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Some identities for a sequence of unnamed polynomials connected with the Bell polynomials

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Abstract

In the paper, by virtue of (1) the Stirling inversion theorem and the binomial inversion theorem, (2) the Faà di Bruno formula and two identities for the Bell polynomials of the second kind, (3) a formula of higher order derivative for the ratio of two differentiable functions, the authors (1) present two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, (2) derive two identities connecting the sequence of unnamed polynomials with the Bell polynomials, (3) recover a known identity connecting the sequence of unnamed polynomials with the Bell polynomials.

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References

  1. Bourbaki, N.: Functions of a Real Variable, Elementary Theory, Translated from the 1976 French Original by Philip Spain. Elements of Mathematics (Berlin). Springer, Berlin (2004). https://doi.org/10.1007/978-3-642-59315-4

    Google Scholar 

  2. Cahill, N.D., D’Errico, J.R., Narayan, D.A., Narayan, J.Y.: Fibonacci determinants. Coll. Math. J. 3, 221–225 (2002). https://doi.org/10.2307/1559033

    Article  MATH  Google Scholar 

  3. Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition. D. Reidel Publishing Co., Dordrecht (1974)

    Book  MATH  Google Scholar 

  4. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, Vol. III. Based on Notes left by Harry Bateman. Reprint of the 1955 Original. Robert E. Krieger Publishing Co., Inc, Melbourne (1981)

    MATH  Google Scholar 

  5. Kim, T., Dolgy, D.V., Kim, D.S., Kwon, H.I., Seo, J.J.: Differential equations arising from certain Sheffer sequence. J. Comput. Anal. Appl. 23(8), 1359–1367 (2017)

    MathSciNet  Google Scholar 

  6. Ma, W.-X.: Bilinear equations, Bell polynomials and linear superposition principle. J. Phys. Conf. Ser. 411(1), 11 (2013). https://doi.org/10.1088/1742-6596/411/1/012021. (Aricle ID 012021)

    Google Scholar 

  7. Ma, W.-X.: Bilinear equations and resonant solutions characterized by Bell polynomials. Rep. Math. Phys. 72(1), 41–56 (2013). https://doi.org/10.1016/S0034-4877(14)60003-3

    Article  MathSciNet  MATH  Google Scholar 

  8. Ma, W.-X.: Trilinear equations, Bell polynomials, and resonant solutions. Front. Math. China 8(5), 1139–1156 (2013). https://doi.org/10.1007/s11464-013-0319-5

    Article  MathSciNet  MATH  Google Scholar 

  9. Qi, F.: Integral representations for multivariate logarithmic polynomials, J. Comput. Appl. Math. 336, 54–62 (2018). https://doi.org/10.1016/j.cam.2017.11.047

  10. Qi, F.: Some properties of the Touchard polynomials, ResearchGate Working Paper (2017). https://doi.org/10.13140/RG.2.2.30022.16967

  11. Qi, F., Čerňanová, V., Semenov, Y. S.: On tridiagonal determinants and the Cauchy product of central Delannoy numbers. ResearchGate Working Paper (2016). https://doi.org/10.13140/RG.2.1.3772.6967

  12. Qi, F., Čerňanová, V., Shi, X.-T., Guo, B.-N.: Some properties of central Delannoy numbers. J. Comput. Appl. Math. 328, 101–115 (2018). https://doi.org/10.1016/j.cam.2017.07.013

    Article  MathSciNet  MATH  Google Scholar 

  13. Qi, F., Guo, B.-N.: A diagonal recurrence relation for the Stirling numbers of the first kind. Appl. Anal. Discrete Math. 12 (2018). https://doi.org/10.2298/AADM170405004Q (in press)

  14. Qi, F., Guo, B.-N.: Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials. Mediterr. J. Math. 14(3), 14 (2017). https://doi.org/10.1007/s00009-017-0939-1. (Article 140)

    Article  MathSciNet  MATH  Google Scholar 

  15. Qi, F., Guo, B.-N.: Several explicit and recursive formulas for the generalized Motzkin numbers. https://doi.org/10.20944/preprints201703.0200.v1 (Preprints 2017, 2017030200, 11 pages)

  16. Qi, F., Guo, B.-N.: Viewing some ordinary differential equations from the angle of derivative polynomials. https://doi.org/10.20944/preprints201610.0043.v1 (Preprints 2016, 2016100043, 12 pages)

  17. Qi, F., Lim, D., Guo, B.-N.: Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM (2018). https://doi.org/10.1007/s13398-017-0427-2 (in press)

  18. Qi, F., Niu, D.-W., Guo, B.-N.: Some identities for a sequence of unnamed polynomials connected with the Bell polynomials. https://doi.org/10.20944/preprints201708.0044.v1 (Preprints 2017 , 2017080044, 10 pages)

  19. Qi, F., Niu, D.-W., Lim, D., Guo, B.-N.: A unified generalization of the Bell numbers and the Touchard polynomials and its properties, ResearchGate Working Paper (2017). https://doi.org/10.13140/RG.2.2.36733.05603

  20. Qi, F., Shi, X.-T., Liu, F.-F., Kruchinin, D.V.: Several formulas for special values of the Bell polynomials of the second kind and applications. J. Appl. Anal. Comput. 7(3), 857–871 (2017). https://doi.org/10.11948/2017054

    MathSciNet  Google Scholar 

  21. Qi, F., Zhao, J.-L., Guo, B.-N.: Closed forms for derangement numbers in terms of the Hessenberg determinants, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM (2018). https://doi.org/10.1007/s13398-017-0401-z (in press)

  22. Qi, F., Zou, Q., Guo, B.-N.: Some identities and a matrix inverse related to the Chebyshev polynomials of the second kind and the Catalan numbers. https://doi.org/10.20944/preprints201703.0209.v2 (Preprints 2017 , 2017030209, 25 pages)

  23. Quaintance, J., Gould, H.W.: Combinatorial Identities for Stirling Numbers. The unpublished notes of H. W. Gould. With a foreword by George E. Andrews. World Scientific Publishing Co., Pte. Ltd., Singapore (2016)

    MATH  Google Scholar 

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Acknowledgements

The authors thank anonymous referees for their careful reading and valuable comments on the original version of this paper.

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Authors and Affiliations

  1. Institute of Mathematics, Henan Polytechnic University, Jiaozuo, 454010, Henan, China

    Feng Qi

  2. College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, 028043, Inner Mongolia, China

    Feng Qi

  3. Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin, 300387, China

    Feng Qi

  4. Department of Mathematics, East China Normal University, Shanghai, 200241, China

    Da-Wei Niu

  5. School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, 454010, Henan, China

    Bai-Ni Guo

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  1. Feng Qi
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  2. Da-Wei Niu
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  3. Bai-Ni Guo
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Correspondence to Feng Qi.

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Qi, F., Niu, DW. & Guo, BN. Some identities for a sequence of unnamed polynomials connected with the Bell polynomials. RACSAM 113, 557–567 (2019). https://doi.org/10.1007/s13398-018-0494-z

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  • DOI: https://doi.org/10.1007/s13398-018-0494-z

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