Continued fractions and Hardy sums

For the classical Dedekind sum d(a, c), Rademacher and Grosswald raised two questions: (1) Is \(\{(a/c,d(a,c))\ \vert \ a/c \in {\mathbf{Q}}^{{\ast}}\}\) dense in R 2? (2) Is \(\{d(a,c)\ \vert \ a/c \in {\mathbf{Q}}^{{\ast}}\}\) dense in R? Using the theory of continued fractions, Hickerson answered these questions affirmatively. In function fields, there exists a Dedekind sum s(a, c) (see Sect. 4) similar to d(a, c). Using continued fractions, we answer the analogous problems for s(a, c).

where 0 ∈ Z is such that − 0 ∈ [0 1), and ∈ N for ≥ 1. As is well-known, the regular continued fraction (RCF) expansion of is finite if and only if ∈ Q. In this case there are two possible expansions, otherwise the expansion is unique. Apart from the RCF expansion there are very many other continued fraction expansions: the continued fraction expansion to the nearest integer, Nakada’s α-expansions, Bosma’s optimal expansion . . . in fact too many to mention (see [6] and [3] for some background information). One particular expansion, which attracted no attention whatsoever, and which is quite different from the continued fraction expansions mentioned above, is Denjoy’s canonical continued fraction expansion (see [2], or [1], p. 275–6 for the original paper by Denjoy). In [2], Denjoy stated that every real number has continued fraction expansions of the form

