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\begin{center}
\vskip 1cm{\LARGE\bf Combinatorial Identities\\ 
for the Tricomi Polynomials\\
\vskip 1cm}
\large
Emanuele Munarini \\
Dipartimento di Matematica \\
Politecnico di Milano \\
Piazza Leonardo da Vinci 32 \\
Milano \\
Italy \\
\href{mailto:emanuele.munarini@polimi.it}{\tt emanuele.munarini@polimi.it}\\
\end{center}

\vskip .2 in

\begin{abstract}
 Using the technique of formal power series,
 we obtain some two-parameter binomial identities for the Tricomi polynomials.
 Moreover, we establish some relations between the Tricomi polynomials,
 the generalized derangement polynomials, and the Touchard polynomials.
 Finally, we obtain a characterization of the rising and falling factorial powers
 by means of a generalized binomial theorem.
\end{abstract}

\section{Introduction}

The \emph{Tricomi polynomials} \cite{Tricomi,CarlitzT,Agrawal} are defined by the formula
\begin{equation}\label{def-T}
 \ell_n^{(\alpha)}(x) = \sum_{k=0}^n { x-\alpha \choose k } (-1)^k \frac{x^{n-k}}{(n-k)!}\, .
\end{equation}
They satisfy the three-term recurrence
$$ (n+1) \ell_{n+1}^{(\alpha)}(x) - ( \alpha + n ) \ell_n^{(\alpha)}(x) + x \ell_{n-1}^{(\alpha)}(x) = 0 $$
with initial values $ \ell_0^{(\alpha)}(x) = 1 $ and $ \ell_1^{(\alpha)}(x) = \alpha $,
and have ordinary generating series
\begin{equation}\label{series-T}
 \ell^{(\alpha)}(x;t) = \sum_{n\geq0} \ell^{(\alpha)}_n(x)\, t^n = (1-t)^{x-\alpha} \ee^{xt} \, .
\end{equation}

The \emph{rising factorials} are defined by the \emph{Pochhammer symbol}
$$ (x)_n = x(x+1)(x+2)\cdots(x+n-1), $$
while the \emph{multiset coefficients} are defined by
$ \Mchoose{x}{n} = \frac{(x)_n}{n!} $.
They have generating series
$$
 \sum_{n\geq0} (x)_n \frac{t^n}{n!} = \frac{1}{(1-t)^x}
 \qquad\text{and\qquad}
 \sum_{n\geq0} \mchoose{x}{n} t^n = \frac{1}{(1-t)^x} \, .
$$
Notice that, by series (\ref{series-T}), we have the relations
\begin{equation}
 \frac{1}{(1-t)^\alpha}\cdot\ell^{(\beta)}(x;t) = \ell^{(\alpha+\beta)}(x;t) \label{Id00}
\end{equation}
and
\begin{equation}
 \ell^{(\alpha)}(x;t)\cdot \ell^{(\beta)}(y;t) = \ell^{(\alpha+\beta)}(x+y;t) \label{Id01}
\end{equation}
corresponding to the identities
\begin{equation}
 \sum_{k=0}^n \mchoose{\alpha}{k} \ell_{n-k}^{(\beta)}(x) = \ell^{(\alpha+\beta)}(x)
\end{equation}
and
\begin{equation}
 \sum_{k=0}^n \ell_k^{(\alpha)}(x)\, \ell_{n-k}^{(\beta)}(y) = \ell_n^{(\alpha+\beta)}(x+y) \, .
\end{equation}

