MULTISECTION METHOD AND FURTHER FORMULAE FOR π. De-Yin Zheng
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1 Indian J. pure appl. Math., 39(: 37-55, April 008 c Printed in India. MULTISECTION METHOD AND FURTHER FORMULAE FOR π De-Yin Zheng Department of Mathematics, Hangzhou Normal University, Hangzhou 30036, Peoples Republic of China deyinzheng@yahoo.com.cn (Received 4 December 006; after final revision 6 September 007; accepted September 007 Applying the multisection series method to Lehmer s expansion formula [7], we devise a general procedure to find infinite series formulae for π. Several recent results due to Almkvist et al. [] are recovered and further new summation identities are established. Key words : Multisection series method, π-formula, central binomial coefficient. Introduction In classical combinatorics, there is the following useful multisection series result for formal power series. Lemma. (Comtet [4, P. 84]. Let f be a formal power series with complex coefficients. f f(x n 0 a nx n and w n : exp(πi/n a nth root of unity with n being a natural number. Then for each integer r with 0 r < n, the following formula holds: a r x r + a r+n x r+n + a r+n x r+n + n n wn kr f(wnx. k (. Applying this lemma to Lehmer s expansion formula [7, Eq. 5] (see also [, P. 385] x + x arcsin x ( x 3/ (x n ( n n this paper will investigate the infinite series expressions of π in form (±α n Λ m(n ( mn mn
2 38 DE-YIN ZHENG where α is a rational number and Λ m (n a polynomial of degree m in n. Several formulae due to Almkvist et al. [] are recovered and further new identities are derived. Throughout the paper, the following almost-trivial identity [8, P3] will frequently be used without explanation: m w sj m wmx s mxj, j 0,,,.... (. xm where m is a natural number and w m exp (πi/m a mth root of unity.. Multisection Method and Two Main Summation Theorems From the almost trivial relation we have the following expansion: a x x a {(x a + a(x a { (x a + a(x a / a ( ( k { (x a + a(x a k a k ( ( k k ( k (x a (a k j k+j k j ( a k (x a n With the U n -sequence being defined by U n (a : j0 the last expansion can be reformulated as follows: a x ( k (ak n ( a k ( ( k. k n k ( a ( ( k k k a, (. k n k (x a n (a n ( a ( ( k k k a. (. k n k Integrating the last equation with respect to x from a to x, we obtain arcsin x arcsin a + a U n (a (x a n+ (a n. (.3 n +
3 MULTISECTION METHOD AND FURTHER FORMULAE FOR π 39 Multiplying the last two equations together leads us to arcsin x x a + arcsin a a a + arcsin a a j0 U j (a (x aj (a j U n (a (x an (a n (x a n+ (a n U n (a (x an (a n U k (a (x a k+ (a k k + k + U k(au n k (a from which we derive further another expansion formula: φ(x : x + x arcsin x ( x 3/ d { arcsin x dx x Rewriting the convolution a + arcsin a a (x a n (a n n + k + U k(au n k (a (x a n (a n+ (n + U n+(a. m0 n + k + U k(au n k (a n + k + k i0 ( a a m ( a a i m i0 ( i we can alternatively express φ(x as i n + k + ( i k i n k j0 ( ( i i i k i ( a a j ( ( j j ( ( m i m i j n k j m i n m k + i where V (m, n φ(x + arcsin a a m i0 k + n + (x an (a n ( ( i i i k i m0 a m V (m, n ( a m+ (.4a U n+ (a (n + (x an (.4b (a n+ ( ( m i m i m i n m k + i. (.5