Previous article Next article Numerical Evaluation of Continued FractionsG. BlanchG. Blanchhttps://doi.org/10.1137/1006092PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Leo A. Aroian, Continued fractions for the incomplete Beta function, Ann. Math. Statistics, 12 (1941), 218–223 MR0005193 0025.31901 CrossrefGoogle Scholar[2] Friedrich L. Bauer, The quotient-difference and epsilon algorithms, Edited by R. E. Langer. Publication no. 1 of the Mathematics Research Center, U.S. Army, the University of Wisconsin, The University of Wisconsin Press, Madison, 1959x+462 MR0102594 Google Scholar[3] Gertrude Blanch, On the computation of Mathieu functions, J. Math. Phys. Mass. Inst. Tech., 25 (1946), 1–20 MR0016690 0061.27702 ISIGoogle Scholar[4] Gertrude Blanch and , Ida Rhodes, Table of characteristic values of Mathieu's equation for large values of the parameter, J. Washington Acad. Sci., 45 (1955), 166–196 MR0071144 Google Scholar[5] Christoffel Jacob Bouwkamp, Theoretische en Numerieke Behandeling van de Buiging door een Ronde Opening, Dissertation, University of Groningen, 1941, 60–, Groningen, Batavia MR0017632 Google Scholar[6] British Assn. for the Adv. Sci., Mathematical Tables, Bessel Functions, Vol. X, Cambridge University Press, 1952, Part II. (Introduction by J. C. P. Miller) 0049.36204 Google Scholar[7] Evelyn Frank, Corresponding type continued fractions, Amer. J. Math., 68 (1946), 89–108 MR0015533 0060.16607 CrossrefISIGoogle Scholar[8] Evelyn Frank, On continued fraction expansions for binomial quadratic surds, Numer. Math., 4 (1962), 85–95, pp. 303–307; 5 (1963), pp. 113–117 10.1007/BF01386298 MR0140181 0107.10303 CrossrefGoogle Scholar[9] E. D. Hellinger and , H. S. Wall, Contributions to the analytic theory of continued fractions and infinite matrices, Ann. of Math. (2), 44 (1943), 103–127 MR0008102 0060.16502 CrossrefGoogle Scholar[10] R. E. Lane, Interpolation by means of continued fractions, Fraternal Actuarial Assoc. Proc., 19 (1944-46), Google Scholar[11] Walter Leighton and , W. J. Thron, On value regions of continued fractions, Bull. Amer. Math. Soc., 48 (1942), 917–920 MR0007069 0060.16405 CrossrefGoogle Scholar[12] Walter Leighton and , W. J. Thron, On the convergence of continued fractions to meromorphic functions, Ann. of Math. (2), 44 (1943), 80–89 MR0007805 0060.16404 CrossrefGoogle Scholar[13] Yudell L. Luke, The Padé table and the $\tau$-method, J. Math. Phys., 37 (1958), 110–127 MR0099114 0084.34604 CrossrefGoogle Scholar[14] Nathaniel Macon and , Margaret Baskervill, On the generation of errors in the digital evaluation of continued fractions, J. Assoc. Comput. Mach., 3 (1956), 199–202 MR0080987 CrossrefISIGoogle Scholar[15] A. A. Markoff, Masters Thesis, On certain applications of algebraic continued fractions, Thesis, St. Petersburg, 1884 Google Scholar[16] J. H. Müller, On the application of continued fractions to the evaluation of certain integrals, with special reference to the incomplete Beta function, Biometrika, 22 (1920-1), 284–297 CrossrefGoogle Scholar[17] National Bureau of Standards, Tables relating to Mathieu functions, Columbia University Press, New York, 1951 0045.40109 Google Scholar[18] Oskar Perron, Die Lehre von den Kettenbrüchen, Chelsea Publishing Co., New York, N. Y., 1950xii+524 MR0037384 0041.18206 Google Scholar[19] A. Pringsheim, Ueber die Konvergenzkreiterien für Kettenbrüche mit komplexem Gliedern, Sitzungsber. der Math-Phys. Klasse der Kgl. Bayer. Akad. Wiss., München, 35 (1905), 359–380 Google Scholar[20] R. D. Richtmyer, , Marjorie Devaney and , N. Metropolis, Continued fraction expansions of algebraic numbers, Numer. Math., 4 (1962), 68–84 10.1007/BF01386297 MR0136574 0101.28101 CrossrefGoogle Scholar[21] Heinz Rutishauser, Der Quotienten-Differenzen-Algorithmus, Mitt. Inst. Angew. Math., ETH, 7, Birkhäuser Verlags, Basel, 1956 0077.11103 Google Scholar[22] Heinz Rutishauser, Stabile Sonderfälle des Quotienten-Differenzen-Algorithmus, Numer. Math., 5 (1963), 95–112 10.1007/BF01385882 MR0175288 0196.48404 CrossrefGoogle Scholar[23] H. Schwerdtfeger, Moebius transformations and continued fractions, Bull. Amer. Math. Soc., 52 (1946), 307–309 MR0015532 0060.16202 CrossrefISIGoogle Scholar[24] W. T. Scott and , H. S. Wall, A convergence theorem for continued fractions, Trans. Amer. Math. Soc., 47 (1940), 155–172 MR0001320 0022.32603 CrossrefGoogle Scholar[25] Kurt Spielberg, Representation of power series in terms of polynomials, rational approximations and continued fractions, J. Assoc. Comput. Mach., 8 (1961), 613–627 MR0130102 0105.11204 CrossrefGoogle Scholar[26] Kurt Spielberg, Polynomial and continued-fraction approximations for logarithmic functions, Math. Comp., 16 (1962), 205–217 MR0145645 0105.11205 CrossrefGoogle Scholar[27] Irene Stegun and , Milton Abramowitz, Generation of Bessel functions on high speed computers, Math. Tables Aids Comput, 11 (1957), 255–257 MR0093939 0084.12101 CrossrefGoogle Scholar[28] T.