Sharp inequalities and asymptotic expansion associated with the Wallis sequence

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1 Deng et al. Journal of Inequalities and Applications 0 0:86 DOI 0.86/s z R E S E A R C H Open Access Sharp inequalities and asymptotic expansion associated with the Wallis sequence Ji-En Deng, Tao Ban * and Chao-Ping Chen * Correspondence: bantao44000@sohu.com School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, Henan 44000, People s Republic of China Abstract We present asymptotic expansion of function involving the ratio of gamma functions and provide a recurrence relation for determining the coefficients of the asymptotic expansion. As a consequence, we obtain asymptotic expansion of the Wallis sequence. Also, we establish sharp inequalities for the Wallis sequence. MSC: Primary 40A0; secondary 33B; 4A60; 6D Keywords: Wallis sequence; gamma function; psi function; polygamma function; inequality; asymptotic expansion Introduction The Wallis sequence to which the title refers is W n = n k= 4k, 4k n N := {,,3,...}.. Wallis discovered that k= 4k 4k = = = π. see [], p.68. Several elementary proofs of. can be found see, for example, [ 4]. An interesting geometric construction produces.[]. Many formulasexist for the representation of π, and a collection of these formulas is listed [6, 7]. For more history of π see [, 8 0]. Some inequalities and asymptotic formulas associated with the Wallis sequence W n can be found see, for example, [4]. Hirschhorn [3]provedthatforn N, π 4n < W n < π 4n Also in [3], Hirschhorn pointed out that if the c j are given by x tanh = 4 j=0..3 c j x j+ j!,.4 0 Deng et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Denget al. Journal of Inequalities and Applications 0 0:86 Page of then, as n, W n π + cj exp = π + c j exp.. n n j+ n n j+ j 0 j=0 Very recently, Lin et al. [7]foundthat c j = j+ B j+ j+ j +j +, j N 0 := N {0},.6 where B n n N 0 are the Bernoulli numbers defined by the following generating function: z e z = z n B n, z <π. n! n=0 Also in [7], Lin et al. derived W n = π 4n + The gamma function is defined for x >0by Ɣx= 0 t x e t dt. 3 64n n 3 3,04n 4 +On, n..7 The logarithmic derivative of Ɣx, denoted by ψx =Ɣ x/ɣx, is called psi or digamma function, and ψ k xk N are called polygamma functions. These functions play an important role in various branches of mathematics as well as in physics and engineering. For the various properties of these functions, please refer to [],pp.-60. Define the function Wxby Wx= π + [ ] Ɣx + x x Ɣx +..8 It is easy to see that W n = Wn. The first aim of present paper is to establish sharp inequalities for W n.moreprecisely, we determine the best possible constants α, β, λ,andμ such that the double inequalities and π π < W n π 4n + α 4n + λ < W n π 4n + β 4n + μ

3 Denget al. Journal of Inequalities and Applications 0 0:86 Page 3 of hold for all n N. The second aim of present paper is to develop the formula.7toproduce a complete asymptotic expansion. More precisely, we provide a recurrence relation for determining the coefficients r j j N 0 suchthat Wx π 4x + j=0 r j x j, x. Lemmas The following lemmas are required in our present investigation. Lemma [6], Corollary Let m, n N. Then for x >0, m j= j Bj j! j + n! x j+n < n ψ n x + ψ n x + n! + x n m < j= j Bj j! j + n! x j+n,. where B n arethebernoullinumbers. It follows from.that,forx >0, x 8x + 64x < ψx + ψ x + < 4 8x6 x 8x +. 64x 4 and x + 4x 3 6x < ψ x + ψ x + < x + 4x 3 6x x..3 7 Lemma For all x, [ ] Ɣx + Ɣx + 3 < x 3 4x + 7 3x 4 3 8x ,048x..4 Proof We consider the function Gxdefined by [ Gx=ln Ɣx + ln Ɣ x + + ln x + ] ln x 3 4x + 7 3x 4 3 8x ,048x From the asymptotic expansion [], p.7: b a Ɣx + a ba + b x =+a Ɣx + b x a b + 3a + b a + b + as x,. x

