Polynomials akin to Bell polynomials I'm wondering whether the following polynomial of a single indeterminate has been studied: take the (partial) Bell polynomial $B_{n,k}$, which is a polynomial in indeterminates $x_1$, $x_2$, …, $x_{n-k+1}$, and replace each indeterminate $x_i$ by the falling factorial $(x)_i=x(x-1) \dots (x-i+1)$. Call this polynomial $N(n,k)$.
I conjecture the following matrix identity. Assemble the polynomials into an infinite lower-triangular matrix $N$. Let $s$ be the lower-triangular matrix of Stirling numbers of the first kind. Let D be the diagonal matrix with entries $x$, $x^2$, $x^3$, ….
Conjecture: $Ns=sD$
 A: As pointed out by Richard Stanley in the comments, from the generating function of the Bell polynomials one finds
$$N(n,k) = \frac{1}{k!}\sum_{i=1}^k(-1)^{k-i}\binom ki (ix) _n.$$
Now use the fact that
$$(ix) _n=\sum _{j=0}^n s(n,j)i^j x^j,$$
and that
$$\frac{1}{k!}\sum _{i=0}^k (-1)^{k-i}\binom{k}{i} i^j=S(j,k),$$
to obtain
$$N(n,k)=\sum _{j=k}^n s(n,j)x^jS(j,k).$$
In other words, $N=sDS$, where $s,S$ are the lower triangular matrices of Stirling numbers of first and second kind, respectively. To finish off, notice that $s^{-1}=S$ by Stirling number duality.
A: 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.