-J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894), J1–J122 MR1508159 Google Scholar[29] Jerome D. Swalen and , Louis Pierce, Remarks on the continued fraction calculations of eigenvalues and eigenvectors, J. Mathematical Phys., 2 (1961), 736–739 10.1063/1.1703766 MR0130109 0108.12503 CrossrefISIGoogle Scholar[30] O. Szász, Ueber die Erhaltung bei independenter Veränderlichkeit aller ihrer Elemente, J. für Math., 147 (1916), 132–160 Google Scholar[31] P. L. Tschebycheff, Sur les fractions continues, Jour. de Math., 8 (1858), 289–323 Google Scholar[32] D. Teichroew, Use of continued fractions in high speed computing, Math. Tables and Other Aids to Computation, 6 (1952), 127–133 MR0049650 0047.12002 CrossrefGoogle Scholar[33] W. J. Thron, Two families of twin convergence regions for continued fractions, Duke Math. J., 10 (1943), 677–685 10.1215/S0012-7094-43-01063-4 MR0009214 0060.16407 CrossrefGoogle Scholar[34] W. J. Thron, Zwillingskonvergenzgebiete für Kettenbrüche $1+K(a\sb{n}/1)$, deren eines die Kreisscheibe $\vert a\sb{2n-1}\vert \leq \rho \sp{2}$ ist, Math. Z., 70 (1958/1959), 310–344 10.1007/BF01558596 MR0105488 0085.05002 CrossrefGoogle Scholar[35] W. J. Thron, Convergence regions for continued fractions and other infinite processes, Amer. Math. Monthly, 68 (1961), 734–750 MR0133444 0103.28304 CrossrefISIGoogle Scholar[36] Edward B. Van Vleck, On the convergence of the continued fraction of Gauss and other continued fractions, Ann. of Math. (2), 3 (1901/02), 1–18 MR1502271 CrossrefGoogle Scholar[37] H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., New York, N. Y., 1948xiii+433 MR0025596 0035.03601 Google Scholar[38] H. S. Wall, Continued fractions and totally monotone sequences, Trans. Amer. Math. Soc., 48 (1940), 165–184 MR0002642 0024.21602 CrossrefGoogle Scholar[39] H. S. Wall, A theorem on arbitrary J-fractions, Bull. Amer. Math. Soc., 52 (1946), 671–679 MR0017395 0060.16604 CrossrefISIGoogle Scholar[40] H. S. Wall and , Marion Wetzel, Contributions to the analytic theory of J-fractions, Tran. Amer. Math. Soc., 55 (1944), 373–392 MR0011339 0060.16503 Google Scholar[41] Marion Wetzel, Applications of continued fractions, U.S. Army Tech. Report, 5, Office of Ordnance Research, 1954, Contract DA-04-200-ORD-177, August Google Scholar[42] Arthur Wouk, Difference equations and J-matrices, Duke Math. J., 20 (1953), 141–159 10.1215/S0012-7094-53-02014-6 MR0058111 0051.07201 CrossrefISIGoogle Scholar[43] P. Wynn, Converging factors for continued fractions. I, II, Numer. Math., 1 (1959), 272–320 10.1007/BF01386391 MR0116158 0092.05101 CrossrefGoogle Scholar[44] P. Wynn, The rational approximation of functions which are formally defined by a power series expansion, Math. Comput., 14 (1960), 147–186 MR0116457 0173.18803 CrossrefGoogle Scholar[45] P. Wynn, A comparison technique for the numerical transformation of slowly convergent series based on the use of rational functions, Numer. Math., 4 (1962), 8–14 10.1007/BF01386291 MR0136500 0138.09901 CrossrefGoogle Scholar[46] P. Wynn, The numerical efficiency of certain continued fraction expansions. IB, Nederl. Akad. Wetensch. Proc. Ser. A 65 = Indag. Math., 24 (1962), 138–148 l MR0139255 0105.10003 Google Scholar[47A] Alexey Nikolaevitch Khovanskii, The application of continued fractions and their generalizations to problems in approximation theory, Translated by Peter Wynn, P. Noordhoff N. V., Groningen, 1963xii + 212 MR0156126 0106.27204 Google Scholar[47B] A. Ya. Khintchine, Continued fractions, Translated by Peter Wynn, P. Noordhoff Ltd., Groningen, 1963iii+101 MR0161834 0117.28503 Google Scholar[48] E. G. Kogbetliantz, Computation of ${\rm Sin}\ N$${\rm Cos}\ N$ and $\root m\of N$ using an electronic computer, IBM J. Res. Develop., 3 (1959), 147–152 MR0102170 CrossrefISIGoogle Scholar[49] E. G. Kogbetliantz, Anthony Ralston and , Herbert S. Wilf, Generation of elementary functionsMathematical methods for digital computers, Wiley, New York, 1960, 7–35 MR0117907 Google Scholar[50] H. J. Maehly, Rational approximations for transcendental functionsInformation processing, UNESCO, Paris, 1960, 57–62 MR0148206 0112.35303 Google Scholar[51A] Hans J. Maehly, Methods for fitting rational approximations. I. Telescoping procedures for continued fractions, J. Assoc. Comput. Mach., 7 (1960), 150–162 MR0116455 0113.32502 CrossrefISIGoogle Scholar[51B] Hans J. Maehly, Methods for fitting rational approximations. II, III, J. Assoc. Comput. Mach., 10 (1963), 257–277 MR0157474 0113.32502 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Emergence of new category of continued fractions from the Sturm–Liouville problem and the Schrödinger equation30 July 2021 | São Paulo Journal of Mathematical Sciences, Vol. 15, No. 2 Cross Ref Computation and Applications of Mathieu Functions: A Historical PerspectiveChris Brimacombe, Robert M. 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Dedekind symbols generalize the classical Dedekind sums (symbols). These symbols are determined uniquely, up to additive constants, by their reciprocity laws. For k ≧ 2, there is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws of degree 2k − 2 and the space of modular forms of weight 2k for the full modular group \(PSL_2 (\mathbb{Z}).\) However, this is not the case when k = 1 as there is no modular form of weight two; nevertheless, there exists a unique (up to a scalar multiple) quasi-modular form (Eisenstein series) of weight two. The purpose of this note is to define the Dedekind symbol associated with this quasi-modular form, and to prove its reciprocity law. Furthermore we show that the odd part of this Dedekind symbol is nothing but a scalar multiple of the classical Dedekind sum. This gives yet another proof of the reciprocity law for the classical Dedekind sum in terms of the quasi-modular form.