4 Denget al. Journal of Inequalities and Applications 0 0:86 Page 4 of we conclude that lim x Gx=0. Differentiating and applying the first inequality in.yield,forx, [ G x= ψx + ψ x + ] 4,096x,048x x 3 384x + x, xx +,048x 4,36x 3 +,088x 70x +443 > x 8x + 64x 4 8x 6 4,096x,048x x 3 384x + x, xx +,048x 4,36x 3 +,088x 70x +443 = 8, ,4x +,606,06x +,69,00x 3 >0. +86,43x ,640x / 64x 6 x +,33 +,040x +8,768x + 6,66x 3 +,048x 4 This leads to [ ] Ɣx + Gx=ln Ɣx + 3 ln x 3 4x + 7 3x 4 3 8x ,048x < lim Gx=0, x. x The proof of Lemma is complete. By., we obtain [ x + ψx + ψ x + ] <x + x 8x + 64x 4 By.4, we get x + [ Ɣx + Ɣx + 3 The proof of Theorem makes use of.6and.7. = 4x 8x + 3x x 4..6 ] > x + x 3 4x + 7 3x 4 3 8x ,048x = 4x 3x + 8x 3 83,048x ,096x..7 Lemma 3 [7] Let a < b. Let f and g be differentiable functions on an interval a, b. Assume that either g >0everywhere on a, b or g <0on a, b. Suppose that f a+ = ga+ = 0 or f b = gb = 0. Then

5 Denget al. Journal of Inequalities and Applications 0 0:86 Page of if f g is increasing on a, b, then f g >0on a, b; if f g is decreasing on a, b, then f g <0on a, b. 3 Sharp inequalities Theorem For all n N, π < W n π 4n + α 4n + β 3. with the best possible constants α = and β = 3 9π 3π 8 = Equality in 3. occurs for n =. Proof The inequality 3.canbewrittenas α Fn<β, where Fx= x+/ [ 4x. Ɣx+ ] Ɣx+/ Using., we conclude that lim x Fx=. Differentiating Fx and applying.4,.6, and.7yield,forx 6, x + [ ] Ɣx + Ɣx + 3 F x { [ = x + ψx + ψ x + ] }[ Ɣx + Ɣx x + [ ] Ɣx + Ɣx + 3 < 4x 8x + 3x x 4 ] x 3 4x + 7 3x 4 3 8x ,048x 4x 3x + 8x 3 83,048x ,096x = 48,77, ,83,96x 6 + 0,4,660x 6 4,94,304x 0 <0. +49,038,464x 6 3 +,,84x 6 4 +,68,880x 6 +49,x 6 6

6 Denget al. Journal of Inequalities and Applications 0 0:86 Page 6 of Straightforward calculation produces F = F = F3 = F4 = F = F6 = 3 9π 3π 8 = , 3π +,04 =.74..., 4π 8,9π +6,44 =.6..., 7π 6,37π +4,88 =.4...,,0π 3,768 89,π +,6,440 43,69π 3,07,94,939π + 0,33, ,693π,097, = , = Thus, the sequence Fn n N is strictly decreasing. This leads to < lim 3 9π Fx<Fn F = x 3π 8, n N. The proof of Theorem is complete. Remark In fact, Elezović et al. [] havepreviouslyshownthat is the best possible constant for a lower bound of W n of the type π. Moreover, the authors pointed 4n+α out that W n = π 4n + Theorem For all n N, π 4n + + O, n. n 3 λ < W n π with the best possible constants 4n + λ = and μ = ln3π/8 ln3/ = Equality in 3. occurs for n =. Proof Inequality 3.canbewrittenas μ 3. λ > x n μ, where the sequence x n n N is defined by x n = ln Ɣn+ n+ ln 4n+ Ɣn+.