4 40 DE-YIN ZHENG Then by Taylor series expansion formula (.4 of φ(x, we have φ (n (a (n +! (a n (n +! (a n V (k, n a k arcsin a (n +! ( a + k+ a (a n+ U n+(a V (k, n a k arcsin a (n +! ( a + k+ a (a n+ ( a a k+ ( k + k + ( k + n k. Denote by D the derivative operator with respect to variable u at u. reformulate the last expression as follows: a n φ (n (a (n +! n + a arcsin a (n +! a n+ V (k, n Dk { k! a u ( ( k + k + D k k + n k k! { We can (.6a a. (.6b u In order to find the π-formula of the following form π r Λ m (n (a mn where Λ m (n ( mn mn m λ p n p (.7 we first consider, according to Lemma., the following multisection series of φ(x: ψ m (x : m m φ(ω s mx p0 (x mn where ω m exp ( πi m (.8 ( mn mn which leads us to the following derivative relation a n ψ m (n (a m (ω s m ma n φ (n (ωma. s (.9 On the other hand, it is not hard to get an alternative expression Λ m (n (a mn ( mn mn m λ p p0 m ( x d pψm λ p (x m dx xa p0 n p ( mn mn m p0 (a mn S(p, nλ p p m p a n ψ (n m (a (.0a (.0b
5 MULTISECTION METHOD AND FURTHER FORMULAE FOR π 4 where we have used the following derivative operator relation [4, P0] ( x d dx p with S(p, n being the second kind Stirling number. S(p, nx n dn dx n (. In view of (.6 and (.9, we derive from (.0 the following quadruplicate sum expression: Λ m (n (a mn ( mn mn + m m A m (p, n, k m m F m (s, k m B m (p, n, k F m (s, kg m (s (.a (.b where A m (p, n, k : (n +! S(p, nv (k, nλ p k! n+p m p, (.3a ( ( k + k + (n +! S(p, nλp B m (p, n, k : k + n k k! n+p+ m p+, (.3b F m (s, k : D k{ ωm s a, G m (s : ωs ma arcsin(ω s ma. (.3c u ω s m a Now we are going to simplify the finite sum m F m(s, k. Replacing x a u in the case j 0 of (. and noting that w m ω m, we have m ω s m a u m a m u m. The kth derivative of the last equation with respect to u at u results in m F m (s, k D k{ m a m u m.
6 4 DE-YIN ZHENG Substituting this relation into (., we establish the following theorem. Theorem. For the A m, B m, F m and G m given by (.3, there holds the following summation formula: Λ m (n (a mn ( mn mn + m m A m (p, n, kd k{ a m u m m B m (p, n, k F m (s, kg m (s. (.4a (.4b Making the replacement a ω m a in the last theorem and noting that ω m ω m, we have an alternative sum expression. Theorem. With the A m, B m, F m and G m being given by (.3, there holds the following summation formula: ( n Λ m(n (a mn ( mn mn + m m A m (p, n, kd k{ + a m u m m B m (p, n, k F m (s +, kg m (s +. (.5a (.5b 3. Further π-formulae corresponding to a /( 4 This section investigates the formulae for π exclusively with a being specified to /( 4. Our strategy consists of the following steps: Simplifying first the inner sums with respect to s displayed in (.4b and (.5b by employing the conjugate property of the summands under the involution s m s. One of these sums generally results in a multiple of π and the others are independent of π. Constructing the linear equation system by letting the triple sum corresponding to the inner π-sum just mentioned equal to one and the other triple sums to zero. Resolving the equation system and finding the solution for {λ k which yields the corresponding formula for π. We first consider the case m 4. Combining the following conjugation relations F 8 (, k F 8 (7, k, G 8 ( G 8 (7; F 8 (3, k F 8 (5, k, G 8 (3 G 8 (5,
