Fibonacci and Continued Fractions
| Scientific Paper | |
|---|---|
| Title | Fibonacci and Continued Fractions |
| Read in full | Link to paper |
| Author(s) | Thomas E Phipps |
| Keywords | Fibonacci, difference equations, continued fractions, generalized continued fractions, infinite matrix products. |
| Published | 2008 |
| Journal | Apeiron |
| Volume | 15 |
| Number | 4 |
| No. of pages | 17 |
| Pages | 534-550 |
Read the full paper here
Abstract
The Fibonacci sequence is used as a "hook" to direct interest toward generalizations.
Overview
This is a mathematics paper rather than a physics one — one of the few in T. E. Phipps, Jr.'s output on this wiki, published in Apeiron in October 2008. Its stated pedagogical device is to use the Fibonacci sequence as bait: a reader drawn in by the golden ratio is led, by a chain of increasingly general observations, into difference equations, continued fractions, asymptotics and infinite matrix products, and finally to an open frontier where, Phipps says, the relevant cases "have never been worked out, as far as I know."
Behind the pedagogy is a substantive and combative thesis. Phipps argues that the standard definition of a continued fraction is a bad definition in a precise technical sense — it inhibits connections between mathematical specialties — and that it has crippled the subject since its inception. A continued fraction is the natural companion of a second-order linear homogeneous difference equation, whose characteristic equation is a quadratic with two roots; it is therefore intrinsically two-valued. The received definition forcibly suppresses this by setting the process remainder to zero (or infinity) at every stage, which forces single-valuedness by fiat. Phipps traces the error to an over-hasty analogy with infinite series, which correspond to first-order difference equations and for which single-valuedness is appropriate. His verdict is unsparing: "A Procrustean definition that imposes single-valuedness cripples the subject of continued fractions for life... The indefinite persistence of the problem could justly be seen as one of the major scandals of modern mathematics."
The argument
The Fibonacci difference equation as two iterations
Writing the ratio Rn = Cn+1/Cn, the Fibonacci recurrence becomes a nonlinear relation in only two subscripts,
RnRn+1 − Rn − k = 0, (1)
whose limiting form, under the stability assumption Rn → Rn+1 → r, is the characteristic quadratic r2 − r − k = 0. For the Fibonacci value k = 1 the roots are r1 = (1 + √5)/2 ≈ 1.618, the golden ratio, and r2 = (1 − √5)/2 ≈ −0.618, whose magnitude is the reciprocal of the first, so that r1r2 = −1. Phipps notes the two geometrical readings — long-to-short and short-to-long side of the golden rectangle — and adds a dry aside that "there is no evidence that this ancient aesthetic prejudice has much to do with contemporary designs, as of movie or TV screens, photo formats, etc."
The paper's key structural point is that equation (1) can be turned into an iteration in two ways, and the two behave oppositely.
Case I isolates Rn: Rn = k/(Rn+1 − 1), which expands into a "continued fraction with remainder". Substituting r1 exactly gives Rn = r1 identically for all n — but the result "is deceptive". The iteration is numerically unstable: any departure from the exact irrational remainder, however remote a decimal place, causes the calculation to blow up and then converge inexorably to the other root, r2. Phipps invites the reader to check it on a pocket calculator, and observes that r2 here acts as a "strange attractor" — a label he immediately disowns as poor terminology, "since there is nothing particularly strange about it".
Case II isolates Rn+1: Rn+1 = 1 + k/Rn. This is the complementary iteration, with the same characteristic equation and therefore the same remainder r, but with reversed stability: it converges to the golden ratio r1 from almost any real starting value, and reaches r2 only if R1 = r2 is inserted exactly.
The consequence Phipps draws is that the conventional continued-fraction "value" of the Case-I expansion is not the golden ratio at all but the negative root, because setting all remainders to zero is a departure from r1 and therefore triggers convergence to the attractor. The golden ratio can be recovered only by what he calls an "exceptional remainder sequence" — inserting r1 exactly at every stage beyond some n0 — and established theory "does not recognize the existence of exceptional remainder sequences". For k = −1 the analysis carries over with conjugate complex roots (1 ± i√3)/2.
Infinite matrix products
Once two-valuedness is accepted, a further equivalence follows, which Phipps credits to Milne-Thomson's The Calculus of Finite Differences. Both cases can be written as products of 2×2 matrices of the form (1 1; k 0) acting on a two-entry vector, the continued-fraction value being read off as a ratio of the resulting entries. Proportionality rather than equality is used throughout, "because the basic difference equation to which all these processes are equivalent is linear and homogeneous; hence it determines its coefficients only within a constant multiplier". The orders of multiplication in the two cases are reversed, matching their reversed convergence properties. The summary claim is that second-order linear homogeneous difference equations, iterations, continued fractions and infinite 2×2 matrix products "amount to different notational disguises of the same 'mathematical object'", all sharing an essential "two-ness" traceable to the quadratic.
