SOME NEW IDENTITIES INVOLVING π,

Transcription

1 SOME NEW IDENTITIES INVOLVING π, HENG HUAT CHAN π AND π. Itroductio The umber π, as we all ow, is defied to be the legth of a circle of diameter. The first few estimates of π were 3 Egypt aroud 9 B.C., 3 8 Babyloia, ad 3 7 < π < 3 Archimedes. 7 Sice the, may mathematicias have devoted their time to the determiatio of the explicit value of π. I this tal, I will preset some recet ew idetities ivolvig π, /π ad /π, with the hope to share with you various mathematical cocepts.. The formulas of Vieta ad Wallis I 65, F. Vieta showed that π = ad i 655, J. Wallis discovered that π = At first glace, these two formulas do ot seem to be related. I 999, over 3 years later, T. Osler [8] succeeded i uitig these two idetities. Osler bega with the simple formula si x = cos x si x. = cos x cos x 4 si x 4 Usig a ot-so-simple formula si θ = θ θ π = = = p cos x cos x cos x p si x p. π θ = π π + θ, π

2 HENG HUAT CHAN with θ = x p, he deduced that Now, Hece, si x x si x x = p = cos x cos x cos x p p π x p π = cos x = + cos x, cos x = = p π + x p. π + cos x, + cos x p = cos x, } {{ } p radicals p cos x π x p π = }{{} p π + x p. π radicals Settig x = π/ we see that whe p = we have Wallis s formula ad whe p, we have Vieta s formula. 3. The formula of D. Bailey, P. Borwei ad S.Plouffe Aroud 996, usig PSLQ algorithm, D. Bailey, P. Borwei ad S. Plouffe discovered the idetity 4 π = 6 i 8i + 8i + 4 8i i + 6 i= This idetity ca be proved by first observig that x x 8 dx = x +8i dx = / i= Hece, 4 6 i 8i + 8i + 4 8i + 5 = 8i + 6 i= Substitutig y = x, we fid that 6y 6 y 4 y 3 + 4y 4 dy = i= 6 i 8i x 3 4 x 4 8x 5 x 8 dx. 4y y dy 4y 8 y dy = π. y + The BBP formula is very useful i the sese that oe ca determie the -th hexadecimal of π without owig the the first -th hexadecimal of the expasio of π.

3 SOME NEW IDENTITIES INVOLVING π, π AND π 3 The idea preseted i [3] appears to be useful i derivig -th decimal places of costats provided a formula for those costat exists with powers of i the deomiator of the series. So far, o formula of the type π = T = i i b i has bee foud. However, we all ow that l9/ has a expasio of the above type ad hece the -th decimal places of l9/ ca be foud without owig the -th decimal of l9/. We ed this sectio with the followig observatio: Let Let a = aa + a +. P N,t = A = ! 3 N+t = A. 5 The it appears that the 6N -th to 6N-th decimal places of /π ca be obtaied from the first sixth decimal places of { 6N P N, N }, 9 {x} deotes the fractioal part of x. The above observatio is derived from a series discovered recetly by Cha, W.C. Liaw ad V. Ta [5]. 4. Gosper s formula ad the wor of G. Almvist, C. Krattethaler ad J.Petersso Bill Gosper has a formula for π that taes the form π = Recetly this formula was used by F. Bellard after the wor of S. Plouffe to compute the -th decimal places of π. The mai tool i their wor is the Chiese Remaider Theorem. Gosper s formula ca be proved as follow: It is well ow that the beta itegral ca be evaluated as Hece x m x dx = 3 = 3 + m!! m + +!. x x dx.

4 4 HENG HUAT CHAN Therefore, Sice = x x dx y = 56y + 97y 3 y 3, we coclude that 5 6 = 8 3 8x 6 56x 5 + 8x 4 97x x 6 x 3 x + 3 dx = π. The above method of proof prompted Almvist, Krattethaler ad Petersso [] to cosider the itegral = m + m p x p x m p dx. They the search for polyomials S of degree d i ad T y of degree d i y so that m + Sy = Lettig y = x p x m p /a, they deduced that S = m a p T y y d+. P x x p x m p dx, a d+ P x is a polyomial of degree md + i x. I order to sum to π oe has to put i restrictios so that x p x m p a has zero at i or + i. Usig LLL-algorithm the authors the costructed may ew series for π. It turs out that Gosper s formula remais to be the simplest of its id. The above experimet led Almvist, Krattethaler ad Petersso to the followig theorem: Theorem 4.. For all, there exist a formula π = S S is a polyomial i of degree 4 with ratioal coefficiets.