7 MULTISECTION METHOD AND FURTHER FORMULAE FOR π 43 with the equation R(z z R(z R(z I(z I(z, we can simplify the finite sum 3 F 8 (s +, kg 8 (s + (3.a R{F 8 (, kg 8 ( + R{F 8 (3, kg 8 (3 R{F 8 (, kr{g 8 ( I{F 8 (, ki{g 8 ( (3.b + R{F 8 (3, kr{g 8 (3 I{F 8 (3, ki{g 8 (3. (3.c Keeping in mind of ω m exp(πi/m, we have no difficulty to express the following real part R{F 8 (s, k F 8 (s, k + F 8 (8 s, k D k{ ω8 sa u For the imaginary part, we have similarly + D k{ ω8 s a u D k{ a u cos sπ 4 a u cos sπ 4 + a4 u, s, 3. (3. I{F 8 (s, k i { F 8 (s, k F 8 (8 s, k D k{ a u sin sπ 4 a u cos sπ 4 + a4 u, s, 3. (3.3 In order to evaluate G 8 (, we need further two special values: arcsin(ω 8 a π 8 + i ln 4 and ω 8 a ω 8 a 5 + i 5. (3.4 The first one in (3.4 can be justified as follows. Using some elementary trigonometric identities, it is not hard to derive
8 44 DE-YIN ZHENG sin {arcsin(ω 8 a ± arcsin(ω 7 8a a a 4 ± a a + a 4. (3.5 For a /( 4, they reduce to the following evaluations: arcsin(ω 8 a + arcsin(ω 7 8a π 4, arcsin(ω 8 a arcsin(ω 7 8a arcsin( 3 i i ln whose sum gives the first value in (3.4. The second value displayed in (3.4 follows from the special case a /( 4 of the identity below: ( ω 8 a ω 8 a ± ω8 7 a ω 4 8 a a a 4 a + a 4 ± a a + a 4. (3.6 Multiplying both values displayed in (3.4 leads us to G 8 ( ω 8a arcsin(ω 8 a π ω 8 a 0 ln ( π 0 + i 40 + ln. (3.7 0 Recalling (3., we have 3 F 8 (s +, kg 8 (s + π 40 Dk{ 4 3 a u ln a u + a 4 u 0 Dk{ + a u a u + a 4 u + R{G 8 (3 D k{ + a u + a u+a 4 u π 5 Dk{ 6 3u 3 8u + u ln 5 I{G 8 (3 D k{ D k{ 8 + u 3 8u + u a u + a u+a 4 u + 64R{G 8 (3 D k{ { u+u R{G 8 (3 I{G 8 (3 D k{ u 3+8u+u (3.8a (3.8b (3.8c (3.8d (3.8e
9 MULTISECTION METHOD AND FURTHER FORMULAE FOR π Case m 4 In this case, identities (.5a-(.5b can be written as ( n Λ 4(n (a 8n ( 8n 4n 4 A 4 (p, n, k D k{ + a 8 u 4 (3.9a + π 5 4 B 4 (p, n, k D k h(, u (3.9b ln 5 4 B 4 (p, n, k D k h(, u (3.9c R{G 8 (3 B 4 (p, n, k D k h(3, u (3.9d { R{G 8 (3 I{G 8 (3 B 4 (p, n, k D k h(4, u (3.9e where h(, u : h(3, u : 6 3u 3 8u + u, h(, u : 8 + u 3 8u + u, (3.0a 3 + 8u + u, h(4, u : u 3 + 8u + u. (3.0b Just like the description at the beginning of this section, we consider the equation system determined by 0 4 A 4 (p, n, k D k{ + a 8 u 4 (3.a δ j 4 B 4 (p, n, k D k h(j, u, j,, 3, 4 (3.b where δ j is the Kronecker symbol with δ j, if j and δ j 0 otherwise. Resolving this equation system with the help of Mathematica we can get the numerical values for
10 46 DE-YIN ZHENG the coefficients {λ 0, λ, λ, λ 3, λ 4 of the polynomial Λ 4 (n, which leads us to the following explicit formula for π. Example 3. ([, Example 3.] We have π r Λ 4 (n ( 4 n ( 8n 4n where r (3. and Λ 4 (n n n n n Case m 8 In this case, the corresponding (.4a-(.4b reads as Λ 8 (n (a 6n ( 6n 8n A 8 (p, n, k D k{ a 6 u 8 B 8 (p, n, k 7 F 8 (s, kg 8 (s. (3.3a (3.3b According the parity of s, split the last sum into two parts 7 F 8 (s, kg 8 (s 3 F 8 (s +, kg 8 (s F 4 (s, kg 4 (s. (3.4 The first sum has been treated previously in (3.8. The second sum can be computed in straightforward way: 3 F 4 (s, kg 4 (s 3 G 4 (s D k{ i s a u G 4 (0 D k{ + a u a 4 u + G 4 ( D k{ a u a 4 u + G 4 ( D k{ + ia u + a 4 u + G 4 (3 D k{ ia u + a 4 u { G 4 (0 + G 4 ( D k{ { a 4 u + G 4 (0 G 4 ( D k{ a u a 4 u { + G 4 ( + G 4 (3 D k{ { + a 4 u + G 4 ( G 4 (3 D k{ ia u + a 4 u.
