On Continued Fractions, Fibonacci Numbers and Electrical Networks

Transcription

1 03 Hawaii University International Conferences Education & Technology Math & Engineering Technology June 0 th to June th Ala Moana Hotel, Honolulu, Hawaii On Continued Fractions, Fibonacci Numbers Electrical Networks Suman Balasubramanian DePauw University

2 On Continued Fractions, Fibonacci Numbers Electrical Networks Suman Balasubramanian Department of Mathematics DePauw University 60 S College Avenue Greencastle, IN 4635 sumanbalasubramanian@depauwedu April, 03 Abstract b A Continued Fraction is of the form, a + c + d where a, b,c,d, e, f,, Z In e+ f this paper, we derive formulas for the n th convergent of the CF s p + p + p+ + p + where p Z The associated number sequences electrical networks + are indicated Preliminaries b A Continued Fraction (CF) is of the form, a + where a, b,c,d, e, f,, Z c + d e+ f While there are many variants to the form of these CF s, we are interested in those whose terms a, b, c, d, e, f,, N The first, second third convergent of the above CF The author thanks Professor Balasubramanian Sivaramakrishnan for bringing the problem to her attention

3 are respectively, a, a + b = ab +, a + b b + = a + c a(bc + ) + c = The numerator bc + bc + c the denominator of the n th convergent of the above CF will be denoted by N n D n respectively On careful scrutiny of the convergents, one can write N n = αn n + βn n D n = γd n + δd n, ([6], [3]), where, α, β, γ, δ are the appropriate terms Let F denote the CF p + p + where p Z G denote the CF +, where p + p+ + + p Z We now divide the rest of the paper into sections containing results for F, G their application to physics, respectively It may be noted that an application to a variant of F has been discussed in [] [] The following results are for CF s F G which are different from those studied earlier Result for F In this section we develop the necessary tools to derive the numerator denominator of the n th convergent of F hence use them to give an explicit expression for the n th convergent We also verify certain well known facts about F for specific values of p using the main result We now need the following Lemmas Lemma [6] For the CF F, the numerator the denominator of the n th convergent are given by N n = pn n + N n with N = p, N = p + D n = pd n + D n, with D = D = p, respectively Lemma [4] For Fibonacci type numbers, the n th term F n satisfies F n = c( φ φ ) n + d( φ 4 + φ ) n with a given F F where, c, d, φ R are appropriate constants Lemma 3 Let A n = N n D n, where, D n = p + 4 (p + p + 4 ) n p + 4 (p p + 4 ) n () p N n = 05( + p + 4 )(p + p + 4 ) n p + 05( p + 4 )(p p + 4 ) n ()

4 Then lim A n = α ( p + p + 4 ) + β ( p p + 4 ), n where α = β = 0 ProofLet A n, N n, D n be defined as above Then, it is easy to verify the result for A n using elementary calculus Theorem 4 For the CF F, the numerator denominator of the n th convergent are given respectively by () () ProofBy Lemma, the CF F has the numerator of the n th convergent, N n satisfying the general recursive equation P(n)=pP(n-)+P(n-) (3) where P () = p P () = p + Similarly, one can see that the denominator, D n of the n th convergent satisfies the same recursive equation (3) with the initial conditions P () = P () = p (For ease of computation, we replace N n D n by P (n) temporarily) From Lemmas 3, one can replace the general expression for N n D n by for appropriate constants c d Case I : P () =, P () = p c( p + p + 4 Substituting n = n = in (4) gives Solving for c d, we get c = = P () = c( p + p + 4 p = P () = c( p + p + 4 Case II : P () = p P () = p + ) n + d( p p + 4 ) n (4) ) + d( p p + 4 ) ) + d( p p + 4 ) p + 4 d = p + 4 Similar to Case I, substituting n = n = in (4), we get p = P () = c( p + p ) + d( p p + 4 )