7 Denget al. Journal of Inequalities and Applications 0 0:86 Page 7 of We are now in a position to show that the sequence x n n N is strictly increasing. To this end, we consider the function f x defined by f x= ln Ɣx + ln Ɣx + lnx + ln 4x+ = f x f x, where f x=ln Ɣx + ln Ɣ x + ln x + and f x=ln 4x +. We conclude from the asymptotic formula of ln Ɣz[], p.7 that f = lim x f x=0. Elementary calculations show that 8f x f x = 64x +64x + [ ψx + ψ x + ] =: f 3 x. x + By using inequalities.and.3, we obtain, for x, f 3 [ψx x=8x ψ x + ] x x +64x + [ ψ x + ψ x + + [ >8x +64 x 8x + 64x 4 8x 6 x + ] x + ] + 64x +64x + [ x + 4x 3 6x + x + = 0 + 4,88x +7,860x +4,896x 3 +,368x 4 +44x 6x 6 x + >0. Hence, f 3 xand f x f f x= f x f x = f x f f x f x are both strictly increasing for x. By Lemma 3, the function is strictly increasing for x. Therefore, the sequence x n is strictly increasing for n. Direct computation would yield x = ln3π/8 ln3/ = , x 7 ln +ln 3+ln π + ln = = ln 9 + ln 3+ln 7 ]

8 Denget al. Journal of Inequalities and Applications 0 0:86 Page 8 of Consequently, the sequence x n n N is strictly increasing. This leads to lim x n > x n x = ln3π/8 n ln3/ for n N. It remains to prove that lim n x n =. 3.3 We conclude from the asymptotic formula of ln Ɣz[], p.7 that f x= +Ox +Ox asx, which implies 3.3. This completes the proofof Theorem. 4 Asymptotic expansion Theorem 3 The function Wx, defined by.8, has the following asymptotic expansion: Wx π 4x + with the coefficients r j given by the recurrence relation j=0 r j x j, x, 4. where and j r 0 =, r j =4 r k q j k+ 4p j+, j N, 4. p j = j + j+ j B j+, j N 4.3 j j jj + j j q j = j k, j N. 4.4 k + 4 k+ j k 8 Here, B n are the Bernoulli numbers. Proof Write 4.as ln π Wx ln r j, x. 4. xj 4x+ j=0 The logarithm of gamma function has asymptotic expansion see [8], p.3: ln Ɣx + t x + t ln x x + lnπ+ n+ B n+ t 4.6 nn + x n n=

9 Denget al. Journal of Inequalities and Applications 0 0:86 Page 9 of as x,whereb n t denotes the Bernoulli polynomials defined by the following generating function: xe tx e x = n=0 B n t xn n!. From 4.6, we obtain, as x, [ ] Ɣx + t /t s x exp Ɣx + s t s Setting s, t=,in4.7 and noting that j= B n 0 = n B n = B n and B n j+ B j+ t B j+ s jj + x j = n B n for n N see [], p.80, we obtain, as x, [ ] Ɣx + x exp Ɣx + j= j+ j B j+ jj + By using the Maclaurin expansion of ln+t,. 4.8 x j ln + t= j t j for < t, j j= we obtain + j exp x j j x j j= as x. 4.9 Applying 4.8and4.9yields with ln π Wx j= p j, x 4.0 xj p j = j + j+ j B j+, j N. 4. j j jj + The Maclaurin expansion of ln + twitht =,yields 4x+ ln 4x + j= j= + j j 4 j x j 8x j 4 j x j j k k 8 k x k

10 Denget al.journal of Inequalities and Applications 0 0:86 Page 0 of That is, with ln 4x + j= j= j 4 j x j k k + j k k 8 k x k j j j k k + 4 k+ j k 8 x. j j= q j x j j j q j = j k, j N. k + 4 k+ j k 8 It follows from 4.that j= p jx j j= q jx j j= j= p j x j p j x j We then obtain j j=0 r j x j r j x j, j=0 k= j j= q k x k, r k q j k x. j p j = r k q j k, j N, j p j = r k q j k + r j q, j. Noting that q = 4,weobtain j r j =4 r k q j k 4p j, j, and an empty sum as usual is understood to be nil. Noting that p = 4,wethenobtain the recurrence relation j r 0 =, r j =4 r k q j k+ 4p j+, j N. The proof of Theorem 3 is complete.

