Construction Of Binomial Sums For π And Polylogarithmic Constants Inspired By BBP Formulas

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1 Applied Mathematics E-Notes, 77, 7-46 c ISSN 67-5 Available free at mirror sites of amen/ Construction Of Binomial Sums For π And Polylogarithmic Constants Inspired By BBP Formulas Boris Gourévitch, Jesús Guillera Goyanes Received 8 October 6 Abstract We present new sums involving binomial coefficients for π and various logarithms and polylogarithms constants. These sums are a generalization of BBP formulas first introduced by D. Bailey, P. Borwein and S. Plouffe in 995. In this paper, we describe how to find and prove such sums using the Beta function at integer and rational arguments. Introduction In 995, David Bailey, Peter Borwein and Simon Plouffe discovered a new seminal formula for π 4 π 6 n 8n + 8n +4 8n +5. 8n +6 n This amazing formula allows extracting the n-th binary digit of π without knowledge of previous digits []. Such sums are called BBP Bailey-Borwein-Plouffe formulas. Many new BBP formulas for π, G Catalan s constant or ζ Riemann Zeta function were subsequently discovered by Adamchik [], Bellard [], Broadhurst [4], Lupas [5] and Huvent [6]. The main tool for finding BBP formulas is an algorithm designed to detect linear combinations on the set of natural numbers N like PSLQ from its use of a partial-sum-of-squares PS vector and lower-diagonal-orthogonal LQ matrix factorization or LLL Lenstra Lenstra Lovász [7], [8]. To then prove such formulas, an integral is evaluated giving rise to logarithms or polylogarithms Li p z n zn, n p [9]. In October, Almkvist and colleagues presented new sums for π involving binomial coefficients such as π n Mathematics Subject Classifications: 4C 7 rue Montera, 75 Paris, boris@pi4.net Av. Cesareo Alierta, esc. izda 4oVA, Zaragoza 5n 6 n n n. 7

2 8 Binomial Sums for π and Polylogarithmic Constants They proved this formula using the so-called Beta method which is based on use of Beta function Br, s xr x s for positive integer values r and s []. In this paper, using an adapted Beta method, we show how to prove some more general sums of the form T q n mn n b mn + + b mn b k mn + k + + b m mn +m where T is a constant often related to π and the natural logarithm lnz. These sums involve binomial coefficients and they are similar to BBP formulas. We therefore call them BBP binomial formulas. One example of such formulas is π 687 n 7n n n n + 6 7n + 7n + 5 7n n n +6 4 that we discovered with LLL algorithm. An explicit proof of this formula is detailed in this paper. Then we show how to generate an infinite set of similar formulas for π and the natural logarithm lnz. We also present outlines for proofs of formulas showing that Beta method is not restrained to integer parameters. Proofs of BBP Binomial Formulas Consider the Eq.. Proof of such formula is reduced to the following proposition: PROPOSITION The Beta Method. There exists a polynomial P X Q m [X] polynomial of degree m with coefficients in the set of rational numbers Q such that qpx T 5 q x p x m p PROOF. The Beta function Br, s is Br, s ΓrΓs Γr + s 6 where Γz tz e t dt is the Gamma function. With parameters m, p, n N, we use Γn n! to obtain mn + mn x x m. 7

3 B. Gourévitch and J. Guillera Goyanes 9 Summing Eq. 7 on n with mn+ mn on the left side, we obtain n q n mn mn + q n n x x m n x p x m p n q q. 8 q x p x m p We can evaluate this integral by partial fraction decomposition. More generally, to obtain the elements in Eq. we write mn + k x +k x m mn+k +k mn k mn + mn +...mn + k. 9 For a given k, the partial fraction decomposition of k mn+mn+...mn+k in n gives a k,j j,..,k such that x k x x m mn ak, mn + + a k, mn a k,k mn + k with a k,j Q. For values k ranging from to m, we linearly combine a k,j j,..,k in order to find the m values b k of Eq.. This is equivalent to solving the system a, a,... a m, b a,... a m, V b a m,k b m Vector V contains the coefficients of the polynomial P X V t. m, such that q n mn n n when we use Eq.. b mn + + b mn P x x x m q n qpx q x p x m p X... X m b m mn +m of degree

