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Some exact constants for the approximation of the quantity in the Wallis’ formula
Journal of Inequalities and Applications volume 2013, Article number: 67 (2013)
Abstract
In this article, a sharp two-sided bounding inequality and some best constants for the approximation of the quantity associated with the Wallis’ formula are presented.
MSC:41A44, 26D20, 33B15.
1 Introduction and main result
Throughout the paper, ℤ denotes the set of all integers, ℕ denotes the set of all positive integers,
and
Here in (1), the floor function denotes the integer which is less than or equal to the number t.
The Euler gamma function is defined and denoted for by
One of the elementary properties of the gamma function is that
In particular,
Also, note that
For the approximation of n!, a well-known result is the following Stirling’s formula:
which is an important tool in analytical probability theory, statistical physics and physical chemistry.
Consider the quantity , defined by (2). This quantity is important in the probability theory - for example, the three events, (a) a return to the origin takes place at time 2n, (b) no return occurs up to and including time 2n, and (c) the path is non-negative between 0 and 2n, have the common probability . Also, the probability that in the time interval from 0 to 2n the particle spends 2k time units on the positive side and time units on the negative side is . For details of these interesting results, one may see [[1], Chapter III].
is closely related to the Wallis’ formula.
The Wallis’ formula
can be obtained by taking
in the infinite product representation of sinx (see [[2], p.10], [[3], p.211])
Since
another important form of Wallis’ formula is (see [[4], pp.181-184])
The following generalization of Wallis’ formula was given in [5].
In fact, by letting
in (9), we have
From (13), we get
for
(12) is a special case of (14). The proof of (12) in [5] involves integrating powers of a generalized sine function.
There is a close relationship between Stirling’s formula and Wallis’ formula. The determination of the constant in the usual proof of Stirling’s formula (7) or Stirling’s asymptotic formula
relies on Wallis’ formula (see [[2], pp.18-20], [[3], pp.213-215], [[4], pp.181-184]).
Also, note that
and Wallis’ sine (cosine) formula (see [[6], p.258])
Some inequalities involving were given in [7–12].
In this article, we give a sharp two-sided bounding inequality and some exact constants for the approximation of , defined by (2). The main result of the paper is as follows.
Theorem 1 For all , ,
The constants and in (20) are best possible.
Moreover,
Remark 1 By saying that the constants and in (20) are best possible, we mean that the constant in (20) cannot be replaced by a number which is greater than and the constant in (20) cannot be replaced by a number which is less than .
2 Lemmas
We need the following lemmas to prove our result.
Lemma 1 ([[13], Theorem 1.1])
The function
is strictly logarithmically concave and strictly increasing from onto .
Lemma 2 ([[13], Theorem 1.3])
The function
is strictly logarithmically concave and strictly increasing from onto .
Lemma 3 ([[6], p.258])
For all ,
where is defined by (2).
Remark 2 Some functions associated with the functions and , defined by (22) and (23) respectively, were proved to be logarithmically completely monotonic in [14–16]. For more recent work on (logarithmically) completely monotonic functions, please see, for example, [17–43].
3 Proof of the main result
Proof of Theorem 1
By Lemma 1, we have
and
The lower and upper bounds in (25) are best possible.
By Lemma 3, (25) and (26) can be rewritten respectively as
and
The constants and in (27) are best possible.
By Lemma 2, we get
and
The lower bound and the upper bound in (29) are best possible.
From (27) and (29), we obtain that for all ,
The constants and in (31) are best possible. From (31) we get that for all ,
The constants and in (32) are best possible.
From (28) and (30), we see that
which is equivalent to (21).
The proof is thus completed. □
References
Feller W: An Introduction to Probability Theory and Its Applications. Wiley, New York; 1966.
Andrews GE, Askey R, Roy R: Special Functions. Cambridge University Press, Cambridge; 1999.
Webster R: Convexity. Oxford University Press, Oxford; 1994.
Mitrinović DS: Analytic Inequalities. Springer, Berlin; 1970.
Piros M: A generalization of the Wallis’ formula. Miskolc Math. Notes 2003, 4: 151–155.
Abramowitz M, Stegun IA: Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover, New York; 1966.
Cao J, Niu D-W, Qi F: A Wallis type inequality and a double inequality for probability integral. Aust. J. Math. Anal. Appl. 2007., 5: Article ID 3
Chen C-P, Qi F: Best upper and lower bounds in Wallis’ inequality. J. Indones. Math. Soc. 2005, 11: 137–141.
Chen C-P, Qi F: Completely monotonic function associated with the gamma function and proof of Wallis’ inequality. Tamkang J. Math. 2005, 36: 303–307.
Chen C-P, Qi F: The best bounds in Wallis’ inequality. Proc. Am. Math. Soc. 2005, 133: 397–401. 10.1090/S0002-9939-04-07499-4
Qi F: Bounds for the ratio of two gamma functions. J. Inequal. Appl. 2010., 2010: Article ID 493058
Qi F, Luo Q-M: Bounds for the ratio of two gamma functions - from Wendel’s and related inequalities to logarithmically completely monotonic functions. Banach J. Math. Anal. 2012, 6: 132–158.
Guo S: Monotonicity and concavity properties of some functions involving the gamma function with applications. J. Inequal. Pure Appl. Math. 2006., 7: Article ID 45
Guo S, Qi F: A logarithmically complete monotonicity property of the gamma function. Int. J. Pure Appl. Math. 2008, 43: 63–68.
