In this paper, by means of the classical Lagrange inversion formula, the authors establish a general nonlinear inverse relation as the solution to the problem proposed in the paper [J. Wang, Nonlinear inverse relations for the Bell polynomials via the Lagrange inversion formula, J. Integer Seq., Vol. 22 (2019), Article 19.3.8]. As applications of this inverse relation, the authors not only find a short proof of another nonlinear inverse relation due to Birmajer, et al. (2012), but also set up a few convolution identities concerning the Mina polynomials.
Abstract
In this paper, by means of the classical Lagrange inversion formula, the authors establish a general nonlinear inverse relation as the solution to the problem proposed in the paper [J. Wang, Nonlinear inverse relations for the Bell polynomials via the Lagrange inversion formula, J. Integer Seq., Vol. 22 (2019), Article 19.3.8]. As applications of this inverse relation, the authors not only find a short proof of another nonlinear inverse relation due to Birmajer, et al. (2012), but also set up a few convolution identities concerning the Mina polynomials.
关键词
Bell polynomial /
convolution identity /
formal power series /
Lagrange inversion formula /
Mina polynomial /
nonlinear inverse relation /
recurrence relation
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Key words
Bell polynomial /
convolution identity /
formal power series /
Lagrange inversion formula /
Mina polynomial /
nonlinear inverse relation /
recurrence relation
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参考文献
[1] Henrici P, Applied and Computational Complex Analysis, Vol.1, John Wiley & Sons, Inc., New York, 1974.
[2] Comtet L, Advanced Combinatorics, Dordrecht, Boston, 1974.
[3] Wang J, Nonlinear inverse relations for Bell polynomials via the Lagrange inversion formula, J. Integer Seq., 2019, 22:Article 19.3.8.
[4] Riordan J, Combinatorial Identities, John Wiley & Sons, Inc., Hoboken, 1968.
[5] Chou W S, Hsu L C, and Shiue P J S, Application of Faà di Bruno's formula in characterization of inverse relations, J. Comput. Appl. Math., 2006, 190:151-169.
[6] Mihoubi M, Partial Bell polynomials and inverse relations, J. Integer Seq., 2010, 13:Article 10.4.5.
[7] Birmajer D, Gil J B, and Weiner M D, Some convolution identities and an inverse relation involving partial Bell polynomials, Electron. J. Combin., 2012, 19:#P34.
[8] Huang J F and Ma X R, Two elementary applications of the Lagrange expansion formula, J. Math. Res. Appl., 2015, 35:263-270.
[9] Birmajer D, Gil J B, and Weiner M D, A family of Bell transformations, Discrete Math., 2019, 342:38-54.
[10] He T X, One-p th Riordan arrays in the construction of identities, J. Math. Res. Appli., 2021, 41:111-126.
[11] Bell E T, Generalized Stirling transforms of sequences, Amer. J. Math., 1940, 62:717-724.
[12] Gessel I M, Lagrange inversion, J. Combin. Theory Ser. A, 2016, 144:212-249.
[13] Gould H W, Some generalizations of Vandermonde's convolution, Amer. Math. Monthly, 1956, 63:84-91.
[14] Hofbauer J, Lagrange inversion, Sém. Lothar. Combin., 1982, 6:B06a. Available at http://www.emis.de/journals/SLC/opapers/s06hofbauer.html.
[15] Merlini D, Sprugnoli R, and Verri M C, Lagrange inversion:When and how, Acta Appl. Math., 2006, 94:233-249.
[16] Taghavian H, A fast algorithm for computing Bell polynomials based on index break-downs using prime factorization, https://arxiv.org/pdf/2004.09283.pdf.
[17] Wang J and Ma X R, Some notices on Mina matrix and allied determinant identities, J. Math. Res. Appli., 2016, 36:253-264.
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脚注
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基金
This research was supported by the National Natural Science Foundation of China under Grant Nos. 11971341 and 12001492, and the Natural Science Foundation of Zhejiang Province under Grant No. LQ20A010004.
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