5 p + = P () = c( p + p + 4 ) + d( p p + 4 ) p p We now see that c = 05( + ) d = 05( p + 4 p + 4 ) It can be observed that substituting p = in F gives the Fibonacci numbers for P (n) As a result, Theorem 4 gives the limiting value of the ratios of successive terms of the numerator the denominator respectively of the n th convergent to be the golden mean Similarly, substituting p = in F gives the Pell numbers for P(n) As a result, Theorem 4 gives the limiting value of the ratios of successive terms of the numerator the denominator respectively of the n th convergent to be the silver mean The Binet s formula for Fibonacci Numbers is given by the expression φn φ n where φ 5 is the golden ratio φ, its conjucgate Analogous to Binet s form, we get the following result for the numerator denominator of F Corollary 5 Letting φ = p + p + 4 of F from 4 are φn φ n φ φ φn+ φ φ φ Proof Clear φ, its conjugate, the numerator denominator n+ respectively 3 Result for G In order to study certain numbers related to the Fibonacci numbers like Lucas numbers etc, consider a variant to the CF F, namely, the CF G defined earlier Lemma 6 For the CF G, the numerator the denominator of the n th convergent are given by N n = N n + N n with N =, N = p + D n = D n + D n, with D = D = p, respectively Any CF of the form of G has the numerator or denominator of the n th convergent looking like c( + 5 ) n + d( 5 ) n, (5) WLOG, assume that N n D n to be of the form (5) for appropriate constants c d 4

6 Theorem 7 For the CF G, the numerator denominator of the n th convergent are respectively N n = p( 5 ) + 5 D n = p( 5 ) ( + 5 ( + 5 ) n + p( 5 + ) 5 ( 5 ) n ) n + p( 5 + ) ( 5 ) n ProofLet G be the CF defined as above By Lemma 6, the numerator denominator satisfy the general recursive equation P (n) = P (n ) + P (n ) (6) with the initial conditions P () =, P () = p + P () = P () = p respectively (As earlier, for the sake of simplicity, we replace N n D n by P (n)) Case I : P () = P () = p + In this case, substituting n = n = in (5) we get, (by Lemma [4]), Solving for c d, we get Case II : P () = P () = p = P () = c( + 5 p + = P () = c( + 5 c = p( 5 ) + 5 d = p( 5 + ) 5 ) + d( 5 ) ) + d( 5 ) As in Case I, substituting n = n = in (5) we get, (by Lemma [4]), = P () = c( + 5 ) + d( 5 ) p = P () = c( + 5 ) + d( 5 ) 5

7 Solving for c d, we get c = p( 5 ) d = p( 5 + ) Observe that for p =, one gets the Lucas Numbers (defined by the sequence L n = L n + L n, with L =, L = 3) from the numerator of the n th convergent whose value is then got from Theorem 7 Analogous to Binet s form as discussed in the previous section, we get the following result for the numerator denominator of G Corollary 8 Letting φ = + 5 φ, its conjugate, the numerator denominator of G from 7 are φn φ n n + φ n φ φ + p φ φn φ n p) φn n φ φ φ φ + ( + φ respectively, φ φ where φφ = ProofClear 4 An application to physics The relationship between continued fractions physics have been long studied as a result of a question first raised by Dwight E Neuenschwer [5] In this section, we study an application of continued fractions to physics Consider the infinite electrical network with resistances in series parallel as shown in Figure () We can see that the following Lemma holds for such a network Lemma 9 The effective resistance between nodes A B is the CF F The truncated networks (obtained by considering the first series resistance, the first series parallel resistance taken together, the first two series resistances taken together with the first resistance 6

8 that is parallel to both, etc) have the effective resistance p, p + p, p + p + p etc with the n th truncated network having the effective resistance between A B to be given by N n D n from Theorem 4 The following corollaries are then immediate Corollary 0 If p =, then the effective resistance between the nodes A B is the golden mean + 5 Corollary If all the resistances are taken to be except the first parallel resistance (which is taken to be ), then the effective resistance between A B is Proof The CF associated with this network is G with p = Let z = + + = 3x + x +, x where x = + 5 On simplification, the result follows 7

9 Figure : An infinite network of series parallel resistors with resistance p p respectively The top left node is A while the bottom left node is B References [] LQ English RR Winters, Continued fractions the harmonic oscillator using feynman path integrals, American Journal of Physics 65 (997), no 5, [] JJ Good, Complex fibonacci lucas numbers, continued fractions, the square root of the golden ratio, Fibonacci Quarterly 3 (993), no, 7 0 [3] GH Hardy EM Wright, An introduction to the theory of numbers, Oxford, Clarendon, 938 [4] Lovasz L, J Pelikan, Vostergombi K, Discrete mathematics, elementary beyond I, Springer Verlag, New York, 00, Fifth Indian Reprint, 008 [5] DE Neuenschwer, Is there a physics application that is best analyzed in terms of continued fractions?, American Journal of Physics 6 (994), no 0, 87 [6] C Smith, A treatise on algebra, Macmillan Co, London, 94 8