11 MULTISECTION METHOD AND FURTHER FORMULAE FOR π 47 Similar to 3., we consider the equation system given by 0 δ j 8 8 A 8 (p, n, k D k{ a 6 u 8 B 8 (p, n, k D k h(j, u, j,,..., 8 (3.5a (3.5b where for j,, 3, 4, h(j, u and h(4 + j, u are given respectively by (3.0 and the following fractions: h(5, u h(7, u a 4 u, h(6, u u a 4 u, (3.6a + a 4 u, h(8, u u + a 4 u. (3.6b Using Mathematica, we can solve the equation system (3.5 and obtain the numerical values of λ p for p 0,,,..., 8. This recovers the following formula due to Almkvist et al. Example 3. ([, Example 3.3] We have π r Λ 8 (n 6 n ( 6n 8n where r (3.7 and Λ 8 (n n n n n n n n n Case m l As we have seen that the π-formula corresponding to the case m 8 in 3.. We now turn to the more general π-formula displayed in (.4a and (.4b for m l with l 4, 5,. For the extreme inner sum of (.4b, we can recursively split it into the two parts as follows:
12 48 DE-YIN ZHENG m F m (s, kg m (s m/ m/ F m (s, kg m (s + F m (s +, kg m (s +. By ω m ω m/, the first part corresponds to the case of m/. We need therefore to consider only the second part. In order to derive the π-formula, we must impose the following equation: m m/ B m (p, n, k F m (s +, kg m (s + 0 which can be replaced by the stronger condition: m B m (p, n, kf m (s +, k 0, s 0,,,..., m. Because this equation system is difficult to solve, we replace it by the following m/ equivalent equations: m m/ B m (p, n, k ωm j(s+ F m (s +, k 0, j 0,,,..., m. Noting that w m ω m, ω m ω m/ and then specifying in (. with m m/ and x a u ω m/, we can write down the following identity: m/ ω j(s+ m a u ω (s+ m ma j u j ( + a m u m/, j 0,,,..., m. Just like the previous procedure, in order to find new π-formulae, we have to resolve the equation system: 0 δ j m m A m (p, n, k D k{ a m u m (3.8a B m (p, n, k D k h m (j, u, j,,..., m. (3.8b
13 MULTISECTION METHOD AND FURTHER FORMULAE FOR π 49 This can be done by putting the initial values for m 3 8 h 8 (j, u : h(j, u, j,,..., 8, (3.9 and successively making the replacements h m (j, u h m/ (j, u, j,,..., m (3.0a h m ( m + j, u uj + a m u m/, j,,..., m. (3.0b With the help of the Mathematica package designed for this task, we can resolve recursively the equation systems for m l with l 4, 5, and compute numerically the coefficients of the polynomials Λ m (n: λ p, p 0,,,..., m for m 6, 3,. Consequently, we can establish π-formulae of the form Λ m (n (a n, m 6, 3,. ( mn mn Here we present the formula corresponding to m 6. Example 3.3 ([, Example 3.4]. We have π r Λ 6 (n 56 n. (3. ( 3n 6n
14 50 DE-YIN ZHENG where r and Λ 6 (n n n n n n n n n n n n n n n n n New π-formulae corresponding to a / 4 Up to now, most of the π-formulae of the form (.7 concern exclusively the value a /( 4 apart from few exceptions (see [, Ex. 3.5] for example. This can be seen from
15 MULTISECTION METHOD AND FURTHER FORMULAE FOR π 5 the formulae established in Section 3 and those due to Almkvist et al. []. A natural question is that are there other values of a such that there hold the corresponding formulae of form (.7. An affirmative answer will be provided in this section. We shall establish several new π-formulae with a / 4. All the procedure developed in the last section applies also in this case. We need only to compute the four h-fractions displayed in (3.0, corresponding to a / 4 with m 4. Recalling (3.5 and (3.6, we have the two special values arcsin(ω 8 a π 4 + i ln( +, ω 8 a ω 8 a + i, (4. which leads directly to the following expression G 8 ( ω 8a arcsin(ω 8 a ω 8 a π 4 ln( + ( π + i 4 + ln( +. (4. From (3., (3. and (3.3, we can restate (3.8 as 3 F 8 (s +, kg 8 (s + π 4 Dk{ a u a u + a 4 u + R{G 8 (3 D k{ + a u + a u + a 4 u ln( + D k{ a u + a 4 u I{G 8 (3 D k{ a u + a u + a 4 u (4.3a (4.3b (4.3c π D k (, u ln( + D k (, u (4.3d + 4R{G 8 (3 D k (3, u + { R(G 8 (3 I(G 8 (3 D k (4, u (4.3e where the four -fractions are given by (, u : (3, u : u u + u, (, u : u + u, (4.4a + u + u, (4, u : u + u + u. (4.4b 4.. Case m 4 The corresponding identity (.5a-(.5b reads
