Power series expansion, decreasing property, and concavity related to logarithm of normalized tail of power series expansion of cosine

Aying Wan; Feng Qi; Aying Wan; Feng Qi

doi:10.3934/era.2024143

Electronic Research Archive

2024, Volume 32, Issue 5: 3130-3144. doi: 10.3934/era.2024143

Previous Article Next Article

Research article

Power series expansion, decreasing property, and concavity related to logarithm of normalized tail of power series expansion of cosine

Aying Wan ¹,
Feng Qi ^{2,3,1
,
,}

1.
Department of Science and Technology, Hulunbuir University, Hulunbuir 021008, China
2.
School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454010, China
3.
Independent researcher, University Village, Dallas 75252, USA

Received: 19 January 2024 Revised: 15 March 2024 Accepted: 07 April 2024 Published: 25 April 2024

In this paper, in view of a determinantal formula for higher order derivatives of the ratio of two differentiable functions, we expand the logarithm of the normalized tail of the power series expansion of the cosine function into a Maclaurin power series expansion whose coefficients are expressed in terms of specific Hessenberg determinants, present the decreasing property and concavity of the normalized tail of the Maclaurin power series expansion of the cosine function, deduce a new determinantal expression of the Bernoulli numbers, and verify the decreasing property for the ratio of the logarithms of the first two normalized tails of the Maclaurin power series expansion of the cosine function.
- Maclaurin power series expansion,
- normalized tail,
- cosine,
- ratio,
- derivative formula,
- Hessenberg determinant,
- decreasing property,
- concavity,
- Bernoulli number,
- logarithm,
- determinantal expression
Citation: Aying Wan, Feng Qi. Power series expansion, decreasing property, and concavity related to logarithm of normalized tail of power series expansion of cosine[J]. Electronic Research Archive, 2024, 32(5): 3130-3144. doi: 10.3934/era.2024143

Related Papers:

Abstract

In this paper, in view of a determinantal formula for higher order derivatives of the ratio of two differentiable functions, we expand the logarithm of the normalized tail of the power series expansion of the cosine function into a Maclaurin power series expansion whose coefficients are expressed in terms of specific Hessenberg determinants, present the decreasing property and concavity of the normalized tail of the Maclaurin power series expansion of the cosine function, deduce a new determinantal expression of the Bernoulli numbers, and verify the decreasing property for the ratio of the logarithms of the first two normalized tails of the Maclaurin power series expansion of the cosine function.

References

[1]	I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 8th edition, D. Zwillinger, V. Moll, editors, Academic Press, Amsterdam, 2014. https://doi.org/10.1016/C2010-0-64839-5
[2]	Y. F. Li, F. Qi, A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine, Open Math., 21 (2023). https://doi.org/10.1515/math-2023-0159
[3]	J. C. Kuang, Applied Inequalities, 4th edition, Shandong Science and Technology Press, Ji'nan, 2010.
[4]	T. Zhang, Z. H. Yang, F. Qi, W. S. Du, Some properties of normalized tails of Maclaurin power series expansions of sine and cosine, Fractal Fract., 8 (2024), In press.
[5]	N. Bourbaki, Elements of Mathematics: Functions of a Real Variable: Elementary Theory, P. Spain, translator, Springer-Verlag, Berlin, 2004. https://doi.org/10.1007/978-3-642-59315-4
[6]	J. Cao, J. L. López-Bonilla, F. Qi, Three identities and a determinantal formula for differences between Bernoulli polynomials and numbers, Electron. Res. Arch., 32 (2024), 224–240. https://doi.org/10.3934/era.2024011 doi: 10.3934/era.2024011
[7]	H. Chen, Bernoulli numbers via determinants, Int. J. Math. Educ. Sci. Tech., 34 (2003), 291–297. http://dx.doi.org/10.1080/0020739031000158335 doi: 10.1080/0020739031000158335
[8]	C. Y. He, F. Qi, Reformulations and generalizations of Hoffman's and Genčev's combinatorial identities, Results Math., 2014 (2014). http://doi.org/10.1007/s00025-024-02160-0
[9]	F. Qi, On signs of certain Toeplitz–Hessenberg determinants whose elements involve Bernoulli numbers, Contrib. Discrete Math., 18 (2023), 48–59. https://doi.org/10.55016/ojs/cdm.v18i2.73022 doi: 10.55016/ojs/cdm.v18i2.73022
[10]	Z. Y. Sun, B. N. Guo, F. Qi, Determinantal expressions, identities, concavity, Maclaurin power series expansions for van der Pol numbers, Bernoulli numbers, and cotangent, Axioms, 12 (2023), 665. https://doi.org/10.3390/axioms12070665 doi: 10.3390/axioms12070665
[11]	T. M. Apostol, Calculus, 2nd edition, Blaisdell Publishing Co., Waltham, 1967.
[12]	G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997.
[13]	Y. W. Li, F. Qi, W. S. Du, Two forms for Maclaurin power series expansion of logarithmic expression involving tangent function, Symmetry, 15 (2023), 1686. https://doi.org/10.3390/sym15091686 doi: 10.3390/sym15091686
[14]	X. L. Liu, H. X. Long, F. Qi, A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing sine, Mathematics, 11 (2023), 3107. https://doi.org/10.3390/math11143107 doi: 10.3390/math11143107

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)