math. Ann. 206 (+H-D=, 365–3.D [+,] H. M. Stark, An introduction to number theory (MIT Mress +H-.= [++] G. 2. C. Walters, ‘6lternative derivation of some regular continued frac8 tions’, J. Austral. Math. Soc. (+H6.=, 3,5–3+3 Alfred J. van der Poorten Centre for Number Theory Research Mac0uarie 1niversity NS2 2134 Australia

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