Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions

Bai-Ni Guo; Dongkyu Lim; Feng Qi; Bai-Ni Guo; Dongkyu Lim; Feng Qi

doi:10.3934/math.2021438

AIMS Mathematics

2021, Volume 6, Issue 7: 7494-7517. doi: 10.3934/math.2021438

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Research article

Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions

Bai-Ni Guo ¹,
Dongkyu Lim ^{2
,
,},
Feng Qi ^{3
,
,}

1.
School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454003, China
2.
Department of Mathematics Education, Andong National University, Andong 36729, South Korea
3.
School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China

Received: 22 February 2021 Accepted: 30 April 2021 Published: 07 May 2021
MSC : Primary: 11B83; Secondary: 11C08, 12E10, 26A39, 33B10, 41A58

In the paper, the authors

1. establish general expressions of series expansions of $ (\arcsin x)^\ell $ for $ \ell\in\mathbb{N} $;

2. find closed-form formulas for the sequence

$ \begin{equation*} {\rm{B}}_{2n,k}\biggl(0,\frac{1}{3},0,\frac{9}{5},0,\frac{225}{7},\dotsc, \frac{1+(-1)^{k+1}}{2}\frac{[(2n-k)!!]^2}{2n-k+2}\biggr), \end{equation*} $

where $ {\rm{B}}_{n, k} $ denotes the second kind Bell polynomials;

3. derive series representations of generalized logsine functions.

The series expansions of the powers $ (\arcsin x)^\ell $ were related with series representations for generalized logsine functions by Andrei I. Davydychev, Mikhail Yu. Kalmykov, and Alexey Sheplyakov. The above sequence represented by special values of the second kind Bell polynomials appeared in the study of Grothendieck's inequality and completely correlation-preserving functions by Frank Oertel.
- general expression,
- closed-form formula,
- arcsine,
- series expansion,
- power,
- special value,
- second kind Bell polynomials,
- series representation,
- generalized logsine function
Citation: Bai-Ni Guo, Dongkyu Lim, Feng Qi. Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions[J]. AIMS Mathematics, 2021, 6(7): 7494-7517. doi: 10.3934/math.2021438

Related Papers:

Abstract

In the paper, the authors

1. establish general expressions of series expansions of $ (\arcsin x)^\ell $ for $ \ell\in\mathbb{N} $;

2. find closed-form formulas for the sequence

$ \begin{equation*} {\rm{B}}_{2n,k}\biggl(0,\frac{1}{3},0,\frac{9}{5},0,\frac{225}{7},\dotsc, \frac{1+(-1)^{k+1}}{2}\frac{[(2n-k)!!]^2}{2n-k+2}\biggr), \end{equation*} $

where $ {\rm{B}}_{n, k} $ denotes the second kind Bell polynomials;

3. derive series representations of generalized logsine functions.

The series expansions of the powers $ (\arcsin x)^\ell $ were related with series representations for generalized logsine functions by Andrei I. Davydychev, Mikhail Yu. Kalmykov, and Alexey Sheplyakov. The above sequence represented by special values of the second kind Bell polynomials appeared in the study of Grothendieck's inequality and completely correlation-preserving functions by Frank Oertel.