Generalizations
Section 4 replaces the constant k with n-dependent coefficients, giving the general second-order equation with coefficients an, bn. Case I becomes Rn = an/(bn + Rn+1) and Case II becomes Rn+1 = −bn + an/Rn, with an asymptotic characteristic equation rn+1rn + bnrn − an ≈ 0. Phipps notes that the Case-I form corresponds to a difference equation with "boundary conditions at infinity" and the Case-II form to boundary conditions at finite n — a distinction that does not appear in conventional treatments — and that the student is thereby "inexorably drawn into a related field, that of asymptotics", where remainders can be approximated to accelerate convergence.
He makes a practical computational point in favour of the matrix representation: extending a continued-fraction calculation by one stage requires "rolling up" the whole process from the bottom, whereas the matrix form needs only the last 2×2 matrix retained in memory, and the Case-II form not even that, extra stages being added by left multiplication. He also remarks that physicists "have tended to shrink from 'three-term recurrence relations'", hunting for substitutions that reduce them to two-term ones — a fear he thinks the matrix method largely dissolves, while conceding that "if Schroedinger's radial wave equation for the H-atom could not be reduced to a two-term recurrence... the pedagogy of elementary quantum mechanics would suffer a severe blow."
The generalisation then runs upward: a third-order difference equation corresponds to a product of 3×3 matrices with a cubic characteristic equation and up to three values, and an mth-order equation to m×m matrices with up to m distinct roots and m cases. Beyond second order, continued fractions themselves must be abandoned — "they are notationally limited to representation on two-dimensional paper" — and one has to rely on the matrix equivalent. The forms appropriate to mixed finite and infinite boundary conditions, he says, have never been worked out.
The appendix
An appendix supplies the proof of the paper's most surprising assertion. Using Wall's notation and the standard result Fn(wn) = (An + wnAn−1)/(Bn + wnBn−1), Phipps proves a short lemma: if the conventional value L = lim An/Bn exists, then Fn(wn) → L + V, where the numerator of V is (An/Bn − An−1/Bn−1) and therefore vanishes in the limit. For almost any remainder sequence — including random numbers — the denominator does not vanish, so V = 0 and convergence is to L. "Thus there is nothing magic about the remainder sequence {0, 0, 0, ...} demanded by the conventional definition." The exceptional sequences that give a different value are exactly those for which the denominator approaches zero at least as fast as the numerator, and "the Fibonacci example establishes that such sequences exist."
Assessment
The mathematics here is correct and the central observation is genuinely worth making. The complementarity of the two iterations — that isolating Rn and isolating Rn+1 from the same relation produces processes whose stable and unstable roots are exchanged — is a clean fact, easily verified, and it does explain in one stroke why the conventional continued-fraction value of the Fibonacci expansion is the negative root rather than the golden ratio that the sequence itself approaches. That is a small paradox many readers will have met and not resolved, and Phipps resolves it properly. The appendix lemma is elementary but does exactly the work asked of it: it shows that the choice of zero remainders is not privileged, merely typical, and that the set of remainder sequences yielding a different limit is non-empty. The equivalence to matrix products, and the practical observation about extending a calculation by left multiplication rather than re-rolling the fraction, are both sound and useful.
The paper is weaker where it moves from mathematics to polemic, and the two are not always kept apart. The claim that the standard definition constitutes "one of the major scandals of modern mathematics" overstates what has been shown. Nothing in the paper is inconsistent with conventional continued-fraction theory; what Phipps has done is to point out that the conventional object is one specialisation, obtained by a particular remainder choice, of a wider two-valued object. That is a case for enlarging the definition, not for calling the existing one false — and the enlargement he wants is, as he himself notes, precisely the identification of continued fractions with second-order difference equations, a correspondence that classical analysis has long recognised even if it does not usually phrase matters in terms of "remainders". The attribution of the situation to "faddism", to "second-rate mathematical minds", and to Hardy's aesthetics is asserted rather than evidenced, and the closing advice to young readers that trends among pure mathematicians are "steadily downhill as far as usefulness to mankind is concerned" is a judgement the paper has not earned. Readers should note that the same aside dismisses stochastic difference equations — coefficients drawn at random from {1, −1} — as having "little practical use", which is not a defensible verdict on random-matrix and random-recurrence methods.
Two smaller gaps. The term "strange attractor" is borrowed and then disowned, but the phenomenon described is ordinary linear stability of a fixed point, not chaotic attraction; using the phrase at all invites confusion the paper then has to work against. And the promising generalisation to mth order stops just short of the interesting question: the paper writes down the m×m matrix form and observes that up to m values are possible, but does not say which boundary data select which root, nor whether the exchange of stability seen between Cases I and II has a systematic higher-order analogue. Phipps flags this honestly as the frontier rather than claiming to have crossed it, which is to his credit.
Read for what it is — a short, pointed essay in classical analysis with a definitional axe to grind — the paper does its job. Its interest to this wiki is chiefly as evidence of how Phipps' methodological temperament, familiar from his physics, operates in a domain where his objection can be checked in full.