11 Denget al.journal of Inequalities and Applications 0 0:86 Page of Here, from 4., we give the following explicit asymptotic expansion: W n = π 4n n n 3 3,04n 4 n +, n, 4. which develops the formula.7 to produce a complete asymptotic expansion. Competing interests The authors declare that they have no competing interests. Authors contributions All authors read and approved the final manuscript. Acknowledgements The authors thank the referees for their careful reading of the manuscript and insightful comments. Received: January 0 Accepted: 8 May 0 References. Berggren, L, Borwein, J, Borwein, P eds.: Pi: A Source Book, nd edn. Springer, New York 000. Amdeberhan, T, Espinosa, O, Moll, VH, Straub, A: Wallis-Ramanujan-Schur-Feynman. Am. Math. Mon. 7, Miller, SJ: A probabilistic proof of Wallis formula for π. Am. Math. Mon., Wästlund, J: An elementary proof of the Wallis product formula for pi. Am. Math. Mon. 4, Myerson, G: The limiting shape of a sequence of rectangles. Am. Math. Mon. 99, Sofo, A: Some representations of π. Aust. Math. Soc. Gaz. 3, Sofo, A: π and some other constants. J. Inequal. Pure Appl. Math. 6, Article electronic 8. Beckmann, P: A History of Pi. St. Martin s, New York Dunham, W: Journey Through Genius: The Great Theorems of Mathematics. Penguin, Baltimore Agarwal, RP, Agarwal, H, Sen, SK: Birth, growth and computation of pi to ten trillion digits. Adv. Differ. Equ. 03, doi:0.86/ Burić, T, Elezović, N, Šimić, R: Asymptotic expansions of the multiple quotients of gamma functions with applications. Math.Inequal.Appl.6, Elezović, N, Lin, L, Vukšić, L: Inequalities and asymptotic expansions for the Wallis sequence and the sum of the Wallis ratio. J. Math. Inequal. 7, Hirschhorn, MD: Comments on the paper: Wallis sequence estimated through the Euler-Maclaurin formula: even from the Wallis product π could be computed fairly accurately by V. Lampret. Aust. Math. Soc. Gaz. 3, Lampret, V: Wallis sequence estimated through the Euler-Maclaurin formula: even from the Wallis product π could be computed fairly accurately. Aust. Math. Soc. Gaz. 3, Lampret, V: An asymptotic approximation of Wallis sequence. Cent. Eur. J. Math. 0, Lin, L: Further refinements of Gurland s formula for π. J. Inequal. Appl. 03, Lin, L, Deng, J-E, Chen, C-P: Inequalities and asymptotic expansions associated with the Wallis sequence. J. Inequal. Appl. 04, Mortici, C: Estimating π from the Wallis sequence. Math. Commun. 7, Mortici, C, Cristea, VG, Lu, D: Completely monotonic functions and inequalities associated to some ratio of gamma function.appl.math.comput.40, Mortici, C: Completely monotone functions and the Wallis ratio. Appl. Math. Lett., Mortici, C: Sharp inequalities and complete monotonicity for the Wallis ratio. Bull. Belg. Math. Soc. Simon Stevin 7, Mortici, C: A new method for establishing and proving new bounds for the Wallis ratio. Math. Inequal. Appl. 3, Mortici, C: New approximation formulas for evaluating the ratio of gamma functions. Math. Comput. Model., Păltănea, E: On the rate of convergence of Wallis sequence. Aust. Math. Soc. Gaz. 34, Abramowitz, M, Stegun, IA eds.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th edn. National Bureau of Standards: Applied Mathematics Series, vol.. Dover, New York Chen, CP, Paris, RB: Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function.appl.math.comput.0, Pinelis, I: L Hospital type rules for monotonicity, with applications. J. Inequal. Pure Appl. Math. 3, Article 00 electronic 8. Luke, YL: The Special Functions and Their Approximations, vol. I. Academic Press, New York 969