4 4 Binomial Sums for π and Polylogarithmic Constants REMARK. Almkvist et al. [] use the relation n n n + xn x n in order to prove the formula in Eq.. In general, they consider formulas of the form Sn π q n n where S Q m [X] and q N. However, such forms of sums lead to the intricate evaluation of fx n mn +Sn x p x n. m p q fx is indeed an integral whose denominator is a power of the one considered in Eq. 8. Our method uses on the left side of Eq. 7, which produces formulas of the general form of mn+ mn Eq.. Proof of Eq. 4 We write Eq. 4 as mn U n π Using Proposition, we are looking for the polynomial P x such that x n x 5n U n P x P x n n n x x 5 P x x x +x 5 x 4 +x + x +. Because n 4 x x + π, it is likely that P x x 5 x 4 +x + x +. Let us verify this conjecture. To obtain the equivalent linear system in, and therefore vector V, we calculate all coefficients a k,i i columns of left matrix in Eq. with help from the computation of the integral in Eq.. For example, for k nd column of matrix, we have x.x n x 5n B5n +, n + n + 7n 7n + 7n + n 5 7n 7 7n n + n Consequently, a, 5 7 and a, 7. The whole linear system is V

5 B. Gourévitch and J. Guillera Goyanes 4 Its solution is V. Finally, we have P X V t X... X 5 which is the expected polynomial. +X +X X 4 + X 5 4 Construction and Prediction of BBP Binomial Formulas qpx Using proposition, we choose a suitable P x that simplifies. Knowing P then gives rise to a BBP formula by using Eq., after combination and q x m p x p simplification. As an example, we find formulas for π choosing P X such that due to +x qpx q x m p x p +x π 4. To find m and p, we have to know when + X divides q X m p X p i.e. when i and i are roots of q X m p X p. Case m p in Eq. is excluded since it only provides classical BBP formulas without binomial coefficients. Let us choose p,m, then i i i + i. Now there exists a formula for π with n n, p and q. Moreover, i and i are fourth roots of, so we can multiply x x by x 4k and keep roots i and i. Thus there exists formulas for π involving C+4kn n, k N. More generally, if KX q X m p X p, { } {i and i are roots of K, p < m} q ±i m p i p,p<m { ± i p m+p q±i m,p<m} { + i p m+p qi m,p<m} by complex conjugate { p } q and p π 4 m + pπ + m π [π],p<m with modulus and argument, where [x] designates modulo x { p q, p < m and p m [8] } { p k, k N, k q, p < m and 7p m [8] }. This idea also leads to other constants using the root ln the polynomial x x +π 9 x+x. For non-alternating sums b n mn+ + b mn bm mn+m with equivalent integral representation q n mn, or +x qpx q x p x m p, we obtain the following table for

6 4 Binomial Sums for π and Polylogarithmic Constants p<m: Polynomial Constant Condition given by roots Existence of a sum for p<m +x π i m p i p q p k, k q, 7p m [8] +x ln m p p q m p +k, q p x + x π p p+m +i q q,mp +6k +x ln z +x ln z+ z +z x ln z For alternating sums n bm with equiva- mn+m lent integral representation m p p q m p +k, q k p z m p z p q m p +k, q z k z p m p p z + z q m p +k, q z p m q n mn b mn+ + b mn z p+k qpx q+x p x m p, we get the following table: Polynomial Constant Condition given by roots Existence of a sum for p<m +x π i m p i p q p k, k q, 7p m + 4 [8] +x ln m p+ p q m p +k +,q p x + x π p+ +i p+m q q,mp ++6k +x ln z +x ln z+ z +z x ln z m p p q m p +k +,q k p z m p z p q m p +k +,q z k z p m p p z + z q m p +k +,q z p z p+k 5 Generalized Beta Function Method Albeit more intricate, some similar formulas can also be found using fractional parameters in Beta function. Considering Beta function and a b, c Q, we have d x + a b x mn+ Γ + a c d b + Γ mn + c d + Γ p + mn + a b + c d + a c b + d mn+ Γ a b Γ c d a b + d c Γ a p+mn+ b + c d Γx + n where x n xx+ x+n is Pochhammer symbol. Γ a Γx b is often known for small integers values of a et b. All following formulas stem from summing this relation on n for particular values of a b, c d. We now emphasize three particular cases of interest.

