Formulae for some classical constants - Simon Plouffe

Formulae for some classical constantsAlexandru Lupaş ∗(to appear in Proceedings of ROGER-2000)The goal of this paper is to present formulas for Apéry Constant , Archimede’sConstant , Logarithm Constant , Catalan’s Constant.Such formulas are useful for high precision calculations frequently appearingin number theory. Also , one motivation for computing digits of some constantsis that these are excellent test of the integrity of computer hardwareand software.The Apery’s Constant is defined as1. The Apéry’s Constant ζ(3)ζ(3) =∞∑k=11k 3 == 1.20205690315959428539973816151144999076498629234049... .The designation of ζ(3) as Apéry’s constant is new but well-deserved. In1979, Roger Apéry stunned the mathematical world with a miraculous proofthat ζ(3) is irrational (see [4], [16]). The above expression is very slow toconverge. Another formula is• Roger Apéry ([4]-1979)(1)ζ(3) = 5 ∞∑(−1) k−1 1( )2k=1 k 3 2kkwhich was the starting point of Apéry’s incredible proof of the irrationalityof ζ(3) . Since 1980, many formulas for ζ(3) were discovered.∗ Ro-Ger University, Calea Dumbrăvii 28-32, 2400-Sibiu, E-Mail : lupas@jupiter.sibiu.ro1

For instance• T.Amderberhan ([2]-1996) proved that(2)(3)ζ(3) = 1 ∞∑(−1) k−1 (56k2 − 32k + 5) ( (k − 1)! ) 34k=1(2k − 1) 2 (3k)!ζ(3) = 118∞∑(−1) k−1 (k!)2 (2k)! · A(k)(4k)!kk=13 · B(k)with ⎧⎪⎨⎪ ⎩A(k) = 5265(k − 1) 4 + 13878(k − 1) 3 + 13761(k − 1) 2 ++6120(k − 1) + 1040B(k) = (3k − 1) 2 (3k − 2) 2• T.Amdeberhan and Doron Zeilberger ([2]-1996), ( [3]-1997)(4)where• (A. Lupaş)(5)withζ(3) = 37523125 + 16250ζ(3) = 1 ∞∑(−1) k−1 Q(k) ( )2k=1 k 5 2k 5kQ(k) = 205k 2 − 160k + 32 .∞∑(−1) k−1 P(k)( )k=1 k 5 2k 5(2kk + 1)5P(k) = 14760k 4 + 28010k 3 + 19505k 2 + 5920k + 672 .The proof of (5) was performed by means of WZ-method (see [17] ,[19] also[2] ,[3] ) followed by a Kummer transformation.Let us remind that a discrete function a(n, k) defined for n, k ∈ {0, 1, ...} isa(n + 1, k)called a Closed Form =(CF) in two variables when the ratios ,a(n, k)a(n, k + 1)are both rational functions. A pair (F, G) of CF functions whicha(n, k)satisfy(6)F (n + 1, k) − F (n, k) = G(n, k + 1) − G(n, k)is a so-called WZ- form. There are many investigations connected with suchWZ-forms (today known as WZ-theory) : see [2], [3] ,[17] ,[19].In the 1990’s , the WZ theory has been the method of choice in resolvingconjectures for hypergeometric identities.The key of the WZ-theory is thefollowing theorem2

Theorem 1 (D. Zeilberger-[17], 1993 ) For any WZ-form (F,G)∞∑∞∑ (G(k, 0) = F (k + 1, k) + G(k, k) ) ,k=0k=0whenever either side converges.Let us remark that all formulas which are listed in (1)-(5)are of the form(7)ζ(3) = K 1 + K 2∞ ∑k=1(−1) k−1 a k , a k > 0where K 1 , K 2 are some real constants. Supposing that we have also anotherrepresentation , namely(8)∞∑ζ(3) = K 1 ′ + K 2′ (−1) k−1 b k , b k > 0k=1we shall say that (7) ≪ (8) iffζ na n = O(ζ n ) , b n = O(η n ) with lim = 0 .n→∞ η nFormula a n = O(ζ n )(1) a n = O ( )14 n·n2√ n(2) a n = O ( 127 n·n 2 )(3) a n = O ( 164 n·n 2 )(4) a n = O ( )11024 n·√n(5) a n = O ( )11024 n·n3√ nTherefore(5) ≪ (4) ≪ (3) ≪ (2) ≪ (1) .3

