Let
\begin{equation*}%\label{Fall-Factorial-Dfn-Eq}
\langle\alpha\rangle_n=
\prod_{k=0}^{n-1}(\alpha-k)=
\begin{cases}
\alpha(\alpha-1)\dotsm(\alpha-n+1), & n\ge1\\
1,& n=0
\end{cases}
\end{equation*}
and
\begin{equation*}
(\alpha)_n=\prod_{\ell=0}^{n-1}(\alpha+\ell)
=
\begin{cases}
\alpha(\alpha+1)\dotsm(\alpha+n-1), & n\ge1\\
1, & n=0
\end{cases}
\end{equation*}
denote the falling and rising factorials of $\alpha\in\mathbb{C}$ respectively.
In Remark 3.1 of the paper [1] below, the formula
\begin{equation}\label{Bell-1-lambda}\tag{1}
B_{n,k}\Biggl(1, 1-\alpha, (1-\alpha)(1-2\alpha),\dotsc, \prod_{\ell=0}^{n-k}(1-\ell\alpha)\Biggr)
=\frac{(-1)^k}{k!} \sum_{\ell=0}^k (-1)^{\ell} \binom{k}{\ell} \prod_{q=0}^{n-1}(\ell-q\alpha)
\end{equation}
for $\alpha\in\mathbb{C}$ and $n\ge k\ge0$ was concluded.
In Theorem 2.1 of the paper [2] below, the formulas
\begin{equation}\label{Bell-fall-Eq}\tag{2}
B_{n,k}(\langle\alpha\rangle_1, \langle\alpha\rangle_2, \dotsc,\langle\alpha\rangle_{n-k+1})
=\frac{(-1)^k}{k!}\sum_{\ell=0}^{k}(-1)^{\ell}\binom{k}{\ell}\langle\alpha\ell\rangle_n
\end{equation}
and
\begin{equation}\label{Bell-fall-Eq-inv}\tag{3}
\sum_{\ell=0}^{k}\frac{B_{n,\ell}(\langle\alpha\rangle_1, \langle\alpha\rangle_2, \dotsc,\langle\alpha\rangle_{n-\ell+1})}{(k-\ell)!}
=\frac{\langle\alpha k\rangle_n}{k!}
\end{equation}
for $n\ge k\ge0$ and $\alpha\in\mathbb{R}$ were established.
As consequences of the formulas \eqref{Bell-fall-Eq} and \eqref{Bell-fall-Eq-inv}, the formulas
\begin{equation}\label{Bell-rise-Eq}\tag{4}
B_{n,k}((\alpha)_1, (\alpha)_2, \dotsc, (\alpha)_{n-k+1})
=\frac{(-1)^k}{k!}\sum_{\ell=0}^{k}(-1)^{\ell}\binom{k}{\ell}(\alpha\ell)_n
\end{equation}
and
\begin{equation}\label{Bell-rise-Eq-inv}\tag{5}
\sum_{\ell=0}^{k}\frac{B_{n,\ell}((\alpha)_1, (\alpha)_2, \dotsc, (\alpha)_{n-\ell+1})}{(k-\ell)!}
=\frac{(\alpha k)_n}{k!}
\end{equation}
for $\alpha\in\mathbb{C}$ and $n\ge k\ge0$ were derived in Corollary 2.1 of the paper [2].
The formulas \eqref{Bell-1-lambda} and \eqref{Bell-fall-Eq} were reviewed in Section 1.3 of the article [3] below.
The formulas \eqref{Bell-1-lambda}, \eqref{Bell-fall-Eq}, and \eqref{Bell-rise-Eq} are equivalent to
\begin{equation}\tag{6}
B_{n,k}(\langle\alpha\rangle_1, \langle\alpha\rangle_2, \dotsc,\langle\alpha\rangle_{n-k+1})
=\sum _{j=k}^n s(n,j)\alpha^jS(j,k)
\end{equation}
for $n\ge k\ge0$ and $\alpha\in\mathbb{R}$ at https://mathoverflow.net/a/88071/147732, where $s(n,j)$ and $S(j,k)$ stand for the Stirling numbers of the first and second kinds respectively.
References
- B.-N. Guo and F. Qi, Viewing some ordinary differential equations from the angle of derivative polynomials, Iran. J. Math. Sci. Inform. 16 (2021), no. 1, 77--95; available online at https://doi.org/10.29252/ijmsi.16.1.77.
- 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.
- 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.
