##### Title: On the Wallis Formula

##### Pages: 30-38

##### Cite as:

Bai-Ni Guo, Feng Qi, On the Wallis Formula, Int. J. Anal. Appl., 8 (1) (2015), 30-38.#### 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 2 N and m∈Z.

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#### References

- M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Dover Publications, New York and Washington, 1972.
- N. Bourbaki, Functions of a Real Variable, Elementary Theory, Translated from the 1976 French original by Philip Spain. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2004.
- J. Cao, D.-W. Niu, and F. Qi, A Wallis type inequality and a double inequality for probability integral, Aust. J. Math. Anal. Appl. 4 (2007), no. 1, Art. 3.
- C.-P. Chen and F. Qi, Best upper and lower bounds in Wallis’ inequality, J. Indones. Math. Soc. (MIHMI) 11 (2005), no. 2, 137–141.
- C.-P. Chen and F. Qi, Completely monotonic function associated with the gamma function and proof of Wallis’ inequality, Tamkang J. Math. 36 (2005), no. 4, 303–307.
- C.-P. Chen and F. Qi, The best bounds in Wallis’ inequality, Proc. Amer. Math. Soc. 133 (2005), no. 2, 397–401.
- C.-P. Chen and F. Qi, The best bounds to (2n)! 22n(n!)2 , Math. Gaz. 88 (2004), 540–542.
- J. T. Chu, A modified Wallis product and some applications, Amer. Math. Monthly 69 (1962), no. 5, 402–404.
- T. Dana-Picard and D. G. Zeitoun, Parametric improper integrals, Wallis formula and Catalan numbers, Internat. J. Math. Ed. Sci. Tech. 43 (2012), no. 4, 515–520.
- B.-N. Guo and F. Qi, A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications, J. Korean Math. Soc. 48 (2011), no. 3, 655–667.
- B.-N. Guo and F. Qi, A class of completely monotonic functions involving the gamma and polygamma functions, Cogent Math. 1 (2014), 1:982896, 8 pages.
- B.-N. Guo and F. Qi, Logarithmically complete monotonicity of a power-exponential function involving the logarithmic and psi functions, Glob. J. Math. Anal. 3 (2015), no. 2, 77–80.
- B.-N. Guo and F. Qi, On the increasing monotonicity of a sequence originating from computation of the probability of intersecting between a plane couple and a convex body, Turkish J. Anal. Number Theory 3 (2015), no. 1, 21–23.
- B.-N. Guo and F. Qi, Sharp inequalities for the psi function and harmonic numbers, Analysis (Berlin) 34 (2014), no. 2, 201–208.
- B.-N. Guo, F. Qi, J.-L. Zhao, and Q.-M. Luo, Sharp inequalities for polygamma functions, Math. Slovaca 65 (2015), no. 1, 103–120.
- S. Guo, J.-G. Xu, and F. Qi, Some exact constants for the approximation of the quantity in the Wallis’ formula, J. Inequal. Appl. 2013, 2013:67, 7 pages.
- T. Hyde, A Wallis product on clovers, Amer. Math. Monthly 121 (2014), no. 3, 237–243.
- D. K. Kazarinoff, On Wallis’ formula, Edinburgh Math. Notes No. 40 (1956), 19–21.
- S. Koumandos, Remarks on a paper by Chao-Ping Chen and Feng Qi, Proc. Amer. Math. Soc. 134 (2006), 1365–1367.
- M. Kovalyov, Removing magic from the normal distribution and the Stirling and Wallis formulas, Math. Intelligencer 33 (2011), no. 4, 32–36.
- V. Krasniqi and F. Qi, Complete monotonicity of a function involving the p-psi function and alternative proofs, Glob. J. Math. Anal. 2 (2014), no. 3, 204–208.
- P. Levrie and W. Daems, Evaluating the probability integral using Wallis’s product formula for π, Amer. Math. Monthly 116 (2009), no. 6, 538–541.
- D. S. Mitrinovi´c, Analytic Inequalities, Springer, Berlin, 1970.
- C. Mortici and F. Qi, Asymptotic formulas and inequalities for the gamma function in terms of the tri-gamma function, Results Math. 66 (2015), in press.
- F. Qi, A completely monotonic function involving the gamma and tri-gamma functions, available online at http://arxiv.org/abs/1307.5407.