In this thesis continued fractions are studied in three directions: semi-regular continued fractions, Nakada’s α-expansions and N-expansions. In Chapter 1 the general concept of a continued fraction is given, involving an operator that yields the partial quotients or digits of a continued fraction expansion. The approximation coefficients θ_n(x) := q²|x-p_n/q_n| are introduced, where p_n/q_n, n ∈ 0, 1, 2, . . ., are the convergents of the continued fraction. Some well-known results on semi-regular continued fractions are given. Finally, the concept of ‘natural extension’ is explained. Chapter 2 is about orders (called patterns) of triplets of three consecutive approximation coefficients θ_(n-1)(x), θ_n(x) and θ_(n+1)(x). The asymptotic frequency of pattern Χ(n) is defined by AF(X(n)) := lim_(N→∞) 1/N #{n ∈ N | 2 ≤ n ≤ N, X(n)}. Starting with the regular continued fraction (RCF), it is shown that, for instance, the asymptotic frequency as n → ∞of the pattern θ_(n-1)(x) l θ_n(x) l θ_(n+1)(x) is smaller than the asymptotic frequency of the pattern θ_n(x) l θ_(n+1)(x) l θ_(n-1)(x). The asymptotic frequencies in the case of the RCF are explicitly given: two of them are 0.1210..., the others are 0.1894... . After this, these patterns are studied of two other semi-regular continued fractions: the optimal continued fraction (OCF) and the nearest integer continued fraction (NICF). The asymptotic frequencies of the OCF prove to be more equally distributed: the two less frequent patterns of the RCF now have the asymptotic frequency 0.1603... , where this is 0.1698... for the other patterns. The asymptotic frequencies of the NICF prove to be different for all six patterns. However, summation of specific pairs yield once 2 · 0.1603... and two times 2 · 0.1698... , thus showing a great correspondence with the OCF. Chapter 3 is dedicated to the natural extension of Nakada’s α-expansions. By meansof singularisations and insertions in these continued fraction expansions, involving the removal or addition of partial quotients 1 in exchange with partial quotients with a minus sign, the interval on which the natural extension of Nakada’s continued fractionmap T_α is given is extended from [√2-1,1) to [(√10-3)/2,1). From our construction it followsthat ­Ω_α, the domain of the natural extension of T_α, is metrically isomorphic to ­g for α ∈ [g², g), where g is the small golden mean. Finally, although ­Ω_α proves to be very intricate and unmanageable for α ∈ [g², (√10-3)/2), the α-Legendre constant L(α) on this interval is explicitly given. In Chapter 4 N-expansions are introduced for natural numbers N larger than 1. These expansions, like semi-regular continued fraction expansions, are also sequences of partial quotients, called orbits, existing in the interval I_α = [α,α+1] for some α ∈ (0,√N-1]. Depending on N and α, there is a finite number of consecutive digits that occur as partial quotient. It appears that there are conditions (that is, combinations of N and α) such that these orbits eventually do not land in certain parts of the interval I_α, called gaps. It is proved that if the number of digits is at least five, no gaps exist. If the number of digits is four, there do not exist gaps for most N, but in the cases that there are α such that I_α contains a gap, there is only one and it covers the lion’s part of I_α. When the number of digits is two or three, the number of gaps varies, but it is possible to give very clear conditions under which there are no gaps.