References

[1]	M. Abramowitz, I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, In: National Bureau of Standards, Applied Mathematics Series, 55, 10th printing, Dover Publications, 1972.
[2]	E. P. Adams, R. L. Hippisley, Smithsonian Mathematical Formulae and Tables of Elliptic Functions, Smithsonian Institute, Washington, D.C., 1922.
[3]	E. Alkan, Approximation by special values of harmonic zeta function and log-sine integrals, Commun. Number Theory Phys., 7 (2013), 515–550. Available from: https://doi.org/10.4310/CNTP.2013.v7.n3.a5.
[4]	B. C. Berndt, Ramanujan's Notebooks, Part I, With a foreword by S. Chandrasekhar, Springer-Verlag, New York, 1985. Available from: https://doi.org/10.1007/978-1-4612-1088-7.
[5]	J. M. Borwein, D. H. Bailey, R. Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Ltd., Natick, MA, 2004.
[6]	J. M. Borwein, 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.
[7]	J. M. Borwein, M. Chamberland, Integer powers of arcsin, Int. J. Math. Math. Sci., 19381 (2007), 10. Available from: https://doi.org/10.1155/2007/19381.
[8]	J. M. Borwein, R. E. Crandall, Closed forms: What they are and why we care, Notices Amer. Math. Soc., 60 (2013), 50–65. Available from: https://doi.org/10.1090/noti936.
[9]	J. M. Borwein, A. Straub, Mahler measures, short walks and log-sine integrals, Theoret. Comput. Sci., 479 (2013), 4–21; Available from: https://doi.org/10.1016/j.tcs.2012.10.025.
[10]	J. M. Borwein, A. Straub, Special values of generalized log-sine integrals, ISSAC 2011–Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, 43–50, ACM, New York, 2011. Available from: https://doi.org/10.1145/1993886.1993899.
[11]	T. J. I. Bromwich, An Introduction to the Theory of Infinite Series, Macmillan Co., Limited, London, 1908.
[12]	C. A. Charalambides, Enumerative Combinatorics, CRC Press Series on Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, FL, 2002.
[13]	C. P. Chen, Sharp Wilker- and Huygens-type inequalities for inverse trigonometric and inverse hyperbolic functions, Integral Transforms Spec. Funct., 23 (2012), 865–873. Available from: https://doi.org/10.1080/10652469.2011.644851.
[14]	J. Choi, Log-sine and log-cosine integrals, Honam Math. J., 35 (2013), 137–146. Available from: https://doi.org/10.5831/HMJ.2013.35.2.137.
[15]	J. Choi, Y. J. Cho, H. M. Srivastava, Log-sine integrals involving series associated with the zeta function and polylogarithms, Math. Scand., 105 (2009), 199–217. Available from: https://doi.org/10.7146/math.scand.a-15115.
[16]	J. Choi, H. M. Srivastava, Explicit evaluations of some families of log-sine and log-cosine integrals, Integral Trans. Spec. Funct., 22 (2011), 767–783. Available from: https://doi.org/10.1080/10652469.2011.564375.
[17]	J. Choi, H. M. Srivastava, Some applications of the Gamma and polygamma functions involving convolutions of the Rayleigh functions, multiple Euler sums and log-sine integrals, Math. Nachr., 282 (2009), 1709–1723. Available from: https://doi.org/10.1002/mana.200710032.
[18]	L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974. Available from: https://doi.org/10.1007/978-94-010-2196-8.
[19]	A. I. Davydychev, M. Yu. Kalmykov, Massive Feynman diagrams and inverse binomial sums, Nuclear Phys. B, 699 (2004), 3–64. Available from: https://doi.org/10.1016/j.nuclphysb.2004.08.020.
[20]	A. I. Davydychev, M. Yu. Kalmykov, New results for the $\varepsilon$-expansion of certain one-, two- and three-loop Feynman diagrams, Nuclear Phys. B, 605 (2001), 266–318. Available from: https://doi.org/10.1016/S0550-3213(01)00095-5.
[21]	J. Edwards, Differential Calculus, 2Eds., Macmillan, London, 1982.
[22]	I. S. Gradshteyn, 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 from: https://doi.org/10.1016/B978-0-12-384933-5.00013-8.
