Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations
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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.
KeywordsBell polynomial Explicit formula Derivative Stirling number Generating function Identity Faà di Bruno formula Differential equation
Mathematics Subject ClassificationPrimary 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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