What are the special values for the Bell polynomials of the second kind $B_{n,k}(1,1,3!!,5!!,7!!,\dotsc,(2(n-k)-1)!!)$? See the picture below. Please help me find a reference to cite. Thank a lot.
Note that, except for alternating signs (-1)k+1, these are precisely the coefficients of the (reverse) Bessel polynomials, i.e.,
1, 1
3, 3, 1
15, 15, 6, 1
etc.
For this (and other connections), see http://oeis.org/A001497.
F. Qi, X.-T. Shi, and F.-F. Liu, Several formulas for special values of the Bell polynomials of the second kind and applications, ResearchGate Technical Report, available online at http://dx.doi.org/10.13140/RG.2.1.3230.1927.
The revised and extended version of this preprint will be formally published in the Journal of Applied Analysis and Computation in 2017.
Feng Qi, Xiao-Ting Shi, Fang-Fang Liu, and Dmitry V. Kruchinin, Several formulas for special values of the Bell polynomials of the second kind and applications, Journal of Applied Analysis and Computation 7 (2017), October, in press.
In Remark 1 of the following preprint, there is a general formula for this question.
Feng Qi and Bai-Ni Guo, Viewing some ordinary differential equations from the angle of derivative polynomials, Preprints 2016, 2016100043, 12 pages; Available online at http://dx.doi.org/10.20944/preprints201610.0043.v1.
Feng Qi, Xiao-Ting Shi, Fang-Fang Liu, and Dmitry V. Kruchinin, Several formulas for special values of the Bell polynomials of the second kind and applications, Journal of Applied Analysis and Computation 7 (2017), no. 3, 857--871; Available online at http://dx.doi.org/10.11948/2017054.
The following formally published papers are related to this question:
[1] Feng Qi, Xiao-Ting Shi, Fang-Fang Liu, and Dmitry V. Kruchinin, Several formulas for special values of the Bell polynomials of the second kind and applications, Journal of Applied Analysis and Computation 7 (2017), no. 3, 857--871; Available online at http://dx.doi.org/10.11948/2017054
[2] Feng Qi and Bai-Ni Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterranean Journal of Mathematics 14 (2017), no. 3, Article 140, 14 pages; available online at https://doi.org/10.1007/s00009-017-0939-1
[3] Feng Qi, Da-Wei Niu, Dongkyu Lim, and Yong-Hong Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, Journal of Mathematical Analysis and Applications 491 (2020), no. 2, Paper No. 124382, 31 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124382
[4] Feng Qi, Da-Wei Niu, Dongkyu Lim, and Bai-Ni Guo, Closed formulas and identities for the Bell polynomials and falling factorials, Contributions to Discrete Mathematics 15 (2020), no. 1, 163--174; available online at https://doi.org/10.11575/cdm.v15i1.68111
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