From a purely combinatorial point of view,
it is more convenient to consider the exponential version of the Tricomi polynomials,
namely the polynomials
\begin{equation}\label{def-TT}
 \Lambda_n^{(\alpha)}(x) = n! \ell_n^{(\alpha)}(x)
 = \sum_{k=0}^n { n \choose k } { x-\alpha \choose k } (-1)^k k!\, x^{n-k}
\end{equation}
satisfying the recurrence
$$ \Lambda_{n+2}^{(\alpha)}(x) - (\alpha+n+1)\,\Lambda_{n+1}^{(\alpha)}(x) + (n+1)\,x\,\Lambda_n^{(\alpha)}(x) = 0 $$
with the initial values $ \Lambda_0^{(\alpha)}(x) = 1 $ and $ \Lambda_1^{(\alpha)}(x) = \alpha $,
and having exponential generating series
\begin{equation}\label{series-TT}
 \Lambda^{(\alpha)}(x;t) = \sum_{n\geq0} \Lambda^{(\alpha)}_n(x)\, \frac{t^n}{n!}
 = (1-t)^{x-\alpha} \ee^{xt} \, .
\end{equation}
For the first values of $ n $, we have the following polynomials:
\begin{align*}
  \Lambda_0^{(\alpha)}(x) &= (\alpha)_0 = 1 \\
  \Lambda_1^{(\alpha)}(x) &= (\alpha)_1 = \alpha \\
  \Lambda_2^{(\alpha)}(x) &= (\alpha)_2 - x \\
  \Lambda_3^{(\alpha)}(x) &= (\alpha)_3 - (2+3\alpha) x \\
  \Lambda_4^{(\alpha)}(x) &= (\alpha)_4 - (6+14\alpha+6\alpha^2) x + 3 x^2 \\
  \Lambda_5^{(\alpha)}(x) &= (\alpha)_5 - (24+70\alpha+50\alpha^2+10\alpha^3) x + (20+15\alpha) x^2 \\
  \Lambda_6^{(\alpha)}(x) &= (\alpha)_6 - (120+404 \alpha + 375 \alpha^2 + 130 \alpha^3 + 15 \alpha^4) x
 + (130 + 165 \alpha + 45 \alpha^2) x^2 - 15 x^3\, .
\end{align*}
Notice that $ \Lambda_n^{(\alpha)}(x) $ is a polynomial of degree $ n $ in $ \alpha $
and is a polynomial of degree at most $ \lfloor n/2 \rfloor $ in $ x $.
Moreover, if $ \alpha \in \NN $, then
$ \Lambda_n^{(\alpha)}(x) $ is a polynomial with integer coefficients.
In particular, we have $ \Lambda_n^{(\alpha)}(0) = (\alpha)_n $.

Identities (\ref{Id00}) and (\ref{Id01}) also hold for the exponential series $ \Lambda^{(\alpha)}(x;t) $
defined by (\ref{series-TT}). This time, we have the identities
\begin{equation*}
 \sum_{k=0}^n { n \choose k } (\alpha)_k\, \Lambda_{n-k}^{(\beta)}(x) = \Lambda^{(\alpha+\beta)}(x)
\end{equation*}
and
\begin{equation*}
 \sum_{k=0}^n { n \choose k } \Lambda_k^{(\alpha)}(x)\, \Lambda_{n-k}^{(\beta)}(y) = \Lambda_n^{(\alpha+\beta)}(x+y) \, .
\end{equation*}

The Tricomi polynomials
$ \Lambda_n^{(\alpha)}(x) = \sum_{k=0}^{\lfloor n/2 \rfloor} \Lambda_{n,k}^{(\alpha)}\; x^k $
are the row polynomials of the (improper) Sheffer matrix
(\cite[p.\ 309]{Aigner2007} \cite{MunariniA,MunariniC,FerrariMunarini})
$$
 \Lambda^{(\alpha)} = \big[ \Lambda_{n,k}^{(\alpha)} \big]_{n,k\geq0}
 = \left( \frac{1}{(1-t)^\alpha},t-\ln\frac{1}{1-t} \right)
$$
where
$$
 \Lambda_{n,k}^{(\alpha)} = \sum_{i=0}^n { n \choose i }
 \sum_{j=0}^{\min(i,k)} { i \choose j } { n-i \brack k-j } (-1)^{k-j} (\alpha)_{i-j}\, ,
$$
where the coefficients $ { n \brack k } $
are the Stirling numbers of the first kind \cite{GrahamKnuthPatashnik}.

In this paper, we obtain some two-parameter binomial identities for the Tricomi polynomials.
Moreover, we establish some relations between the Tricomi polynomials,
the generalized derangement polynomials and the Touchard polynomials.
Finally, we obtain a characterization of the rising and falling factorial powers
by means of a generalized binomial theorem.