G.Fee and Simon Plouffe used (4) in their evaluation of ζ(3) with 520,000digits (see [3]). Later, by means of the same formula (4) B.Haible andT.Papanikolau computed ζ(3) to 1,000,000 digits.Open problem : If T n (x) = cos (n · arccos x) is the Chebychev’s polynomialof the first kind, and (see (5) )⎧⎪⎨⎪⎩P (j) = 14760j 4 + 28010j 3 + 19505j 2 + 5920j + 672S k = 1 k∑(−1) j−1 P (j)( )6250j=1 j 5 2j 5(2jj + 1)5A n = 1( )n∑ n n + k4 k S kT n (3)k=1n + k 2kB n = 3752 + A 3125 nthen prove or disprove that.|ζ(3) − B n | = O ( 1 )5939 n · n 3√ n, (n → ∞)The following proposition generalize an identity proved by R.Ap’ery ( x =0 , [4]) .Theorem 2 For x ≥ 0n∑k=11(k + x) 3 = 5 2+ 1 2n∑k=1n∑k=1(−1) k−1k 3 ( 2k+xk)( ) k+xk(−1) kk 3 ( n+k+xkA more general function than ζ(s) isζ(s, x) =∞∑k=0()2x(3k + 2x)1 − +5(k + x) 2)( ) n+x.k1(k + x) s , Re(s) > 1 , x > 0 .Corollary 1 If x > 0 , thenζ(3, x + 1) = 5 2∞∑k=0(−1) k−1( )(k 3 2k+x k+xk k)()2x(3k + 2x)1 −5(k + x) 2.4

2. The constants ln 2 , G , πTheorem 3 Let x > 0 , y > 0 . Then(9)∞∑(−1) k(k=0(x) k(x + 2y) k) 2=( ) 3= y ∞∑ (−1) k (2y + 1)2k 10(k + y) 2 + (6x + 1)(k + y) + x 22k=016 k ((y + 1)k (x + 2y) 2k+1 ) ) 2k + y.Proof. Define( ) 3(2y +A(j) = A(j, x, y) = y (−1)j 1)2j16 j ( ) 2(y + 1)jand⎧⎪⎨G(n, k) =⎪⎩λ(n, k) = (−1) n((x) n(x + 2y) n+k) 21λ(n, 2j)A(j) , k = 2jy+j2(y + j + x + n)λ(n, 2j + 1)A(j) , k = 2j + 1.FurtherF (n, k) =⎧⎪ ⎨⎪⎩λ(n,2j)A(j) , k = 2j2(y+j).(n + x + 3y + 3j + 1)λ(n, 2j + 1)A(j) , k = 2j + 1 2If G(n, k) and F (n, k) are defined as above , thenF (n + 1, k) − F (n, k) = G(n, k + 1) − G(n, k) ,that is (F, G) is a WZ-form. By applying WZ-method we find (9).Further, by G is denoted the Catalan constant G which is defined asG = β(2) =∞∑k=0(−1) k(2k + 1) 2 = 0.91596554... ,5

∞∑ (−1) kβ(x) :=being the Dirichlet’s beta function .k=0(2k + 1)xSome particular cases of (9) arex y Identity a k The order of a n(n → ∞)1∑1 π = 4 − ∞ (−1) k−1 (a 2k k )2 kk=140k 2 +16k+1( 4k2k) 22k(4k+1) 2 O ( 1)64 n·√n1 1 ln 2 = 3 − ∑ ∞ (−1) k−1 1a2 kk=116 k ( 2kk) 5k+1k(k+ 1) O ( )12 4 n·n 23121∑G = ∞ (−1) k−1 a2 kk=1(Catalan’ s Constant)k2 8k·(40k 2 −24k+3)3·(4k2k) 2 ( 2k k )64(2k−1)O ( )114 n·√n1 1 π 2 ∑= ∞ (−1) k−1 1a2 kk 3 ( 2k k ) · 14k2 −9k+2O ( 13(2k−1)k=14 n·n 3 2)The mathematicians have assumed that there is no shortcut to determiningjust the n−th digit of π . Thus it came as a great surprise when such ascheme was in 1997 discovered [6] (see (11) from below ). This formula wasdiscovered empirically, using months of PSLQ computations , see [8]. Thisis likely the first instance in history that a signifiant new formula for π wasdiscovered by a computer. In the following, we present an identity whichgives a proof of (11).Proposition 1 If |z + 1| < √ 2 , r ∈ C , then1 − 2z − z2∞∑π + 4 arctan z + (2 + 8r) ln = (4 + 8r)1 + z 2 k=0− 8r(10)− (2 + 8r)∞∑k=0∞∑k=0− (1 + 2r)(1 + z) 8k+28k + 2(1 + z) 8k+48k + 4∞∑k=0(1 + z) 8k+68k + 6116 k − 4r ∞ ∑k=0(1 + z) 8k+38k + 3116 k − (1 + 2r) ∞ ∑116 k + r ∞ ∑6k=0k=0(1 + z) 8k+18k + 1116 k −(1 + z) 8k+58k + 5(1 + z) 8k+78k + 7116 k −116 k .116 k −

