International Journal of Analysis and Applications ISSN 2291-8639 Volume 8, Number 1 (2015), 30-38 http://www.etamaths.com ON THE WALLIS FORMULA BAI-NI GUO1,∗, FENG QI2,3 Abstract. By virtue of complex methods and tools, the authors express the famous Wallis formula as a sum involving binomial coefficients, establish the expansions for sink x and cosk x in terms of cos(mx), find the general formulas for the derivatives of sink x and cosk x, and recover the general multiple-angle formulas for sin(kx) and cos(kx), where k ∈ N and m ∈ Z. 1. Introduction It is well known [8, 9, 16, 18, 23] that (1.1) In = ∫ π/2 0 cosn x d x = ∫ π/2 0 sinn x d x = (n− 1)!! n!! × {π 2 for n even 1 for n odd for n ∈ N, where n!! denotes a double factorial. Usually we call (1.1) the Wallis cosine or sine formula, or simply say, the Wallis formula, in the literature. In mathematical analysis, the Wallis formula (1.1) is derived generally by integrating by parts and mathematical induction. The formula (1.1) may also be represented by In = √ π Γ((n + 1)/2) nΓ(n/2) = √ π 2 Γ((n + 1)/2) Γ((n + 2)/2) , where Γ(x) stands for the classical Euler gamma function which may defined by Γ(z) = ∫ ∞ 0 tz−1e−t d t, <(z) > 0. The Wallis ratio is defined [42] as Wn = (2n− 1)!! (2n)!! = (2n)! 22n(n!)2 = 1 √ π Γ ( n + 1 2 ) Γ(n + 1) , n ∈ N. It is clear that for n ∈ N (1.2) Wn = 2 π I2n = 1 22n ( 2n n ) and I2n−1I2n = π 4n . There have existed plenty of literature about bounding the Wallis ratio. See, for example, [4, 5, 6, 7, 9, 16, 17, 19, 20, 22, 42, 43, 47]. 2010 Mathematics Subject Classification. Primary 33B10; Secondary 26A06, 26A09, 33B15. Key words and phrases. Wallis formula; sine; cosine; derivative; multiple-angle formula. c©2015 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 30 ON THE WALLIS FORMULA 31 In [18], the Wallis formula (1.1) was generalized as I(t) = ∫ π/2 0 cost x d x = ∫ π/2 0 sint x d x = √ π 2 Γ((t + 1)/2) Γ((t + 2)/2) , t ≥ 0. See also [27, Section 2.3] and [48, 49]. In [2, p. 123], it was claimed that if Im,n is a primitive of sin m x cosn x for m,n ∈ R, then Im+2,n = − sinm+1 x cosn+1 x m + n + 2 + m + 1 m + n + 2 Im,n is a primitive of sinm+2 x cosn x if m + n + 2 6= 0. With the aid of this formula the formula (1.1) may be recovered. In [3, 10], by establishing double inequalities for I2n−1 and I2n, the double in- equality √ π√ 1 + (9π/16 − 1)/n ≤ ∫ √n − √ n e−x 2 d x < √ π√ 1 − 3/(4n) was obtained for n ∈ N. As a result, the probability integral∫ ∞ 0 e−x 2 d x = √ π 2 was recovered. For more information, please refer to [2, p. 123], [22, 34] and related references therein. In [13, 44], among other things, the sequence nI2n for n ∈ N, which originates from computation of the probability of intersecting between a plane couple and a convex body, was proved to be increasing. For recent developments on the gamma function and the ratios of two gamma functions, please refer to the papers [11, 12, 14, 15, 21, 24, 25, 26, 29, 30, 32, 33, 35, 36, 37, 40, 41, 45, 46], the expository and survey articles [27, 28, 38, 39] and closely related references therein. The aims of this paper are, by virtue of complex methods and tools, to express the sequence I2n−1 as a sum involving binomial coefficients and to recover the identity (1.2). As by-products, the expansions for sink x and cosk x in terms of cos(mx) for m ∈ Z, the derivatives for sink x and cosk x, and the general multiple- angle formulas for sin(kx) and cos(kx) are established and recovered. 