To obtain the mentioned two-parameter binomial identities,
we will use (as we did in \cite{MunariniCal},
in order to extended a similar identity involving the derangement numbers)
the following theorem in the context of formal series:
\begin{theorem}[Taylor's formula]\label{thm-Taylor}
 For any formal power series $ f(t) $, the exponential generating series
 of the successive derivatives $ D_t^m f(t) $,
 where $ D_t = \frac{\dd}{\dd t} $ denotes the formal derivative with respect to $t$, is
 \begin{equation}\label{formula-Taylor}
  \sum_{m\geq0} D^m_t f(t)\,\frac{u^m}{m!} = f(t+u) \, .
 \end{equation}
\end{theorem}

Notice that this theorem is valid both when $f(t)$ is an exponential series and when $f(t)$ is an ordinary series.
Moreover, the $m$-derivative of an exponential series $ f(t) = \sum_{n\geq0} f_n \frac{t^n}{n!} $ is
\begin{equation}\label{series-DSer1}
 D^m f(t) = \sum_{n\geq0} f_{n+m} \frac{t^n}{n!}
\end{equation}
while the $m$-derivative of an ordinary series $ f(t) = \sum_{n\geq0} f_n\, t^n $ is
\begin{equation}\label{series-DSer2}
 D^m f(t) = m! \sum_{n\geq0} { m+n \choose n } f_{n+m}\, t^n \, .
\end{equation}

\section{Tricomi polynomials}

We start by computing the successive derivatives of the generating series of the Tricomi polynomials.
\begin{lemma}\label{lemma-DmT}
 For every $ m \in \NN $, we have the identity
 \begin{equation}\label{id-DmT}
  D^m_t \ell^{(\alpha)}(x;t) =
  m! \sum_{k=0}^m { x-\alpha \choose k } (-1)^k \frac{x^{m-k}}{(m-k)!}\; \ell^{(\alpha+k)}(x;t)
 \end{equation}
 or, equivalently,
 \begin{equation}\label{id-DmTT}
  D^m_t \Lambda^{(\alpha)}(x;t) =
  \sum_{k=0}^m { m \choose k } { x-\alpha \choose k } (-1)^k k!\, x^{m-k}\; \Lambda^{(\alpha+k)}(x;t) \, .
 \end{equation}
\end{lemma}
\begin{proof}
 By applying Taylor's formula (\ref{formula-Taylor}) to series (\ref{series-T}), we have
 \begin{align*}
   \sum_{m\geq0} D^m_t \ell^{(\alpha)}(x;t)\,\frac{u^m}{m!}
   &= \ell^{(\alpha)}(x;t+u) \\
   &= (1-t-u)^{x-\alpha}\, \ee^{x(t+u)} \\
   &= (1-t)^{x-\alpha} \Big(1-\frac{u}{1-t}\Big)^{x-\alpha}\, \ee^{xt} \ee^{xu} \\
   &= \ell^{(\alpha)}(x;t)\; \Big(1-\frac{u}{1-t}\Big)^{x-\alpha}\, \ee^{xu} \\
   &= \sum_{m\geq0}
      \left[ \sum_{k=0}^m { x-\alpha \choose k } (-1)^k
      \frac{m!\,x^{m-k}}{(m-k)!}\, \frac{\ell^{(\alpha)}(x;t)}{(1-t)^k} \right] \frac{u^m}{m!}\, .
 \end{align*}
 Hence, by identity (\ref{Id00}), we obtain identity (\ref{id-DmT})
 (and, consequently, identity (\ref{id-DmTT})).
\end{proof}

As an immediate consequence of Lemma \ref{lemma-DmT}
and formulas (\ref{series-DSer1}) and (\ref{series-DSer2}),
we have the following theorem.
\begin{theorem}
 For every $ m,n \in \NN $, we have the identities
 \begin{equation}
  { m+n \choose n } \ell^{(\alpha)}_{m+n}(x) =
  \sum_{k=0}^m { x-\alpha \choose k } (-1)^k \frac{x^{m-k}}{(m-k)!}\; \ell^{(\alpha+k)}_n(x) \label{id-m+nT}
 \end{equation}
 and
 \begin{equation}
  \Lambda^{(\alpha)}_{m+n}(x) =
  \sum_{k=0}^m { m \choose k } { x-\alpha \choose k } (-1)^k k!\, x^{m-k}\; \Lambda_n^{(\alpha+k)}(x) \, .
 \end{equation}
\end{theorem}