Various properties of classical Dedekind sums S(h, q) have been investigated by many authors. For example, Wenpeng Zhang, On the mean values of Dedekind sums, J. Théor. Nombres Bordx, 8 (1996), 429–442, studied the asymptotic behavior of the mean value of Dedekind sums, and H.Rademacher and E.Grosswald, Dedekind Sums, The Carus Mathematical Monographs No. 16, The Mathematical Association of America, Washington, D.C., 1972, studied the related properties. In this paper, we use the algebraic method to study the computational problem of one kind of mean value involving the classical Dedekind sum and the quadratic Gauss sum, and give several exact computational formulae for it.

In this paper, we use the analytic method and the properties of Gauss sums to study the hybrid mean value problem involving the classical Dedekind sums and exponential sums, and give several interesting identities and asymptotic formulae for it. Keywords: Dedekind sum; exponential sum; hybrid mean value; identity MR(2010) Subject Classification: 11L40; 11F20 / CLC number: O156.4 Document code: A Article ID: 1000-0917(2014)04-0527-07

We display a number with a surprising continued fraction expansion and show that we may explain that expansion as a specialisation of the continued fraction expansion of a formal series: A series ΣchX-h has a continued fraction expansion with partial quotients polynomials in X of positive degree (other, perhaps than the 0-th partial quotient). Simple arguments, let alone examples, demonstrate that it is noteworthy if those partial quotients happen to have rational integer coefficients only. In that special case one may replace the variable X by an integer ≥ 2; that is: one may 'specialise' and thereby proceed to obtain the regular continued fraction expansion of values of the series. And that is significant because, generally, it is difficult to obtain the explicit continued fraction expansion of a number presented in different shape. Our example leads to a series with a specialisable continued fraction expansion and, a little surprisingly, our arguments suggest that the phenomenon of specialisability for series of the kind appearing here may be reserved to just the special subclass of series we happen to have stumbled upon.

F. Schweiger introduced the continued fraction with even partial quotients. We will show a relation between closed geodesics for the theta group (the subgroup of the modular group generated by z+2 and -1 / z) and the continued fraction with even partial quotients. Using thermodynamic formalism, Tauberian results and the above-mentioned relation, we obtain the asymptotic growth number of closed trajectories for the theta group. Several results for the continued fraction expansion with even partial quotients are obtained; some of these are analogous to those already known for the usual continued fraction expansion related to the modular group, but our proofs are by necessity in general technically more difficult.

Numbers whose continued fraction expansion contains only small digits have been extensively studied. In the real case, the Hausdorff dimension σ M of the reals with digits in their continued fraction expansion bounded by M was considered, and estimates of σ M for M→∞ were provided by Hensley (J. Number Theory 40:336–358, 1992). In the rational case, first studies by Cusick (Mathematika 24:166–172, 1997), Hensley (In: Proc. Int. Conference on Number Theory, Quebec, pp. 371–385, 1987) and Vallée (J. Number Theory 72:183–235, 1998) considered the case of a fixed bound M when the denominator N tends to ∞. Later, Hensley (Pac. J. Math. 151(2):237–255, 1991) dealt with the case of a bound M which may depend on the denominator N, and obtained a precise estimate on the cardinality of rational numbers of denominator less than N whose digits (in the continued fraction expansion) are less than M(N), provided the bound M(N) is large enough with respect to N. This paper improves this last result of Hensley towards four directions. First, it considers various continued fraction expansions; second, it deals with various probability settings (and not only the uniform probability); third, it studies the case of all possible sequences M(N), with the only restriction that M(N) is at least equal to a given constant M 0; fourth, it refines the estimates due to Hensley, in the cases that are studied by Hensley. This paper also generalises previous estimates due to Hensley (J. Number Theory 40:336–358, 1992) about the Hausdorff dimension σ M to the case of other continued fraction expansions. The method used in the paper combines techniques from analytic combinatorics and dynamical systems and it is an instance of the Dynamical Analysis paradigm introduced by Vallée (J. Théor. Nr. Bordx. 12:531–570, 2000), and refined by Baladi and Vallée (J. Number Theory 110:331–386, 2005).

Relations between theta-functions Hardy sums Eisenstein and Lambert series in the transformation formula of [formula omitted