16 5 DE-YIN ZHENG ( n Λ 4(n (a 8n ( 8n 4n 4 A 4 (p, n, k D k{ + a 8 u 4 (4.5a + π 4 B 4 (p, n, k D k (, u (4.5b ln( + 4 B 4 (p, n, k D k (, u (4.5c 4 + 4R{G 8 (3 B 4 (p, n, k D k (3, u (4.5d { 4 + R{G 8 (3 I{G 8 (3 B 4 (p, n, k D k (4, u. (4.5e Resolving, with the help of Mathematica, the following equation system 0 δ j 4 4 A 4 (p, n, k D k{ + a 8 u 4, B 4 (p, n, k D k (j, u, j,, 3, 4, we find the coefficients {λ 0, λ, λ, λ 3, λ 4 of the polynomial Λ 4 (n which yields the following new π-formula. Example 4. where r and π r ( n Λ 4(n8 n (4.6 ( 8n 4n Λ 4 (n n 78547n n n Case m 8 The same method in 3. works also in this case. The only difference lies in the fact that the corresponding equation system consists of (3.5 but with the h(j, u being replaced by (j, u within the four equations labeled by j,, 3, 4. The solution of this last equation system results in another new summation formula for π.
17 MULTISECTION METHOD AND FURTHER FORMULAE FOR π 53 Example 4. We have where r and π r Λ 8 (n8 4n (4.7 ( 6n 8n Λ 8 (n n n n n n n n n Case m l In order to search more general formulae of the form (.7, we need only first to replace h(j, u for j,, 3, 4 in (3.8 by (j, u displayed in (4.4 and then resolve the system. We produce one formula just for exemplifying the existence of the formula of this type. Example 4.3 We have π r Λ 6 (n8 8n. (4.8 where r and ( 3n 6n Λ 6 (n n n n n n n n n n 9
18 54 DE-YIN ZHENG n n n n n n n 6. Concluding Remarks The multisection series method has been successfully used to cover numerous π-formulas corresponding to m 4, 8,, 6, 4. In particular, for m l, we have developed a general iteration procedure which allows us theoretically to search more π-formulae. This last procedure applies also to other m-values. For the limit of space, we are not going to reproduce further long and tedious formulae. Acknowledgement The author thanks professor Wenchang Chu for the encouragements and suggestions during the preparation of this work. The author is very grateful to the referees for their various suggestions that led to an improvement of this paper. This work has been done under the auspices of the project on Fundamental Mathematics supported by Hangzhou City. References. G. Almkvist, C. Krattenthaler and J. Petersson Some New Formulas for π, Experimental Math., (003, J. M. Borwein and P. B. Borwein, Pi and the AGM, John Wiley & Sons, New York ( W. Chu, Summations on Trigonometric Functions, Appl. Math. & Comput., 4 (003, L. Comtet Advanced Combinatorics, Reidel, Boston, Mass. ( C. Krattenthaler, Advanced Determinant Calculus, Séminaire Lotharingien Combin., 4 (999 ( The Andrews Festschrift, Article B4q, 67 pp. ar xiv, math
19 MULTISECTION METHOD AND FURTHER FORMULAE FOR π C. Krattenthaler, Advanced Determinant Calculus: A Complement, Linear Algebra appl., 4 (005, D. H. Lehmer, Interesting Series Involving the Central Binomial Coefficient, Amer. Math. Monthly, 9 (985, J. Riordan, Combinatorial Identities, John Wiley & Sons, New York (968.
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