\begin{remark}
 Notice that Agrawal \cite{Agrawal} obtained the following different relation
 $$
  { m+n \choose n } \ell^{(\alpha)}_{m+n}(x) =
  \sum_{k=0}^{\min(m,n)} \mchoose{\alpha-x+n}{k} \ell^{(\alpha+n+k)}_{m-k}(x)\, \ell^{(\alpha-m+k)}_{n-k}(x),
 $$
 which can also be rewritten as
 $$
  \Lambda^{(\alpha)}_{m+n}(x) = \sum_{k=0}^{\min(m,n)} { m \choose k } { n \choose k }
  (\alpha-x+n)_k\, k!\, \Lambda^{(\alpha+n+k)}_{m-k}(x)\, \Lambda^{(\alpha-m+k)}_{n-k}(x) \, .
 $$
\end{remark}

More generally, Lemma \ref{lemma-DmT} implies the following two-parameter identities.
\begin{theorem}
 For every $ m,n \in \NN $, we have the identity
 \begin{equation}\label{id-CallanT}
  \sum_{k=0}^n { m+k \choose k } \ell^{(\alpha)}_{m+k}(x)\, \ell^{(\beta)}_{n-k}(y) =
  \sum_{k=0}^m { x-\alpha \choose k } (-1)^k \frac{x^{m-k}}{(m-k)!}\, \ell^{(\alpha+\beta+k)}_n(x+y) \, .
 \end{equation}
 Equivalently, we have the identity
 \begin{equation}\label{id-CallanTT}
  \sum_{k=0}^n { n \choose k } \Lambda^{(\alpha)}_{m+k}(x)\, \Lambda^{(\beta)}_{n-k}(y) =
  \sum_{k=0}^m { m \choose k } { x-\alpha \choose k } (-1)^k k!\, x^{m-k} \Lambda^{(\alpha+\beta+k)}_n(x+y) \, .
 \end{equation}
\end{theorem}
\begin{proof}
 By identity (\ref{id-DmT}) and property (\ref{Id01}), we have
 \begin{align*}
  \ell^{(\beta)}(y;t)\, \frac{1}{m!} D^m_t \ell^{(\alpha)}(x;t)
   &= \sum_{k=0}^m { x-\alpha \choose k } (-1)^k \frac{x^{m-k}}{(m-k)!}\; \ell^{(\alpha+k)}(x;t)\,\ell^{(\beta)}(y;t) \\
   &= \sum_{k=0}^m { x-\alpha \choose k } (-1)^k \frac{x^{m-k}}{(m-k)!}\; \ell^{(\alpha+\beta+k)}(x+y;t)
 \end{align*}
 from which we have identity (\ref{id-CallanT}) (and identity (\ref{id-CallanTT})).
\end{proof}