Analogies of Dedekind sums in function fields

The obvious way to compute the continued fraction of a real number α > 1 is to compute a very accurate numerical approximation of α, and then to iterate the well-known truncate-and-invert step which computes the next partial quotient a = bαc and the next complete quotient α′ = 1/(α − a). This method is called the basic method. In the course of this process precision is lost, and one has to take precautions to stop before the partial quotients become incorrect. Lehmer [6] gives a safe stopping-criterion, and a trick to reduce the amount of multi-length arithmetic. Schonhage [12] describes an algorithm for computing the greatest common divisor of u and v and the related continued fraction expansion of u/v in O(n log n log log n) steps if both u and v do not exceed 2. A disadvantage of this approach is that if we wish to extend the list of partial quotients computed from an initial approximation of α, we have to compute a more accurate initial approximation of α, compute the new complete quotient using this new approximation and the partial quotients already computed from the old approximation, and then extend the list of partial quotients using that new complete quotient (we notice that Shiu in [13, p.1312] incorrectly states that all the previous calculations have to be repeated, but the partial quotients computed so far don’t have to be recomputed). Bombieri and Van der Poorten [1], and Shiu [13] have recently proposed a remedy for this problem. They give a formula for computing a rational approximation of the next complete convergent from the first n partial quotients. From that complete convergent about n new partial quotients can be computed. So each step gives an approximate doubling of the number of partial quotients. To start the method, a few partial quotients have to be computed with the basic or indirect method. In [1] this method is proposed for algebraic numbers (which are zeros of polynomials) of degree ≥ 3, whereas Shiu also applies it to more general numbers, namely to transcendental numbers that can be defined as the zero of a function for which the logarithmic derivative at a rational point can be computed with arbitrary precision. This includes numbers like π, log π, and log 2. For each of thirteen different numbers, Shiu computes 10000 partial quotients. Their frequency distributions are compared with the one which almost all numbers should obey, according to the Khintchine– Levy theory [3, 7]. No significant deviations from this theory are reported. Shiu calls his method the direct method. Curiously, Shiu does not refer to what we would call the polynomial method for algebraic numbers [2, 5, 11] of degree ≥ 3, which computes the partial quotients of α using only the coefficients of its defining polynomial. Moreover, Shiu gives neither implementational

In this note it is shown, by a counterexample, that the familiar theorem on the continued fraction expansions of equivalent numbers does not hold when these notions are extended to complex numbers. Two real numbers x, x' are called equivalent, x-x', if (I) x' = (ax + b)l(cx +d) forsomea, b, c, deZ, ad-bc =1. Consider also the continued fraction (CF) expansion of a real number (2) x = (ao, al, , an-1, xJ), xn,1 = an-1 + 1/xn, where xn is the nth complete quotient of x. It is a standard theorem in CF's that x-x' if and only if, in the CF expansions of x and x', there exist m, n such that am+k=an+k for all k>O-more briefly, xm=xn. Hurwitz, in a paper [2] on the CF (where the partial quotients an may be negative) proved that essentially the same result carries over. That is, x-x' if and only if there exist m, n such that xm= ?x , where these are complete quotients of the nearest integer CF's. Hurwitz also defined [1] a complex generalization of the nearest integer CF (it might be called the Gaussian CF) by which a complex number x is expanded in a simple CF as in (2) with partial quotients an in Z[i]. There is an analogous notion of equivalent numbers as in (1), where a, b, c, d eZ[i], ad-bc=+I, +i. Although this complex CF has many analogies to real CF's, the expected theorem on equivalent numbers fails, as shown by a COUNTEREXAMPLE. Let Q2 = j(i+ (43 +28i)1/2), A = (5-i+Q2)/(4-i), B=(3+2i+Q2)/4. Then A--B, in fact A=(2B-i)/(B-i). However the CF expansions of A and B, which are periodic, are distinct: A = (2 + i, 3i, -1 + 2i, -I + 2i,3, -2-i), B = (2 + i, -2 + i, -2 + i, I 2i, I 2i, I + 2i). Thus A m = + Bn, or even + iBn, never holds. Received by the editors May 8, 1973. A MS (MOS) subject classifications (1970). Primary IOF20; Secondary 12A05.

This paper presents the numerical analysis of the discrete, approximated Fractional Order PID Controller (FOPID). The fractional parts of the controller are approximated with the use of the most known methods: Fractional Order Backward Difference (FOBD) and Continuous Fraction Expansion (CFE). CFE is simpler and faster than the FOBD method, but its accuracy is not always satisfying. For both approximations optimum sample time was found by minimizing of the cost function Integral Absolute Error (IAE). Additionally, to optimize of CFE its parameter a was applied. Results of numerical tests show that the FOPID using FOBD is more accurate in the sense of IAE cost function for FOPI and FOPID controllers, but CFE is more accurate for FOPD controller. Next, the FOBD requires to use of smaller sample time to obtain of good accuracy than CFE. This allows to conclude that FOPD controller using CFE can be applied in time critical applications at bounded platforms, for example in robotics or numerical control.