2. Main results Now we are in a position to establish and recover our main results and by- products. Theorem 2.1. For n ∈ N, we have (2.1) I2n−1 = (−1)n+1 22n−1 2n−1∑ k=0 (−1)k 2n− 2k − 1 ( 2n− 1 k ) . First proof. Let i = √ −1 be the imaginary unit. Then for n ∈ N we have I2n−1 = ∫ π/2 0 ( eix + e−ix 2 )2n−1 d x 32 GUO AND QI = 1 22n−1 ∫ π/2 0 2n−1∑ `=0 ( 2n− 1 ` ) ei`xe−i(2n−1−`)x d x = 1 22n−1 2n−1∑ `=0 ( 2n− 1 ` )∫ π/2 0 ei(2`−2n+1)x d x = 1 22n−1 2n−1∑ `=0 ( 2n− 1 ` ) 1 i(2`− 2n + 1) [ ei(2`−2n+1)π/2 − 1 ] = 1 22n−1 2n−1∑ `=0 ( 2n− 1 ` ) 1 2`− 2n + 1 i [ 1 −ei(2`−2n+1)π/2 ] = 1 22n−1 2n−1∑ `=0 ( 2n− 1 ` ) 1 2`− 2n + 1 sin (2`− 2n + 1)π 2 = 1 22n−1 2n−1∑ `=0 ( 2n− 1 ` ) 1 2`− 2n + 1 cos[(`−n)π] = 1 22n−1 2n−1∑ `=0 ( 2n− 1 ` ) (−1)`−n 2`− 2n + 1 . The formula (2.1) follows. � Second proof. For n ∈ N, we have In = ∫ π/2 0 ( eix −e−ix 2i )n d x = 1 2n ∫ π/2 0 [ ei(x−π/2) −e−i(x+π/2) ]n d x = 1 2n ∫ π/2 0 n∑ `=0 (−1)n−` ( n ` ) ei`(x−π/2)e−i(n−`)(x+π/2) d x = 1 2n n∑ `=0 (−1)n−` ( n ` )∫ π/2 0 ei[(2`−n)x−nπ/2] d x = 1 2n n∑ `=0 (−1)n−` ( n ` )∫ π/2 0 cos [ (2`−n)x−n π 2 ] d x. Therefore, it follows that I2n−1 = −1 22n−1 2n−1∑ `=0 (−1)` ( 2n− 1 ` )∫ π/2 0 cos [ (2`− 2n + 1)x− (2n− 1) π 2 ] d x = (−1)n 22n−1 2n−1∑ `=0 (−1)` ( 2n− 1 ` )∫ π/2 0 sin[(2`− 2n + 1)x] d x = (−1)n+1 22n−1 2n−1∑ `=0 (−1)` ( 2n− 1 ` ) 1 2`− 2n + 1 [ cos (2`− 2n + 1)π 2 − 1 ] ON THE WALLIS FORMULA 33 = (−1)n 22n−1 2n−1∑ `=0 (−1)` ( 2n− 1 ` ) 1 2`− 2n + 1 . The proof is completed. � Corollary 2.1. For ` ∈ N, we have cos` x = 1 2` ∑̀ q=0 ( ` q ) cos[(2q − `)x],(2.2) sin` x = (−1)` 2` ∑̀ q=0 (−1)q ( ` q ) cos [ (2q − `)x− ` 2 π ] ,(2.3) and ∑̀ q=0 ( ` q ) sin[(2q − `)x] = 0,(2.4) ∑̀ q=0 (−1)q ( ` q ) sin [ (2q − `)x− ` 2 π ] = 0.(2.5) Proof. From the second proof of Theorem 2.1, we conclude that cos` x = 1 2` (eix + e−ix)` = 1 2` ∑̀ q=0 ( ` q ) eqixe−(`−q)ix = 1 2` ∑̀ q=0 ( ` q ) e(2q−`)ix = 1 2` ∑̀ q=0 ( ` q ) {cos[(2q − `)x] + i sin[(2q − `)x]}. Equating the real and imaginary parts in the above equality gives equalities (2.2) and (2.4). Similarly, we have sin` x = 1 (2i)` ∑̀ q=0 (−1)`−q ( ` q ) eqixe−(`−q)ix = (−1)` (2i)` ∑̀ q=0 (−1)q ( ` q ) e(2q−`)ix = (−1)` 2` e−πi`/2 ∑̀ q=0 (−1)q ( ` q ) e(2q−`)ix = (−1)` 2` ∑̀ q=0 (−1)q ( ` q ) e[(2q−`)x−π`/2]i = (−1)` 2` ∑̀ q=0 (−1)q ( ` q ){ cos [ (2q − `)x− ` 2 π ] + i sin [ (2q − `)x− ` 2 π ]} . Hence, we obtain equalities (2.3) and (2.5). � Corollary 2.2. For m,k ∈ N, we have dm cosk x d xm = 1 2k k∑ q=0 ( k q ) (2q −k)m cos [ π 2 m + (2q −k)x ] ,(2.6) dm sink x d xm = (−1)k 2k k∑ q=0 (−1)q ( k q ) (2q −k)m cos [ (m−k) π 2 + (2q −k)x ] ,(2.7) 34 GUO AND QI and k∑ q=0 ( k q ) (2q −k)m sin [ π 2 m + (2q −k)x ] = 0, k∑ q=0 (−1)q ( k q ) (2q −k)m sin [ (m−k) π 2 + (2q −k)x ] = 0. Proof. These identities follow from directly differentiating on all the sides of the identities in Corollary 2.1. � Remark 2.1. The formulas (2.6) and (2.7) were established and applied in the paper [31]. Theorem 2.2. For n ∈ N, we have (2.8) I2n = π 22n+1 ( 2n n ) . First proof. A direct calculation reveals that I2n = ∫ π/2 0 ( eix + e−ix 2 )2n d x = 1 22n ∫ π/2 0 2n∑ `=0 ( 2n ` ) ei`xe−i(2n−`)x d x = 1 22n 2n∑ `=0 ( 2n ` )∫ π/2 0 ei(2`−2n)x d x = 1 22n [( n−1∑ `=0 + 2n∑ `=n+1 )( 2n ` )∫ π/2 0 ei(2`−2n)x d x + π 2 ( 2n n )] = π 22n+1 ( 2n n ) + 1 22n ( n−1∑ `=0 + 2n∑ `=n+1 )( 2n ` ) 1 i(2`− 2n) [ ei(2`−2n)π/2 − 1 ] = π 22n+1 ( 2n n ) + 1 22n ( n−1∑ `=0 + 2n∑ `=n+1 )( 2n ` ) i 2`− 2n [ 1 −ei(2`−2n)π/2 ] = π 22n+1 ( 2n n ) + 1 22n ( n−1∑ `=0 + 2n∑ `=n+1 )( 2n ` ) 1 2(`−n) sin 2(`−n)π 2 = π 22n+1 ( 2n n ) . Consequently, the formula (2.8) is proved. � Second proof. By virtue of (2.3), it follows that I2n = 1 22n 2n∑ `=0 (−1)` ( 2n ` )∫ π/2 0 cos[(2`− 2n)x−nπ] d x = (−1)n 22n 2n∑ `=0 (−1)` ( 2n ` )∫ π/2 0 cos[(2`− 2n)x] d x ON THE WALLIS FORMULA 35 = (−1)n 22n [ (−1)n ( 2n n ) π 2 + ( n−1∑ `=0 + 2n∑ `=n+1 ) (−1)` ( 2n ` ) 1 2`− 2n sin (2`− 2n)π 2 ] = π 22n+1 ( 2n n ) . As a result, the formula (2.8) is proved. � Third proof. Letting ` = 2n and integrating from 0 to π 2 on both sides of (2.2) arrive at the formula (2.8). � Remark 2.2. In [2, p. 100], the formula (2.8) was proved alternatively. 3. General multiple-angle formulas for sine and cosine Let i = √ −1 be the imaginary unit. Then ik =   i, k = 1 + 4`, −1, k = 2 + 4`, −i, k = 3 + 4`, 1, k = 4 + 4`, where k ∈ N and ` ≥ 0. The quantity ik may also be computed by ik = (−1) 1 2 [ k−1−(−1) k 2 ] i 1−(−1)k 2 and ik = ekπi/2 = cos kπ 2 + i sin kπ 2 . It is well known [1, p. 72] that the first few multiple-angle formulas are sin(2x) = 2 sin x cos x, cos(2x) = cos2 x− sin2 x = 2 cos2 x− 1 = 1 − 2 sin2 x, sin(3x) = 3 sin x− 4 sin3 x = 4 sin x sin ( π 3 + x ) sin ( π 3 −x ) , cos(3x) = 4 cos3 x− 3 cos x = 4 cos x cos ( π 3 + x ) cos ( π 3 −x ) , sin(4x) = 8 cos3 x sin x− 4 cos x sin x, cos(4x) = 8 cos4 x− 8 cos2 x + 1. Theorem 3.1. For k ≥ 2, the general multiple-angle formulas for the sine and cosine functions are sin(kx) = k∑ `=0 ( k ` ) sin `π 2 sin` x cosk−` x and cos(kx) = k∑ `=0 ( k ` ) cos `π 2 sin` x cosk−` x. Proof. By the formula ekxi = cos(kx) + i sin(kx), we have ekxi = ( exi )k = (cos x + i sin x)k 36 GUO AND QI = k∑ `=0 ( k ` ) i` sin` x cosk−` x = k∑ `=0 ( k ` )[ cos `π 2 + i sin `π 2 ] sin` x cosk−` x = k∑ `=0 ( k ` ) cos `π 2 sin` x cosk−` x + i k∑ `=0 ( k ` ) sin `π 2 sin` x cosk−` x. Further equating the real and imaginary parts yields the required general multiple- angle formulas for the sine and cosine functions. 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Math. 7 (2006), no. 2, Art. 56. 1School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China 2College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China 3Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China ∗Corresponding author