- F. Qi, A completely monotonic function related to the q-trigamma function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 76 (2014), no. 1, 107–114.
- F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010 (2010), Article ID 493058, 84 pages.
- F. Qi, Bounds for the ratio of two gamma functions: from Gautschi’s and Kershaw’s inequalities to complete monotonicity, Turkish J. Anal. Number Theory 2 (2014), no. 5, 152–164.
- F. Qi, Complete monotonicity of a function involving the tri- and tetra-gamma functions, Proc. Jangjeon Math. Soc. 18 (2015), no. 2, 253–264.
- F. Qi, Complete monotonicity of functions involving the q-trigamma and q-tetragamma functions, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Math. RACSAM. 109 (2015), in press.
- F. Qi, Derivatives of tangent function and tangent numbers, available online at http://arxiv.org/abs/1202.1205.
- F. Qi, Integral representations and complete monotonicity related to the remainder of Burnside’s formula for the gamma function, J. Comput. Appl. Math. 268 (2014), 155–167.
- F. Qi, Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions, Math. Inequal. Appl. 18 (2015), no. 2, 493–518.
- F. Qi, L.-H. Cui, and S.-L. Xu, Some inequalities constructed by Tchebysheff ’s integral inequality, Math. Inequal. Appl. 2 (1999), no. 4, 517–528.
- F. Qi and B.-N. Guo, Necessary and sufficient conditions for a function involving divided differences of the di- and tri-gamma functions to be completely monotonic, available online at http://arxiv.org/abs/0903.3071.
- F. Qi and B.-N. Guo, Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function, ResearchGate Technical Report, available online at http://dx.doi.org/10.13140/2.1.2733.3928.
- F. Qi and W.-H. Li, A logarithmically completely monotonic function involving the ratio of gamma functions, available online at http://arxiv.org/abs/1303.1877.
- F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to Elezovi´c-Giordano-Peˇcari´c’s theorem, J. Inequal. Appl. 2013, 2013:542, 20 pages.
- F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions—From Wendel’s and related inequalities to logarithmically completely monotonic functions, Banach J. Math. Anal. 6 (2012), no. 2, 132–158.
- F. Qi and Q.-M. Luo, Complete monotonicity of a function involving the gamma function and applications, Period. Math. Hungar. 69 (2014), no. 2, 159–169.
- F. Qi and B.-N. Guo, A note on additivity of polygamma functions, available online at http://arxiv.org/abs/0903.0888.
- F. Qi and C. Mortici, Some best approximation formulas and inequalities for the Wallis ratio, Appl. Math. Comput. 253 (2015), 363–368.
- F. Qi and C. Mortici, Some inequalities for the trigamma function in terms of the digamma function, available online at http://arxiv.org/abs/1503.03020.
- F. Qi, C. Mortici, and B.-N. Guo, Some properties of a sequence arising from computation of the intersecting probability between a plane couple and a convex body, ResearchGate Research, available online at http://dx.doi.org/10.13140/RG.2.1.1176.0165.
- F. Qi and S.-H. Wang, Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions, Glob. J. Math. Anal. 2 (2014), no. 3, 91–97.
- F. Qi and X.-J. Zhang, Complete monotonicity of a difference between the exponential and trigamma functions, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 21 (2014), no. 2, 141–145.
- J. W¨astlund, An elementary proof of the Wallis product formula for pi, Amer. Math. Monthly 114 (2007), no. 10, 914–917.
- G. N. Watson, A note on gamma functions, Proc. Edinburgh Math. Soc. 11 (1958/1959), no. 2, Edinburgh Math Notes No. 42 (misprinted 41) (1959), 7–9.
- Y.-Q. Zhao and Q.-B. Wu, Wallis inequality with a parameter, J. Inequal. Pure Appl. Math. 7 (2006), no. 2, Art. 56.