Guo S, Qi F, Srivastava HM: Supplements to a class of logarithmically completely monotonic functions associated with the gamma function. Appl. Math. Comput. 2008, 197: 768–774. 10.1016/j.amc.2007.08.011
Guo S, Srivastava HM: A class of logarithmically completely monotonic functions. Appl. Math. Lett. 2008, 21: 1134–1141. 10.1016/j.aml.2007.10.028
Alzer H, Batir N: Monotonicity properties of the gamma function. Appl. Math. Lett. 2007, 20: 778–781. 10.1016/j.aml.2006.08.026
Batir N: On some properties of the gamma function. Expo. Math. 2008, 26: 187–196. 10.1016/j.exmath.2007.10.001
Chen C-P, Qi F: Logarithmically completely monotonic functions relating to the gamma function. J. Math. Anal. Appl. 2006, 321: 405–411. 10.1016/j.jmaa.2005.08.056
Grinshpan AZ, Ismail ME-H: Completely monotonic functions involving the gamma and q -gamma functions. Proc. Am. Math. Soc. 2006, 134: 1153–1160. 10.1090/S0002-9939-05-08050-0
Guo B-N, Qi F: A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications. J. Korean Math. Soc. 2011, 48: 655–667.
Guo S: Some properties of completely monotonic sequences and related interpolation. Appl. Math. Comput. 2013, 219: 4958–4962. 10.1016/j.amc.2012.11.073
Guo S: Some classes of completely monotonic functions involving the gamma function. Int. J. Pure Appl. Math. 2006, 30: 561–566.
Guo S, Qi F: A class of logarithmically completely monotonic functions associated with the gamma function. J. Comput. Appl. Math. 2009, 224: 127–132. 10.1016/j.cam.2008.04.028
Guo S, Qi F: A class of completely monotonic functions related to the remainder of Binet’s formula with applications. Tamsui Oxf. J. Math. Sci. 2009, 25: 9–14.
Guo S, Qi F, Srivastava HM: A class of logarithmically completely monotonic functions related to the gamma function with applications. Integral Transforms Spec. Funct. 2012, 23: 557–566. 10.1080/10652469.2011.611331
Guo S, Qi F, Srivastava HM: Necessary and sufficient conditions for two classes of functions to be logarithmically completely monotonic. Integral Transforms Spec. Funct. 2007, 18: 819–826. 10.1080/10652460701528933
Guo S, Srivastava HM: A certain function class related to the class of logarithmically completely monotonic functions. Math. Comput. Model. 2009, 49: 2073–2079. 10.1016/j.mcm.2009.01.002
Qi F: A class of logarithmically completely monotonic functions and application to the best bounds in the second Gautsch-Kershaw’s inequality. J. Comput. Appl. Math. 2009, 224: 538–543. 10.1016/j.cam.2008.05.030
Qi F: Three classes of logarithmically completely monotonic functions involving gamma and psi functions. Integral Transforms Spec. Funct. 2007, 18: 503–509. 10.1080/10652460701358976
Qi F, Cui R-Q, Chen C-P, Guo B-N: Some completely monotonic functions involving polygamma functions and an application. J. Math. Anal. Appl. 2005, 310: 303–308. 10.1016/j.jmaa.2005.02.016
Qi F, Guo B-N: Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic. Adv. Appl. Math. 2010, 44: 71–83. 10.1016/j.aam.2009.03.003
Qi F, Guo B-N: A logarithmically completely monotonic function involving the gamma function. Taiwan. J. Math. 2010, 14: 1623–1628.
Qi F, Guo B-N: Some logarithmically completely monotonic functions related to the gamma function. J. Korean Math. Soc. 2010, 47: 1283–1297. 10.4134/JKMS.2010.47.6.1283
Qi F, Guo B-N: Wendel’s and Gautschi’s inequalities: refinements, extensions, and a class of logarithmically completely monotonic functions. Appl. Math. Comput. 2008, 205: 281–290. 10.1016/j.amc.2008.07.005
Qi F, Guo B-N: A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw’s double inequality. J. Comput. Appl. Math. 2008, 212: 444–456. 10.1016/j.cam.2006.12.022
Qi F, Guo B-N, Chen C-P: Some completely monotonic functions involving the gamma and polygamma functions. J. Aust. Math. Soc. 2006, 80: 81–88. 10.1017/S1446788700011393
Qi F, Guo S, Guo B-N: Complete monotonicity of some functions involving polygamma functions. J. Comput. Appl. Math. 2010, 233: 2149–2160. 10.1016/j.cam.2009.09.044
Sevli H, Batir N: Complete monotonicity results for some functions involving the gamma and polygamma functions. Math. Comput. Model. 2011, 53: 1771–1775. 10.1016/j.mcm.2010.12.055
Shemyakova E, Khashin SI, Jeffrey DJ: A conjecture concerning a completely monotonic function. Comput. Math. Appl. 2010, 60: 1360–1363. 10.1016/j.camwa.2010.06.017
Koumandos S, Pedersen HL: Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler’s gamma function. J. Math. Anal. Appl. 2009, 355: 33–40. 10.1016/j.jmaa.2009.01.042
Srivastava HM, Choi J: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam; 2012.
Srivastava HM, Guo S, Qi F: Some properties of a class of functions related to completely monotonic functions. Comput. Math. Appl. 2012, 64: 1649–1654. 10.1016/j.camwa.2012.01.016
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The authors thank the editor and the referees for their valuable suggestions to improve the quality of this paper. The present investigation was supported, in part, by the Natural Science Foundation of Henan Province of China under Grant 112300410022.
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Guo, S., Xu, JG. & Qi, F. Some exact constants for the approximation of the quantity in the Wallis’ formula. J Inequal Appl 2013, 67 (2013). https://doi.org/10.1186/1029-242X-2013-67
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DOI: https://doi.org/10.1186/1029-242X-2013-67