7 B. Gourévitch and J. Guillera Goyanes 4 5. Case : a b, c d We have q x q x p x m n x n x x mn q n q n x x mn q n Γ + Γmn + Γ p + mn + n + n!mn +!mn!. 4 q n mn +!! n EXAMPLE. Case p,m. We have π n xx + x x +x x n n x x +x x x x n n x n x n n x x +x x n x x n 5 So now using Eq. 4, we have x x +x x n x n x n [,,, ] x n+[ 5,,, ] T x n Γ n + [ 7, 5,, ] T Γn + [,,, ] Γ 4n + [ 9, 7, 5, ] T 6n!4n!n!4n 885 n!8n! 5 8n n n +5 + in simplified notations. Finally with Eq. 5 and Eq. 6, we have 6n 9 π n n n 885 8n n n n 4n 4n n 5 8n n Case : a + c Z positive and negative integers b d This case allows us to obtain special values of Γ a b Γ c d equal to π multiplied by an algebraic number. In some cases, π is absent and we obtain amazing formulas for algebraic numbers.. 6

8 44 Binomial Sums for π and Polylogarithmic Constants EXAMPLE. The following relations are based on Beta values for a b 4, c d or a b, c d or a b 5, c d n n 7 n n 4π n n 7 n n 8π which gives the formula 8 n x n 4 x n π x n 5 x n 4π n n n 54 n n +. n x n x n x n x n 5. Case : a + c Q\N rationals but not naturals b d We obtain Pochhammer symbols in sums then factors Γ a b Γ c d Γ a b + c in the outcome. d EXAMPLE. Considering x n 4 x n Γ n + 4 Γ n + Γ n x n+ 4 x n Γ n Γ n + Γ n and the following integral n x n xn 4 x n 4 π x xx + we obtain finally n n 4 n 8 n n n n 4 n 8n + + Γ 8n K π where Kk π/ dθ is the complete elliptic integral of the first kind, thus k sin θ making the connection with singular values of the elliptic integral. 6 Higher Order Constants One point of interest is to generalize this method to higher order constants. A constant is of order p if it is equal to a linear combination in N of p-th order polylogarithms. For example, π or ln are of order whereas π ln or G are of order and ζ or π

9 B. Gourévitch and J. Guillera Goyanes 45 are of order. A BBP binomial sum involves a constant of order p, as for BBP sums, if the polynomial in n in the denominator is of degree p. For example, we have n n n + ln + ζ. 7 n + n n n However, the straightforward application of integral representation does not completely help since, for order for instance, we have q n mn mn + n q dy. q y m x m x m p The evaluation is not often easy, there is no known proof of Eq. 7 to our knowledge. Nevertheless, there is evidence for existence of BBP binomial formulas for higher order constants. Acknowledgements. The authors would like to thank Typhenn Brichieri-Colombi, Professor Sui Sun Cheng and the anonymous reviewer for their helpful comments and corrections. References [] D. Bailey, P. Borwein and S. Plouffe, On The Rapid Computation of Various Polylogarithmic Constants, Math. Comp., 66997, 9 9. [] V. Adamchik, π : A -Year Search Changes Direction, Education and Research, 5996, 9 [] F. Bellard, Computation of the n th digit of π in any base in O n, preprint 997, n.ps [4] D. J. Broadhurst, Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ and ζ 5, preprint 998, [5] A. Lupas, Some BBP functions, preprint, Sibiu University, Romania. [6] G. Huvent, Formules BBP, Seminar ANO, IRMA, Lille University, huvent.pdf [7] A. K. Lenstra, H. W. Jr. Lenstra, L. Lovász, Factoring polynomials with rational coefficients, Mathematische Annalen 6498, [8] Pari-GP, software pour PC, MacOS et Unix, documentation at [9] L. Lewin, Polylogarithms and Associated Functions, North-Holland, New York, 98.

10 46 Binomial Sums for π and Polylogarithmic Constants [] G. Almkvist, C. Krattenthaler and J. Petersson, Some new formulas for π, preprint, Matematiska Institutionen, Lunds Universitet, Sweden. and Institut fur Mathematik der Universitat Wien, Austria,