Applicable Analysis and Discrete Mathematics 2022 Volume 16, Issue 2, Pages: 427-466
https://doi.org/10.2298/AADM210401017G
Full text (
515 KB)
Cited by
MacLaurin’s series expansions for positive integer powers of inverse (hyperbolic) sine and tangent functions, closed-form formula of specific partial Bell polynomials, and series representation of generalized logsine function
Guo Bai-Ni 
(School of Mathematics and Informatics, Henan Polytechnic University Jiaozuo, Henan, China), bai.ni.guo@gmail.com
Lim Dongkyu (Department of Mathematics Education Andong National University Andong, Republic of Korea), dgrim84@gmail.com, dklim@anu.ac.kr
Qi Feng (School of Mathematical Sciences Tiangong University Tianjin, China), qifeng618@gmail.com
In the paper, the authors find series expansions and identities for positive
integer powers of inverse (hyperbolic) sine and tangent, for composite of
incomplete gamma function with inverse hyperbolic sine, in terms of the
first kind Stirling numbers, apply a newly established series expansion to
derive a closed-form formula for specific partial Bell polynomials and to
derive a series representation of generalized logsine function, and deduce
combinatorial identities involving the first kind Stirling numbers.
Keywords: Maclaurin’s series expansion, inverse sine function, partial Bell polynomial, generalized logsine function, first kind Stirling number
Show 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, Washington, 1972.
E. P. Adams and R. L. Hippisley, Smithsonian Mathematical Formulae and Tables of Elliptic Functions, Smithsonian Institute, Washington, D.C., 1922.
B. C. Berndt, Ramanujan’s Notebooks, Part I, With a foreword by S. Chandrasekhar, Springer-Verlag, New York, 1985; available online at https://doi.org/10.1007/978-1-4612-1088-7.
J. M. Borwein, D. H. Bailey, and R. Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Ltd., Natick, MA, 2004.
J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1987.
J. M. Borwein and M. Chamberland, Integer powers of arcsin, Int. J. Math. Math. Sci. 2007, Art. ID 19381, 10 pages; available online at https://doi.org/10.1155/2007/19381.
K. N. Boyadzhiev, Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials, Adv. Appl. Discrete Math. 1 (2008), no. 2, 109-122; available online at https://dx.doi.org/10.17654/08AADM00102-109.
T. J. I. Bromwich, An Introduction to the Theory of Infinite Series, Macmillan and Co., Limited, London, 1908.
Yu. A. Brychkov, Power expansions of powers of trigonometric functions and series containing Bernoulli and Euler polynomials, Integral Transforms Spec. Funct. 20 (2009), no. 11-12, 797-804; available online at https://doi.org/10.1080/10652460902867718.
C. A. Charalambides, Enumerative Combinatorics, CRC Press Series on Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, FL, 2002.
S. Chavan, M. Kobayashi, and J. Layja, Integral evaluation of odd Euler sums, multiple t-value t(3, 2, . . . , 2) and multiple Zeta value ζ(3, 2, . . . , 2), arXiv (2021), available online at https://arxiv.org/abs/2111.07097.
C.-P. Chen, Sharp Wilker- and Huygens-type inequalities for inverse trigonometric and inverse hyperbolic functions, Integral Transforms Spec. Funct. 23 (2012), no. 12, 865-873; available online at https://doi.org/10.1080/10652469.2011.644851.
W. Chu and D. Zheng, Infinite series with harmonic numbers and central binomial coefficients, Int. J. Number Theory 5 (2009), no. 3, 429-448; available online at https://doi.org/10.1142/S1793042109002171.
L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974; available online at https://doi.org/10.1007/978-94-010-2196-8.
A. I. Davydychev and M. Yu. Kalmykov, New results for the ε-expansion of certain one-, two- and three-loop Feynman diagrams, Nuclear Phys. B 605 (2001), no. 1-3, 266-318; available online at https://doi.org/10.1016/S0550-3213(01)00095-5.
J. Edwards, Differential Calculus, 2nd ed., Macmillan, London, 1982.
J. Edwards, Differential Calculus for Beginners, Macmillan Co., Limited, London, 1899.
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Eighth edition, Revised from the seventh edition, Elsevier/ Academic Press, Amsterdam, 2015; available online at https://doi.org/10.1016/B978-0-12-384933-5.00013-8.
