AIMS Mathematics, 2019, 4(2): 170-175. doi: 10.3934/math.2019.2.170

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Simplifying coefficients in differential equations associated with higher order Bernoulli numbers of the second kind

1 School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China
2 College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China
3 Department of Mathematics, East China Normal University, Shanghai 200241, China
4 School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454010, China

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In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.
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# References

1. L. Comtet, Advanced Combinatorics: The art of finite and infinite Expansions, Revised and Enlarged Edition, Dordrecht and Boston: D. Reidel Publishing Co., 1974.

2. D. V. Dolgy, L. C. Jang, D. S. Kim, et al. Differential equations associated with higher-order Bernoulli numbers of the second kind revisited, J. Anal. Appl. 14 (2016), 107-121.

3. B. N. Guo, F. Qi, Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, J. Comput. Appl. Math., 272 (2014), 251-257.

4. B. N. Guo, F. Qi, Some identities and an explicit formula for Bernoulli and Stirling numbers, J. Comput. Appl. Math., 255 (2014), 568-579.

5. F. Qi, A new formula for the Bernoulli numbers of the second kind in terms of the Stirling numbers of the first kind, Publ. I. Math., 100 (2016), 243-249.

6. F. Qi, A simple form for coefficients in a family of nonlinear ordinary differential equations, Adv. Appl. Math. Sci., 17 (2018), 555-561.

7. F. Qi, A simple form for coefficients in a family of ordinary differential equations related to the generating function of the Legendre polynomials, Adv. Appl. Math. Sci., 17 (2018), 693-700.

8. F. Qi, Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind, Filomat, 28 (2014), 319-327.

9. F. Qi, Notes on several families of differential equations related to the generating function for the Bernoulli numbers of the second kind, Turkish J. Anal. Number Theory, 6 (2018), 40-42.

10. F. Qi, Simple forms for coefficients in two families of ordinary differential equations, Glob. J. Math. Anal., 6 (2018), 7-9.

11. F. Qi, B. N. Guo, Simplification of coefficients in two families of nonlinear ordinary differential equations, Turkish J. Anal. Number Theory, 6 (2018), 116-119.

12. F. Qi, Simplifying coefficients in a family of nonlinear ordinary differential equations, Acta Commentat. Univ. Tartuensis Math., 22 (2018), 293-297.

13. F. Qi, Simplifying coefficients in a family of ordinary differential equations related to the generating function of the Laguerre polynomials, Appl. Appl. Math., 13 (2018), 750-755.

14. F. Qi, Simplifying coefficients in differential equations related to generating functions of reverse Bessel and partially degenerate Bell polynomials, Bol. Soc. Parana. Mat., 39 (2021), in press.

15. F. Qi, B. N. Guo, A diagonal recurrence relation for the Stirling numbers of the first kind, Appl. Anal. Discrete Math., 12 (2018), 153-165.

16. F. Qi, B. N. Guo, Explicit formulas and recurrence relations for higher order Eulerian polynomials, Indagat. Math., 28 (2017), 884-891.

17. F. Qi, D. Lim, B. N. Guo, Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations, RACSAM, 113 (2019), 1-9.

18. F. Qi, D. Lim, B. N. Guo, Some identities related to Eulerian polynomials and involving the Stirling numbers, Appl. Anal. Discrete Math., 12 (2018), 467-480.

19. F. Qi, D. W. Niu, B. N. Guo, Simplifying coefficients in differential equations associated with higher order Bernoulli numbers of the second kind, Preprints, 2017. Available from: http://dx.doi.org/10.20944/preprints201708.0026.v1.

20. F. Qi, D. W. Niu, B. N. Guo, Some identities for a sequence of unnamed polynomials connected with the Bell polynomials, RACSAM, 112 (2018), in press. Available from: https://doi.org/10.1007/s13398-018-0494-z.

21. F. Qi, J. L. Wang, B. N. Guo, Notes on a family of inhomogeneous linear ordinary differential equations, Adv. Appl. Math. Sci., 17 (2018), 361-368.

22. F. Qi, J. L. Wang, B. N. Guo, Simplifying and finding ordinary differential equations in terms of the Stirling numbers, Korean J. Math., 26 (2018), 675-681.

23. F. Qi, J. L.Wang, B. N. Guo, Simplifying differential equations concerning degenerate Bernoulli and Euler numbers, Trans. A. Razmadze Math. Inst., 172 (2018), 90-94.

24. F. Qi, X. J. Zhang, An integral representation, some inequalities, and complete monotonicity of the Bernoulli numbers of the second kind, B. Korean Math. Soc., 52 (2015), 987-998.

25. F. Qi, J. L. Zhao, Some properties of the Bernoulli numbers of the second kind and their generating function, B. Korean Math. Soc., 55 (2018), 1909-1920.

26. J. Quaintance, H. W. Gould, Combinatorial Identities for Stirling Numbers: The Unpublished Notes of H. W. Gould, Singapore: World Scientific Publishing Co. Pte. Ltd., 2016.

27. J. L. Zhao, J. L. Wang, F. Qi, Derivative polynomials of a function related to the Apostol-Euler and Frobenius-Euler numbers, J. Nonlinear Sci. Appl., 10 (2017), 1345-1349.