[23]	B. N. Guo, D. Lim, F. Qi, Series expansions of powers of the arcsine function, closed forms for special Bell polynomials of the second kind, and series representations of generalized logsine functions, arXiv (2021). Available from: https://arXiv.org/abs/2101.10686v1.
[24]	B. N. Guo, D. Lim, F. Qi, Series expansions of powers of the arcsine function, closed forms for special values of the second kind Bell polynomials, and series representations of generalized logsine functions, arXiv (2021). Available from: https://arXiv.org/abs/2101.10686v2.
[25]	E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, USA, 1975.
[26]	A. Hoorfar, F. Qi, Sums of series of Rogers dilogarithm functions, Ramanujan J., 18 (2009), 231–238. Available from: http://dx.doi.org/10.1007/s11139-007-9043-7.
[27]	L. B. W. Jolley, Summation of Series, 2Eds., Dover Books on Advanced Mathematics Dover Publications, Inc., New York, 1961.
[28]	M. Yu. Kalmykov, A. Sheplyakov, lsjk——a C++ library for arbitrary-precision numeric evaluation of the generalized log-sine functions, Computer Phys. Commun., 172 (2005), 45–59. Available from: https://doi.org/10.1016/j.cpc.2005.04.013.
[29]	S. Kanemitsu, H. Kumagai, M. Yoshimoto, On rapidly convergent series expressions for zeta- and $L$-values, and log sine integrals, Ramanujan J., 5 (2001), 91–104. Available from: https://doi.org/10.1023/A:1011449413387.
[30]	K. S. Kölbig, Explicit evaluation of certain definite integrals involving powers of logarithms, J. Symbolic Comput., 1 (1985), 109–114. Available from: https://doi.org/10.1016/S0747-7171(85)80032-8.
[31]	K. S. Kölbig, On the integral $\int_{0}^{\pi/2}\log^n\cos x\log^p\sin x{\rm{d}} x$, Math. Comp., 40 (1983), 565–570. Available from: https://doi.org/10.2307/2007532.
[32]	A. G. Konheim, J. W. Wrench Jr., M. S. Klamkin, A well-known series, Amer. Math. Monthly, 69 (1962), 1011–1011.
[33]	D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly, 92 (1985), 449–457. Available from: http://dx.doi.org/10.2307/2322496.
[34]	L. Lewin, Polylogarithms and associated functions, With a foreword by A. J. Van der Poorten, North-Holland Publishing Co., New York-Amsterdam, 1981. Available from: https://doi.org/10.1090/S0273-0979-1982-14998-9.
[35]	F. Oertel, Grothendieck's inequality and completely correlation preserving functions——a summary of recent results and an indication of related research problems, arXiv (2020). Available from: https://arXiv.org/abs/2010.00746v1.
[36]	F. Oertel, Grothendieck's inequality and completely correlation preserving functions——a summary of recent results and an indication of related research problems, arXiv (2020). Available from: https://arXiv.org/abs/2010.00746v2.
[37]	K. Onodera, Generalized log sine integrals and the Mordell-Tornheim zeta values, Trans. Am. Math. Soc., 363 (2011), 1463–1485. Available from: https://doi.org/10.1090/S0002-9947-2010-05176-1.
[38]	D. Orr, Generalized Log-sine integrals and Bell polynomials, J. Comput. Appl. Math., 347 (2019), 330–342. Available from: https://doi.org/10.1016/j.cam.2018.08.026.
[39]	F. Qi, A new formula for the Bernoulli numbers of the second kind in terms of the Stirling numbers of the first kind, Publ. Inst. Math. (Beograd) (N.S.), 100 (2016), 243–249. Available from: https://doi.org/10.2298/PIM150501028Q.
[40]	F. Qi, Diagonal recurrence relations for the Stirling numbers of the first kind, Contrib. Discrete Math., 11 (2016), 22–30. Available from: https://doi.org/10.11575/cdm.v11i1.62389.
[41]	F. Qi, Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind, Filomat, 28 (2014), 319–327. Available from: https://doi.org/10.2298/FIL1402319O.
[42]	F. Qi, Integral representations and properties of Stirling numbers of the first kind, J. Number Theory, 133 (2013), 2307–2319. Available from: http://dx.doi.org/10.1016/j.jnt.2012.12.015.