B.-N. Guo, D. Lim, and F. Qi, Maclaurin’s series expansions for positive integer powers of inverse (hyperbolic) sine and related functions, specific values of partial Bell polynomials, and two applications, arXiv (2021), available online at https://arxiv.org/abs/2101.10686v8.
B.-N. Guo, D. Lim, and F. Qi, Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions, AIMS Math. 6 (2021), no. 7, 7494-7517; available online at https://doi.org/10.3934/math.2021438.
E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, USA, 1975.
Y. Hong, B.-N. Guo, and F. Qi, Determinantal expressions and recursive relations for the Bessel zeta function and for a sequence originating from a series expansion of the power of modified Bessel function of the first kind, CMES Comput. Model. Eng. Sci. 129 (2021), no. 1, 409-423; available online at https://doi.org/10.32604/cmes.2021.016431.
G. J. O. Jameson, The incomplete gamma functions, Math. Gaz. 100 (2016), no. 548, 298-306; available online at https://doi.org/10.1017/mag.2016.67.
S. Jin, B.-N. Guo, and F. Qi, Partial Bell polynomials, falling and rising factorials, Stirling numbers, and combinatorial identities, CMES Comput. Model. Eng. Sci. (2022), in press; available online at https://dx.doi.org/10.32604/cmes.2022.019941.
L. B. W. Jolley, Summation of Series, 2nd revised ed., Dover Books on Advanced Mathematics Dover Publications, Inc., New York, 1961.
M. Yu. Kalmykov and A. Sheplyakov, lsjk-a C++ library for arbitrary-precision numeric evaluation of the generalized log-sine functions, Computer Phys. Commun. 172 (2005), no. 1, 45-59; available online at https://doi.org/10.1016/j.cpc.2005.04.013.
M. Kobayashi, Integral representations for local dilogarithm and trilogarithm functions, Open J. Math. Sci. 5 (2021), no. 1, 337-352; available online at https: //doi.org/10.30538/oms2021.0169.
A. G. Konheim, J. W. Wrench Jr., and M. S. Klamkin, A well-known series, Amer. Math. Monthly 69 (1962), no. 10, 1011-1011.
D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly 92 (1985), no. 7, 449-457; available online at http://dx.doi.org/10.2307/2322496.
L. Lewin, Polylogarithms and Associated Functions, With a foreword by A. J. Van der Poorten, North-Holland Publishing Co., New York-Amsterdam, 1981; available online at https://doi.org/10.1090/S0273-0979-1982-14998-9.
M. Milgram, A new series expansion for integral powers of arctangent, Integral Transforms Spec. Funct. 17 (2006), no. 7, 531-538; available online at https: //doi.org/10.1080/10652460500422486 or https://arxiv.org/abs/math/0406337.
V. H. Moll and C. Vignat, On polynomials connected to powers of Bessel functions, Int. J. Number Theory 10 (2014), no. 5, 1245-1257; available online at https://doi.org/10.1142/S1793042114500249.
F. Qi, Derivatives of tangent function and tangent numbers, Appl. Math. Comput. 268 (2015), 844-858; available online at http://dx.doi.org/10.1016/j.amc.2015.06.123.
F. Qi, Diagonal recurrence relations for the Stirling numbers of the first kind, Contrib. Discrete Math. 11 (2016), no. 1, 22-30; available online at https://doi.org/10.11575/cdm.v11i1.62389.
F. Qi, Explicit formulas for partial Bell polynomials, Maclaurin’s series expansions of real powers of inverse (hyperbolic) cosine and sine, and series representations of powers of Pi, Research Square (2021), available online at https://doi.org/10.21203/rs.3.rs-959177/v3.
F. Qi, Monotonicity results and inequalities for the gamma and incomplete gamma functions, Math. Inequal. Appl. 5 (2002), no. 1, 61-67; available online at http: //dx.doi.org/10.7153/mia-05-08.
F. Qi, Series expansions for any real powers of (hyperbolic) sine functions in terms of weighted Stirling numbers of the second kind, arXiv (2022), available online at https://arxiv.org/abs/2204.05612v1.
F. Qi, Taylor’s series expansions for real powers of functions containing squares of inverse (hyperbolic) cosine functions, explicit formulas for special partial Bell polynomials, and series representations for powers of circular constant, arXiv (2021), available online at https://arxiv.org/abs/2110.02749v2.
