Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations

  • Feng QiEmail author
  • Dongkyu Lim
  • Bai-Ni Guo
Original Paper


In the paper, the authors discuss the Bell polynomials and a sequence of polynomials applied to the theory of differential equations. Concretely speaking, the authors find four explicit formulas for these polynomials and for derivatives of generating functions of these polynomials, establish four identities between these two kinds of polynomials, and significantly simplify some known results.


Bell polynomial Explicit formula Derivative Stirling number Generating function Identity Faà di Bruno formula Differential equation 

Mathematics Subject Classification

Primary 11B83 Secondary 11B73 11C08 26A06 26A09 33B10 



The first author was partially supported by the National Natural Science Foundation of China under Grant No. 11361038 and the second author was partially supported by China Postdoctoral Science Foundation under Grant No. 2016M591379.

The authors appreciate the anonymous referees for their careful corrections to and valuable comments on the original version of this paper.


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Copyright information

© Springer-Verlag Italia S.r.l. 2017

Authors and Affiliations

  1. 1.Institute of MathematicsHenan Polytechnic UniversityJiaozuoChina
  2. 2.College of MathematicsInner Mongolia University for NationalitiesTongliaoChina
  3. 3.Department of Mathematics, College of ScienceTianjin Polytechnic UniversityTianjinChina
  4. 4.School of Mathematical SciencesNankai UniversityTianjinChina
  5. 5.School of Mathematics and InformaticsHenan Polytechnic UniversityJiaozuoChina

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