Several recursive and closed-form formulas for some specific values of partial Bell polynomials
Year 2022,
, 528 - 537, 30.12.2022
Wei-shih Du
,
Dongkyu Lım
,
Feng Qi
Abstract
In this paper, the authors derive several recursive and closed-form formulas for some specific values of partial Bell polynomials.
References
- [1] T. Amdeberhan, X. Guan, L. Jiu, V.H. Moll, and C. Vignat, A series involving Catalan numbers: Proofs and demonstra-
tions, Elem. Math. 71 (2016), no. 3, 109-121; available online at https://doi.org/10.4171/EM/306.
- [2] C.A. Charalambides, Enumerative Combinatorics, CRC Press Series on Discrete Mathematics and its Applications, Chap-
man & Hall/CRC, Boca Raton, FL, 2002.
- [3] 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.
- [4] 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, Else-
vier/Academic Press, Amsterdam, 2015; available online at https://doi.org/10.1016/B978-0-12-384933-5.00013-8.
- [5] B.-N. Guo, D. Lim, and F. Qi, 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, Appl. Anal. Discrete Math. 17 (2023), no. 1, in press; available online at https://doi.org/10.2298/
AADM210401017G.
- [6] 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.
- [7] 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.
- [8] 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. 132 (2022), no. 3, 781-799; available online at https://dx.doi.org/10.
32604/cmes.2022.019941.
- [9] Y. L. Luke, The Special Functions and Their Approximations, Vol. I, Mathematics in Science and Engineering, Vol. 53,
Academic Press, New York-London, 1969.
- [10] 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 online at https://arxiv.org/abs/2010.00746v2.
- [11] 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.
- [12] F. Qi, Taylor's series expansions for real powers of two functions containing squares of inverse cosine function, closed-form
formula for specific partial Bell polynomials, and series representations for real powers of Pi, Demonstr. Math. 55 (2022),
no. 1, in press; available online at https://doi.org/10.1515/dema-2022-0157.
- [13] F. Qi, C.-P. Chen, and D. Lim, Several identities containing central binomial coeficients 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.
- [14] 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.
- [15] 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.
- [16] F. Qi and P. Taylor, Several series expansions for real powers and several formulas for partial Bell polynomials of sinc
and sinhc functions in terms of central factorial and Stirling numbers of second kind, arXiv (2022), available online at
https://arxiv.org/abs/2204.05612v4.
- [17] F. Qi and M. D. Ward, Closed-form formulas and properties of coe?cients in Maclaurin's series expansion of Wilf's
function composited by inverse tangent, square root, and exponential functions, arXiv (2022), available online at https:
//arxiv.org/abs/2110.08576v2.
- [18] F. Qi, Q. Zou, and B.-N. Guo, The inverse of a triangular matrix and several identities of the Catalan numbers, Appl.
Anal. Discrete Math. 13 (2019), no. 2, 51-?541; available online at https://doi.org/10.2298/AADM190118018Q.
- [19] 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.
Year 2022,
, 528 - 537, 30.12.2022
Wei-shih Du
,
Dongkyu Lım
,
Feng Qi
References
- [1] T. Amdeberhan, X. Guan, L. Jiu, V.H. Moll, and C. Vignat, A series involving Catalan numbers: Proofs and demonstra-
tions, Elem. Math. 71 (2016), no. 3, 109-121; available online at https://doi.org/10.4171/EM/306.
- [2] C.A. Charalambides, Enumerative Combinatorics, CRC Press Series on Discrete Mathematics and its Applications, Chap-
man & Hall/CRC, Boca Raton, FL, 2002.
- [3] 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.
- [4] 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, Else-
vier/Academic Press, Amsterdam, 2015; available online at https://doi.org/10.1016/B978-0-12-384933-5.00013-8.
- [5] B.-N. Guo, D. Lim, and F. Qi, 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, Appl. Anal. Discrete Math. 17 (2023), no. 1, in press; available online at https://doi.org/10.2298/
AADM210401017G.
- [6] 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.
- [7] 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.
- [8] 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. 132 (2022), no. 3, 781-799; available online at https://dx.doi.org/10.
32604/cmes.2022.019941.
- [9] Y. L. Luke, The Special Functions and Their Approximations, Vol. I, Mathematics in Science and Engineering, Vol. 53,
Academic Press, New York-London, 1969.
- [10] 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 online at https://arxiv.org/abs/2010.00746v2.
- [11] 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.
- [12] F. Qi, Taylor's series expansions for real powers of two functions containing squares of inverse cosine function, closed-form
formula for specific partial Bell polynomials, and series representations for real powers of Pi, Demonstr. Math. 55 (2022),
no. 1, in press; available online at https://doi.org/10.1515/dema-2022-0157.
- [13] F. Qi, C.-P. Chen, and D. Lim, Several identities containing central binomial coeficients 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.
- [14] 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.
- [15] 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.
- [16] F. Qi and P. Taylor, Several series expansions for real powers and several formulas for partial Bell polynomials of sinc
and sinhc functions in terms of central factorial and Stirling numbers of second kind, arXiv (2022), available online at
https://arxiv.org/abs/2204.05612v4.
- [17] F. Qi and M. D. Ward, Closed-form formulas and properties of coe?cients in Maclaurin's series expansion of Wilf's
function composited by inverse tangent, square root, and exponential functions, arXiv (2022), available online at https:
//arxiv.org/abs/2110.08576v2.
- [18] F. Qi, Q. Zou, and B.-N. Guo, The inverse of a triangular matrix and several identities of the Catalan numbers, Appl.
Anal. Discrete Math. 13 (2019), no. 2, 51-?541; available online at https://doi.org/10.2298/AADM190118018Q.
- [19] 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.