F. Qi, C.-P. Chen, and D. Lim, Several identities containing central binomial coefficients and derived from series expansions of powers of the arcsine function, Results Nonlinear Anal. 4 (2021), no. 1, 57-64; available online at https://doi.org/10.53006/rna.867047.
F. Qi and B.-N. Guo, A diagonal recurrence relation for the Stirling numbers of the first kind, Appl. Anal. Discrete Math. 12 (2018), no. 1, 153-165; available online at https://doi.org/10.2298/AADM170405004Q.
F. Qi and B.-N. Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterr. J. Math. 14 (2017), no. 3, Art. 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1.
F. Qi and B.-N. Guo, Integral representations of the Catalan numbers and their applications, Mathematics 5 (2017), no. 3, Article 40, 31 pages; available online at https://doi.org/10.3390/math5030040.
F. Qi and S.-L. Guo, Inequalities for the incomplete gamma and related functions, Math. Inequal. Appl. 2 (1999), no. 1, 47-53; available online at http://dx.doi.org/10.7153/mia-02-05.
F. Qi and J.-Q. Mei, Some inequalities of the incomplete gamma and related functions, Z. Anal. Anwendungen 18 (1999), no. 3, 793-799; available online at http://dx.doi.org/10.4171/ZAA/914.
F. Qi, D.-W. Niu, D. Lim, and B.-N. Guo, Closed formulas and identities for the Bell polynomials and falling factorials, Contrib. Discrete Math. 15 (2020), no. 1, 163-174; available online at https://doi.org/10.11575/cdm.v15i1.68111.
F. Qi, D.-W. Niu, D. Lim, and Y.-H. Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, J. Math. Anal. Appl. 491 (2020), no. 2, Article 124382, 31 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124382.
F. Qi, X.-T. Shi, and F.-F. Liu, Expansions of the exponential and the logarithm of power series and applications, Arab. J. Math. (Springer) 6 (2017), no. 2, 95-108; available online at https://doi.org/10.1007/s40065-017-0166-4.
F. Qi and M. D. Ward, Closed-form formulas and properties of coefficients in Maclaurin’s series expansion of Wilf’s function, arXiv (2021), available online at https://arxiv.org/abs/2110.08576v1.
F. Qi, G.-S. Wu, and B.-N. Guo, An alternative proof of a closed formula for central factorial numbers of the second kind, Turk. J. Anal. Number Theory 7 (2019), no. 2, 56-58; available online at https://doi.org/10.12691/tjant-7-2-5.
F. Qi and M.-M. Zheng, Explicit expressions for a family of the Bell polynomials and applications, Appl. Math. Comput. 258 (2015), 597-607; available online at https: //doi.org/10.1016/j.amc.2015.02.027.
J. Quaintance and H. W. Gould, Combinatorial Identities for Stirling Numbers. The unpublished notes of H. W. Gould. With a foreword by George E. Andrews. World Scientific Publishing Co. Pte. Ltd., Singapore, 2016.
I. J. Schwatt, An Introduction to the Operations with Series, Chelsea Publishing Co., New York, 1924; available online at http://hdl.handle.net/2027/wu.89043168475.
I. J. Schwatt, Notes on the expansion of a function, Phil. Mag. 31 (1916), 590-593.
M. R. Spiegel, Some interesting series resulting from a certain Maclaurin expansion, Amer. Math. Monthly 60 (1953), no. 4, 243-247; available online at https://doi.org/10.2307/2307433.
M. Z. Spivey, The Art of Proving Binomial Identities, Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 2019; available online at https://doi.org/10.1201/9781351215824.
R. Sprugnoli, Riordan Array Proofs of Identities in Gould’s Book, University of Florence, Italy, 2006.
N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996; available online at http://dx.doi.org/10.1002/9781118032572.
C.-F. Wei, Integral representations and inequalities of extended central binomial coefficients, Math. Methods Appl. Sci. (2022), in press; available online at https: //doi.org/10.1002/mma.8115.
H. S. Wilf, generatingfunctionology, Third edition. A K Peters, Ltd., Wellesley, MA, 2006.
C. Xu, Duality of weighted sum formulas of alternating multiple T-values, arXiv (2006), available online at https://arxiv.org/abs/2006.02967v3.
B. Zhang and C.-P. Chen, Sharp Wilker and Huygens type inequalities for trigonometric and inverse trigonometric functions, J. Math. Inequal. 14 (2020), no. 3, 673-684; available online at https://doi.org/10.7153/jmi-2020-14-43.