Nonlinear Inverse Relations of the Bell Polynomials via the Lagrange Inversion Formula (II)

doi:10.1007/s11424-022-1300-8

Journal of Systems Science and Complexity ›› 2023, Vol. 36 ›› Issue (1): 96-116.DOI: 10.1007/s11424-022-1300-8

Previous Articles Next Articles

Nonlinear Inverse Relations of the Bell Polynomials via the Lagrange Inversion Formula (II)

MA Xinrong¹, WANG Jin²

1. Department of Mathematics, Soochow University, Suzhou 215006, China;
2. Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received:2021-08-15 Revised:2021-11-17 Online:2023-01-25 Published:2023-02-09
Supported by:
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.

PDF ( 4 ) Cited

MA Xinrong, WANG Jin. Nonlinear Inverse Relations of the Bell Polynomials via the Lagrange Inversion Formula (II)[J]. Journal of Systems Science and Complexity, 2023, 36(1): 96-116.

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.

Key words: Bell polynomial, convolution identity, formal power series, Lagrange inversion formula, Mina polynomial, nonlinear inverse relation, recurrence relation

[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.

No related articles found!

Viewed

Full text

Abstract