[43]	F. Qi, C. P. Chen, D. Lim, Five identities involving the product or ratio of two central binomial coefficients, arXiv (2021). Available from: https://arXiv.org/abs/2101.02027v1.
[44]	F. Qi, C. P. Chen, D. Lim, Several identities containing central binomial coefficients and derived from series expansions of powers of the arcsine function, Results Nonlinear Anal., 4 (2021), 57–64.
[45]	F. Qi, B. N. Guo, A diagonal recurrence relation for the Stirling numbers of the first kind, Appl. Anal. Discrete Math., 12 (2018), 153–165. Available from: https://doi.org/10.2298/AADM170405004Q.
[46]	F. Qi, 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), 14. Available from: https://doi.org/10.1007/s00009-017-0939-1.
[47]	F. Qi, B. N. Guo, Integral representations of the Catalan numbers and their applications, Mathematics, 5 (2017), 31. Available from: https://doi.org/10.3390/math5030040.
[48]	F. Qi, D. Lim, Closed formulas for special Bell polynomials by Stirling numbers and associate Stirling numbers, Publ. Inst. Math. (Beograd) (N.S.), 108 (2020), 131–136. Available from: https://doi.org/10.2298/PIM2022131Q.
[49]	F. Qi, D. Lim, B. N. Guo, Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 113 (2019), 1–9. Available from: https://doi.org/10.1007/s13398-017-0427-2.
[50]	F. Qi, D. Lim, Y. H. Yao, Notes on two kinds of special values for the Bell polynomials of the second kind, Miskolc Math. Notes, 20 (2019), 465–474. Available from: https://doi.org/10.18514/MMN.2019.2635.
[51]	F. Qi, P. Natalini, P. E. Ricci, Recurrences of Stirling and Lah numbers via second kind Bell polynomials, Discrete Math. Lett., 3 (2020), 31–36.
[52]	F. Qi, D. W. Niu, D. Lim, B. N. Guo, Closed formulas and identities for the Bell polynomials and falling factorials, Contrib. Discrete Math., 15 (2020), 163–174. Available from: https://doi.org/10.11575/cdm.v15i1.68111.
[53]	F. Qi, D. W. Niu, D. Lim, Y. H. Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, J. Math. Anal. Appl., 491 (2020), Article 124382, 31. Available from: https://doi.org/10.1016/j.jmaa.2020.124382.
[54]	F. Qi, X. T. Shi, F. F. Liu, D. V. Kruchinin, Several formulas for special values of the Bell polynomials of the second kind and applications, J. Appl. Anal. Comput., 7 (2017), 857–871. Available from: https://doi.org/10.11948/2017054.
[55]	F. Qi, M. M. Zheng, Explicit expressions for a family of the Bell polynomials and applications, Appl. Math. Comput., 258 (2015), 597–607. Available from: https://doi.org/10.1016/j.amc.2015.02.027.
[56]	I. J. Schwatt, An Introduction to the Operations with Series, Chelsea Publishing Co., New York, 1924. Available from: http://hdl.handle.net/2027/wu.89043168475.
[57]	N. N. Shang, H. Z. Qin, The closed form of a class of integrals involving log-cosine and log-sine, Math. Pract. Theory, 42 (2012), 234–246. (Chinese)
[58]	M. R. Spiegel, Some interesting series resulting from a certain Maclaurin expansion, Amer. Math. Monthly, 60 (1953), 243–247. Available from: https://doi.org/10.2307/2307433.
[59]	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 from: http://dx.doi.org/10.1002/9781118032572.
[60]	H. S. Wilf, generatingfunctionology, Third edition. A K Peters, Ltd., Wellesley, MA, 2006.
[61]	R. Witula, E. Hetmaniok, D. Słota, N. Gawrońska, Convolution identities for central binomial numbers, Int. J. Pure App. Math., 85 (2013), 171–178. Available from: https://doi.org/10.12732/ijpam.v85i1.14.
[62]	B. Zhang, C. P. Chen, Sharp Wilker and Huygens type inequalities for trigonometric and inverse trigonometric functions, J. Math. Inequal., 14 (2020), 673–684. Available from: https://doi.org/10.7153/jmi-2020-14-43.

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