A proof of the above asertion is given in [15].If in (10) we select (z, r) = (0, − 1 2 ) thenπ =∞∑k=0Likewise, using the identityone findsπ =∞∑k=0( 48k + 2 + 28k + 3 + 28k + 4 − 1 ) 12(8k + 7) 16 . karctan 1 + arctan z = arctan t , t := 1 + z1 − z, (z < 1) .( 28k + 1 + 28k + 2 + 18k + 3 − 12(8k + 5) − 12(8k + 6) − 1 ) 14(8k + 7) 16 . kLet us remark that for (z, r) = (0, 0) we find from (10) the remarkableBBP formula (attributed to David Bailey , Peter Borwein and SimonPlouffe) for π , that is(11)π =∞∑k=0( 48k + 1 − 28k + 4 − 18k + 5 − 1 ) 1.8k + 6 16 kAlso when (z, r) = (0, r), r ∈ C one finds another formula for π , namely=∞∑k=0( 4 + 8r8k + 1 −8r8k + 2 −π =4r8k + 3 − 2 + 8r8k + 4 − 1 + 2r8k + 5 − 1 + 2r8k + 6 +r ) 18k + 7 16 kwhich was discovered by Victor Adamchik and Stan Wagon in their interestingHTML paper [1] .References[1] V. Adamchik , S. Wagon , Pi : A 2000 -Year Search Changes Direction,see http : //www.members.wri.com/victor/articles/pi/pi.html[2] T. Amderberhan , Faster and faster convergent series for ζ(3) ,Electronic J. Combinatorics 3 (1996).[3] T. Amderberhan , D. Zeilberger , Hypergeometric series acceleration viathe WZ method, Electronic J. Combinatorics(Wilf Festschrifft Volume ) 4 (2) (1997).7

[4] R. Apéry , Irrationalité de ζ(2) et ζ(3) , Astérisque 61 (1979) 11-13.[5] D.H. Bailey ,The Computation of Pi to 29,360,000 Decimal Digits UsingBorweins’ Quartically Convergent Algorithm ,Mathematics of Computation, Jan. (1988) 283-296.[6] D.H. Bailey , P.B. Borwein , S. Plouffe , On the Rapid Computation ofVarious Polylogarithmic Constants ,Mathematics of Computation 66 (1997) 903-913.[7] D.H.Bailey , J.M. Borwein , S.Plouffe , The Quest for Pi ,The Mathematical Intelligencer 19(1) (1997) 50-57.[8] D.H. Bailey , D.J. Broadhurst , Parallel Integer Relation Detection :Techniques and Applications , Mathematics of Computation (to appear),see also http://front.math.ucdavis.edu/math.NA/9905048[9] J.M. Borwein , D.M. Bradley , R.E.Crandall , Computational strategiesfor the Riemann zeta function , CECM preprint 98:118 (1998).[10] P. Borwein , An Efficient Algorithm for the Riemann Zeta Function ,(to appear).[11] D.J. Broadhurst , Polylograithmic ladders , hypergeometric series andthe ten milionth digits of ζ(3) and ζ(5) , (Preprint) (1998).[12] R.E. Crandall ,Topics in Advanced Scientific Computation,Springer, New York, 1995.[13] S. Finch , Favorite Mathematical Constants ,http://www.mathsoft.com/asolve/constant/apery/apery.html.[14] S. Finch , The Miraculous Bailey-Borwein-Plouffe Pi Algorithm, (forthcoming),http : //www.mathsoft.com/asolve/plouffe/plouffe.html[15] A. Lupaş , Some BBP-Functions, DVI- paper , (see [13]-[14] )[16] A. van der Poorten , A Proof that Euler missed ... Apéry’s proof of theirrationality of ζ(3) , Math.Intelligencer 1 (1979) , 196-203.[17] D. Zeilberger , Closed Form (pun intended !) ,Contemporary Mathematics 143 (1993) 579-607.[18] H.S. Wilf , D. Zeilberger , Rational functions certify combinatorial identities,Jour. Amer.Math.Soc. 3 (1990) 1147-158.[19] H. Wilf , Accelerated series for universal constants ,by the WZ method , Discr.Math.Theoret.Comp.Sci. 3 (1999) 189-192 .8