Relations Among Bell Polynomials, Central Factorial Numbers, and Central Bell Polynomials

Relations Among Bell Polynomials, Central Factorial Numbers, and Central Bell Polynomials

In the note, by virtue of the Faà di Bruno formula and two identities for the Bell polynomials of thesecond kind, the authors derive three relations among the Bell polynomials, central factorial numbers ofthe second kind, and central Bell polynomials.

___

  • [1] Butzer, P. L., Schmidt, M., Stark, E. L. and Vogt, L., Central factorial numbers; their main properties and some applications, Numer. Funct. Anal. Optim. 10 (1989), no. 5-6, 419–488; available online at https://doi.org/ 10.1080/01630568908816313.
  • [2] Charalambides, C. A., Enumerative Combinatorics, CRC Press Series on Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, FL, 2002.
  • [3] Comtet, L., Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974; available online at https://doi.org/10.1007/978-94-010-2196-8.
  • [4] Guo, B.N. and Qi, F., An explicit formula for Bell numbers in terms of Stirling numbers and hypergeometric functions, Glob. J. Math. Anal. 2 (2014), no. 4, 243–248; available online at http://dx.doi.org/10.14419/gjma. v2i4.3310.
  • [5] Kim, D. S., Jang, G.W., Dolgy, D. V. and Kim, T., An expression for central Bell polynomials, Adv. Stud. Contemp. Math. 29 (2019), no. 2, 257–262; available online at http://dx.doi.org/10.17777/ascm2019.29.2.257.
  • [6] Merca, M., Connections between central factorial numbers and Bernoulli polynomials, Period. Math. Hungar. 73 (2016), no. 2, 259–264; available online at https://doi.org/10.1007/s10998-016-0140-5.
  • [7] Qi, F., An explicit formula for the Bell numbers in terms of the Lah and Stirling numbers, Mediterr. J. Math. 13 (2016), no. 5, 2795–2800; available online at https://doi.org/10.1007/s00009-015-0655-7.
  • [8] Qi, F., Integral representations for multivariate logarithmic polynomials, J. Comput. Appl. Math. 336 (2018), 54–62; available online at https://doi.org/10.1016/j.cam.2017.11.047.
  • [9] Qi, F., On multivariate logarithmic polynomials and their properties, Indag. Math. (N.S.) 29 (2018), no. 5, 1179–1192; available online at https://doi.org/10.1016/j.indag.2018.04.002.
  • [10] Qi, F., Simplifying coefficients in differential equations related to generating functions of reverse Bessel and partially degenerate Bell polynomials, Bol. Soc. Paran. Mat. 39 (2021), no. 4, in press; available online at http://dx.doi. org/10.5269/bspm.41758.
  • [11] Qi, F., Some inequalities and an application of exponential polynomials, Math. Inequal. Appl. 22 (2019), in press; available online at https://doi.org/10.20944/preprints201708.0079.v2.
  • [12] Qi, F., Some inequalities for the Bell numbers, Proc. Indian Acad. Sci. Math. Sci. 127 (2017), no. 4, 551–564; available online at https://doi.org/10.1007/s12044-017-0355-2.
  • [13] Qi, F., Cer ˇnanová, V. and Semenov, Y. S., ˇ Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 81 (2019), no. 1, 123–136.
  • [14] Qi, F. and 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 (2017), no. 3, Article 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1.
  • [15] Qi, F., Lim, D. and 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 113 (2019), no. 1, 1–9; available online at https://doi.org/10.1007/s13398-017-0427-2.
  • [16] Qi, F., Lim, D. and Yao, Y.H., Notes on two kinds of special values for the Bell polynomials of the second kind, Miskolc Math. Notes 20 (2019), no. 1, 465–474; available online at https://doi.org/10.18514/MMN.2019.2635.
  • [17] Qi, F., Niu, D.W. and Guo, B.N., Some identities for a sequence of unnamed polynomials connected with the Bell polynomials, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 113 (2019), no. 2, 557–567; available online at https://doi.org/10.1007/s13398-018-0494-z.
  • [18] Qi, F., Niu, D.W., Lim, D. and Guo, B.N., Some properties and an application of multivariate exponential polynomials, HAL archives (2018), available online at https://hal.archives-ouvertes.fr/hal-01745173.
  • [19] Qi, F., Niu, D.W., Lim, D. and Yao, Y.H., Special values of the Bell polynomials of the second kind for some sequences and functions, HAL archives (2018), available online at https://hal.archives-ouvertes.fr/ hal-01766566.
  • [20] Qi, F. and Zheng, M.M., Explicit expressions for a family of the Bell polynomials and applications, Appl. Math. Comput. 258 (2015), 597–607; available online at https://doi.org/10.1016/j.amc.2015.02.027.
  • [21] Wei, C.F. and Qi, F., Several closed expressions for the Euler numbers, J. Inequal. Appl. 2015, 2015:219, 8 pages; available online at https://doi.org/10.1186/s13660-015-0738-9.