5 SOME NEW IDENTITIES INVOLVING π, π AND π 5 The proof of this theorem requires very clever methods of computig determiats. A example of such idetity is π = S 4 S = is 5. Ramauja s series for /π I [9], Ramauja recorded a total of 7 series for /π. Oe of these series 4 π = 6 + 3! 3 4. Such series coverge rapidly to /π ad ca be used to compute π as well. I fact, the Chudovsys discovered that 643 3/ π = 6!! 3 3! ad used it to compute over a billio digits of π. The Chudovsys formula exists i the above form because the rig of algebraic itegers of Q 63 is a pricipal ideal domai ad that the correspodig j-ivariat is a algebraic iteger. This is the fastest coverget series to /π of the form C π = α A + Bβ, with α, A, B, β Q. The Borweis were the first to provide proofs to most of Ramauja s series to /π. I both Chudovsys ad Borweis wor, a crucial igrediet is the Clause s formula F a, b; a + b + ; z = 3 F a, b, a + b, a + b + ; a + b; z, mf m a,, a m ; b,, b m ; z := a a m z b b m!. Recetly, Sog Heg, ZhiGuo ad I [4] discovered that Clause s formula is ot eeded for the derivatio of Ramauja s series. Our wor is ispired

6 6 HENG HUAT CHAN by a series discovered by Taeshi Sato, amely, 5 5. π j = j j 3 = j= Sice o Clause type formula exists for the above series, ew method eeds to be used for the derivatio of series such as 5.. Our ew method yields the followig compaio of 5.: π = = j= j j j j j We ed this sectio with two more ew series arisig from this ew method. The first is aother compaio of 5. that I discovered together with H. Verrill after the wor of Almvist ad Zudili []: = 3!! = 9 8 3π. The secod oe is oe I discovered recetly with Loo Ko Pig: 3 + = m 3 m m 3 m 6 π 9 8 j i G 3, j= i= e π/3 G = e π + e π + + e π + e 4 π + e 3 π + e 6 π + = The Wilf-Zeilberger algorithm ad Guillera s series for /π A discrete fuctio A, is hypergeometric if A +, A, ad A, + A, are both ratioal fuctios. A pair of fuctios F, ad G, is said to be of WZ after Wilf ad Zeilberger if F ad G are hypergeometric ad F +, F, = G, + G,.

7 SOME NEW IDENTITIES INVOLVING π, π AND π 7 I this case, Wilf ad Zeilberger showed that there exists a ratioal fuctio C, such that G, = C, F,. The fuctio C, is called the certificate of F, G. Defiig Zeilberger showed that H, = F +, + + G, +, H, = G,. Ehad ad Zeilberger [6] was the first to use the above method to derive a oe page proof of Ramauja s series see before i the previous sectio π = 4 + 3! 3. Motivated by his wor, J. Guillera [7] foud may ew WZ-pairs F, G ad derived ew series ot oly for /π but for /π. Oe of the most elegat formulas of Guillera is = 3 π. This is proved usig Zeilberger s method by exhibitig the WZ-pair, amely, ad F, = 5 3 U, 4 G, = U, U, = cosπ Γ + Γ6 + Γ3 + π 5 Γ + + Γ 3 + Motivated by the elegace of the above series, Guillera discovered empirically usig PSLQ algorithm may series of type C π = C + D + E α H. Most of the series he discovered caot be proved by the WZ-method, which apparetly ca oly be applied whe H is a power of. A example of Guillera s series which remais to be proved is 8 5 π = ! Ehad is a computer

8 8 HENG HUAT CHAN Refereces [] G. Almvist, C. Krattethaler, ad J. Petersso, Some ew formulas for π, Experimet. Math. 3, o. 4, [] G. Almvist, W. Zudili, Differetial equatios, mirror maps ad zeta values, preprit. [3] D. Bailey, P. Borwei ad S. Plouffe, O the rapid computatio of various polylogarithmic costats, Math. Comp , o. 8, [4] H.H. Cha, S.H. Cha ad Z.-G. Liu, Domb s umbers ad Ramauja-Sato type series for /π, Adv. Math. 86 4, o., [5] H.H. Cha, W.C. Liaw ad V. Ta Ramauja s class ivariat λ ad a ew class of series for /π. J. Lodo Math. Soc., 64, o., [6] S. Ehad, D. Zeilberger, A WZ proof of Ramauja s formula for π, Geometry, aalysis ad mechaics, 7 8, World Sci. Publishig, River Edge, NJ, 994 [7] J. Guillera, Some biomial series obtaied by the WZ-method, Adv. i Appl. Math. 9, o. 4, F [8] T. Osler, The uio of Vieta s ad Wallis s products for π, Amer. Math. Mothly, 6 999, [9] S. Ramauja, Modular equatios ad approximatios to π, Quart. J. Math. Oxford, 45 94,