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A problem of interest that has come up for me recently is solving the following.

$$\frac{d^{n}}{dt^{n}}e^{g(t)}$$

There is a formula for a general $n$-th order derivative of a composition as shown above: http://mathworld.wolfram.com/FaadiBrunosFormula.html

In terms of the Bell Polynomials, we can write

$$\frac{d^{n}}{dt^{n}}e^{g(t)}=e^{g(t)}\sum_{k=0}^{n}B_{n,k}(g'(t),g''(t),\cdots)$$

And the Bell polynomials of the second kind are shown in the Wolfram link above. I am wondering if there is a closed-form solution for the sum of the series of Bell Polynomials.

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$B_{n}$ is the complete exponential Bell polynomial.

$$\sum_{k=1}^n B_{n,k}(g'(t),g''(t),...)=B_{n}(g'(t),g''(t),...)$$

$$B_{0,0}=1,\ B_0=1$$

$$\frac{d^{n}}{dt^{n}}e^{g(t)}=e^{g(t)}B_{n}(g'(t),g''(t),...)$$

For certain $g$, there are general closed-form expressions for $B_{n,k}$ and $B_{n}$ and therefore for $\frac{d^n}{dt^n}e^{g(t)}$ - often with constants that are combinatorial numbers, e.g. Bell numbers, Stirling numbers, partition numbers.

Examples:

$$B_{n,k}(a^t)=S_{n,k}\ln(a)^n(a^t)^k,\ \ B_n(a^t)=\sum_{k=1}^nS_{n,k}\ln(a)^n(a^t)^k$$

$$\frac{d^n}{dt^n}e^{a^t}=e^{a^t}\sum_{k=1}^nS_{n,k}\ln(a)^n(a^t)^k$$

$S_{n,k}$: Stirling numbers of the second kind

$$B_{n,k}(\log_a(t))=s_{n,k}\ln(a)^{-k}t^{-n},\ \ B_n(\log_a(t))=\sum_{k=1}^ns_{n,k}\ln(a)^{-k}t^{-n}$$

$$\frac{d^n}{dt^n}e^{\log_a(t)}=e^{\log_a(t)}\sum_{k=1}^ns_{n,k}\ln(a)^{-k}t^{-n}$$

$s_{n,k}$: Stirling numbers of the first kind

In 2012, I wrote a still unpublished article "On partial Bell polynomials for the higher derivatives of composed functions".

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  • $\begingroup$ (1) 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 vol. 491 (2020), no. 2, Paper No. 124382, 31 pages; available online at doi.org/10.1016/j.jmaa.2020.124382. (2) 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 vol. 15 (2020), no. 1, 163--174; available online at doi.org/10.11575/cdm.v15i1.68111. $\endgroup$
    – qifeng618
    Sep 27 at 11:08

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