A CONNECTION BETWEEN π AND φ. 1. Introduction. 10n n 2

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1 A CONNECTION BETWEEN π AND φ MICHAEL D. HIRSCHHORN Abstract. We fid a expressio for π as a limit ivolvig the golde ratio, φ. We prove that +k k φ+ π 4. Itroductio as, where φ is the golde ratio, ad cosequetly, π = lim 4 +k / φ +. k I order to carry out this program, we use the methods developed i two earlier papers o the Apéry umbers [], [3].. The domiat term +k The first step is to fid the value of k for which the term is a maximum. We k do this by settig +k +k + =. k k+ This yields!+k! k! 3 k! =!+k+! k +! 3 k!, k + 3 = +k+ k. If we suppose k = θ, where θ is to be determied, ad divide by 3, we fid θ+ 3 = +θ+ θ. If we let, this becomes θ 3 = +θ θ, θ θ = 0.

2 MICHAEL D. HIRSCHHORN It follows that θ = = φ, where φ is the golde ratio. Thus, the value of k that we seek is give by where θ = φ. At k θ, the value of the term is +θ H = θ =!+θ! θ! 3 θ! after cosiderable simplificatio. Now, k θ +θ +θ π π+θ e e πθ θ θ 3 π θ θ θ e e 3 +θ +θ +θ = π θ 3 θ θ 3θ θ θ θ = θ ad +θ = θ so +θ θ 3 θ = +θ θ θ 3 = θ θ 4 = θ 4 = φ4 ad +θ +θ θ 3θ θ θ = θ +θ θ 3θ θ θ = θ = φ. So H φ+4. π 3 See fig.. At poits ear θ, the terms of the sum are give by +θ+k +θ+k / +θ = H θ+k θ+k θ!+θ+k! /!+θ! = H θ+k! 3 θ k! θ! 3 θ! = H +θ+ +θ+k θ θ k+ θ+ 3 θ+ 3 θ+k 3

3 A CONNECTION BETWEEN π AND φ 3 sice ad Figure. The case = 000, showig the poits k, +k k for 80 k 660, together with the vertical x = φ, 0 y φ+4 = H + + +θ { Hexp Hexp { k } = Hexp { k φ3 k +θ + θ +θ + +k θ θ + 3 θ } +θ +θ k θ k θ 3k θ + 3 θ +θ = θ + 3 θ +θ +θ = + = 4 = θ k θ k θ 3k = θ = +3θ θ3 θ π 3 θ 3 + k θ 3 k θ + +k 3 θ + +k = +3θ θ θ θ θ = +θ +θ φ = φ +φ+ = 4φ+3 +3 = + = + = φ 3. Thus, the terms are essetially distributed ormally, with σ = φ 3, ad the sum is give by +k } H exp { x k σ dx = H σ π }

4 4 MICHAEL D. HIRSCHHORN φ+4 π 3 φ 3 4 π as claimed. See fig.. = φ+ π 4, Figure. The case = 000, showig the poits k, +k k for 0 { k 690, together with the approximatig ormal, y = φ+4 exp φ3 x }. π 3 φ Let s = k +k see 4 that s satisfies the recurrece 3. The correctio term. It was stated by Apéry [] ad proved by A. va der Poorte + s + ++3s s = 0, + s s s = 0. We ow suppose that s = C Φ + a + a + a 3 3 +, where C = φ π 4 ad Φ = φ, ad substitute ito the recurrece, to obtai + C+ Φ + + a + + a + + a

5 A CONNECTION BETWEEN π AND φ C Φ + a + a + a C Φ + a + a + a = 0. If we ow divide by CΦ, ad multiply by, we fid + Φ + a + + a + + a a + a + a Φ + a + a + a = 0. If we set = u, + = u +u, = u, ad expad i powers of u, we fid u Φ+u+a u+a a u +a 3 a +a u 3 + +u+3u +a u+a u +a 3 u 3 + Φ +u+u +u 3 + +a u+a +a u +a 3 +a +a u 3 + = 0. We ow set the coefficiets of the powers of u equal to zero, ad solve for a, a, a 3 ad so o. The costat term ad the coefficiet of u are automatically zero, because we had Φ correct ad the factor correct. The coefficiet of u is Φa a +a +3 Φ a +a + = 0, We fid +Φ a 3+Φ = a = 3+Φ +Φ = = 0 =. 0 + If we cotiue i the same way, we fid ad so o. a = 3 0, a 3 = , a 4 = 437 0, 60 This completes the proof.

6 6 MICHAEL D. HIRSCHHORN ad 4. The recurrece A. va der Poorte s proof [] goes as follows. If we defie the it is easy to verify that fk = k +6+3k +9+ k +k g = k +k fk fk = + g+ ++3g g. The recurrece follows o summig over k from 0 to +. Followig the work of Sister Celie Fasemyer ad Petrovsek, Wilf ad Zeilberger [4], the discovery of such idetities is routie. Refereces [] R. Apéry, Irratioalité de ζ, ζ3, Astérisque 6979, 3. [] M. D. Hirschhor, Estimatig the Apéry umbers, Fiboacci Quarterly 00, 9 3. [3] M. D. Hirschhor, Estimatig the Apéry umbers II, Fiboacci Quarterly, to appear. [4] M. Petrovsek, H. Wilf ad D. Zeilberger, A=B, A. K. Peters Ltd., Wellesley MA, 996. [] A. va der Poorte, A proof that Euler missed, Mathematical Itelligecer 979, AMS Classificatio Numbers: 4A60 School of Mathematics ad Statistics, UNSW, Sydey, Australia 0