My question is expressed by the picture below. May you provide me a formula and a reference to cite? Thank a lot.
This is a good question.
Bell polynomials of the second kind are considered in great detail in
A. Aboud, J.-P. Bultel, A. Chouria, J.-G. Luque, O. Mallet, Bell polynomials in combinatorial Hopf algebras, arXiv 1402, 2015, no. 2960, 1-24:
http://arxiv.org/pdf/1402.2960.pdf
See Section 2.1, starting on page 3.
Dear All
This formula has been employed in the following papers and preprints. Hope you have some interests in them.
Best regards
Feng
(1) Feng Qi and Robin J. Chapman, Two closed forms for the Bernoulli polynomials, Journal of Number Theory 159 (2016), 89--100; Available online at http://dx.doi.org/10.1016/j.jnt.2015.07.021.
(2) Feng Qi, Explicit formulas for the convolved Fibonacci numbers, ResearchGate Working Paper (2016), available online at http://dx.doi.org/10.13140/RG.2.2.36768.17927.
(3) Feng Qi, Xiao-Ting Shi, and Bai-Ni Guo, Two explicit formulas of the Schroder numbers, Integers 16 (2016), Paper No. A23, 15 pages.
(4) Feng Qi, Xiao-Ting Shi, and Bai-Ni Guo, Two explicit formulas of the Schroder numbers, ResearchGate Working Paper (2016), available online at http://dx.doi.org/10.13140/RG.2.1.2676.3283.
(5) Feng Qi and Bai-Ni Guo, Some properties of the Hermite polynomials and their squares and generating functions, Preprints 2016, 2016110145, 14 pages; Available online at http://dx.doi.org/10.20944/preprints201611.0145.v1.
(6) Feng Qi and Bai-Ni Guo, Explicit and recursive formulas, integral representations, and properties of the large Schroder numbers, Kragujevac Journal of Mathematics 41 (2017), no. 1, 121--141.
(7) Feng Qi, Xiao-Ting Shi, and Bai-Ni Guo, Integral representations of the large and little Schroder numbers, ResearchGate Working Paper (2016), available online at http://dx.doi.org/10.13140/RG.2.1.1988.3288.
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