\begin{remark}\label{rem-main}
 If $ y = -x $, then
 $$ \ell^{(\alpha+\beta+k)}_n(x+y) = \ell^{(\alpha+\beta+k)}_n(0) = \mchoose{\alpha+\beta+k}{n} $$
 and identity (\ref{id-CallanT}) becomes
 \begin{equation}
  \sum_{k=0}^n { m+k \choose k } \ell^{(\alpha)}_{m+k}(x)\, \ell^{(\beta)}_{n-k}(-x) =
  \sum_{k=0}^m \mchoose{\alpha+\beta+k}{n} { x-\alpha \choose k } (-1)^k \frac{x^{m-k}}{(m-k)!} \, .
 \end{equation}
 Moreover, if $ \beta = y = -x $, then
 $$ \ell^{(-x)}_n(-x) = (-1)^n \frac{x^n}{n!} $$
 and
 $$
  \ell^{(\alpha+\beta+k)}_n(x+y) = \ell^{(\alpha+\beta+k)}_n(0)
  = \mchoose{\alpha-x+k}{n} = (-1)^n { x-\alpha-k \choose n }.
 $$
 So, identity (\ref{id-CallanT}) becomes
 \begin{equation}
  \sum_{k=0}^n { m+k \choose k } (-1)^k \frac{x^{n-k}}{(n-k)!}\; \ell^{(\alpha)}_{m+k}(x) =
  \sum_{k=0}^m { x-\alpha \choose k }  { x-\alpha-k \choose n } (-1)^k \frac{x^{m-k}}{(m-k)!} \, .
 \end{equation}
 Similarly, if $ y = 0 $, then identity (\ref{id-CallanT}) becomes
 \begin{equation}
  \sum_{k=0}^n { m+k \choose k } \mchoose{\beta}{n-k} \ell^{(\alpha)}_{m+k}(x) =
  \sum_{k=0}^m { x-\alpha \choose k } (-1)^k \frac{x^{m-k}}{(m-k)!}\, \ell^{(\alpha+\beta+k)}_n(x)\, .
 \end{equation}
 Finally, if $ x = y = 0 $, then identity (\ref{id-CallanT}) becomes
 \begin{equation}
  \sum_{k=0}^n { m+k \choose k } \mchoose{\alpha}{m+k} \mchoose{\beta}{n-k} =
  \mchoose{\alpha}{m} \mchoose{\alpha+\beta+m}{n} \, .
 \end{equation}
 Equivalently, this identity can be easily rewritten as
 \begin{equation}\label{thm-bin-rising}
  \sum_{k=0}^n { n \choose k } (\alpha)_{m+k} (\beta)_{n-k} =
  (\alpha)_{m} (\alpha+\beta+m)_{n} \, .
 \end{equation}
 For $ m = 0 $, we recover the fact that the rising factorials
 form a polynomial sequence of binomial type \cite{MullinRota,RotaKO,JoniRotaSagan},
 that is, that they satisfy the binomial identity
 $$ \sum_{k=0}^n { n \choose k } (\alpha)_k (\beta)_{n-k} = (\alpha+\beta)_{n} \, . $$
 Notice that replacing $ \alpha $ and $ \beta $ by $ -\alpha $ and $ -\beta $, respectively,
 then identity (\ref{thm-bin-rising}) becomes
 \begin{equation}\label{thm-bin-falling}
  \sum_{k=0}^n { n \choose k } \falling{\alpha}{m+k}\; \falling{\beta}{n-k} =
  \falling{\alpha}{m}\; \falling{(\alpha+\beta-m)}{n}
 \end{equation}
 where the polynomials $ \falling{x}{n} = x(x-1)(x-2)\cdots(x-n+1) $
 are the \emph{falling factorials}.
\end{remark}

\section{Generalized derangement polynomials}

The \emph{generalized derangement numbers} $ d^{(\nu)}_n $
and the \emph{generalized arrangement numbers} $ a^{(\nu)}_n $
are defined \cite{MunariniCal} by the formulas
\begin{align}
 & d_n^{(\nu)} = \sum_{k=0}^n { \nu+n-k \choose n-k } \frac{n!}{k!}\, (-1)^k \label{def-dG} \\
 & a_n^{(\nu)} = \sum_{k=0}^n { \nu+n-k \choose n-k } \frac{n!}{k!} \label{def-aG}
\end{align}
and have exponential generating series
\begin{align}
 & d^{(\nu)}(t) = \sum_{n\geq0} d^{(\nu)}_n \frac{t^n}{n!}
 = \frac{\ee^{-t}}{(1-t)^{\nu+1}} \label{series-dn} \\
 & a^{(\nu)}(t) = \sum_{n\geq0} a^{(\nu)}_n \frac{t^n}{n!}
 = \frac{\ee^{t}}{(1-t)^{\nu+1}} \, . \label{series-an}
\end{align}
For $ \nu = 0 $, we have the ordinary derangement numbers $ d_n $
\cite[p.\ 182]{Comtet} (\seqnum{A000166} in the OEIS {Sloane})
and the ordinary arrangement numbers $ a_n $
\cite[p.\ 75]{Comtet} \seqnum{A000522}.

The \emph{generalized derangement polynomials} \cite{MunariniCal}
are the \emph{Appell polynomials} \cite{Appell,MunariniA,Roman}
associated with the generalized derangement numbers, namely
$$
 D_n^{(\nu)}(x) = \sum_{k=0}^n { n \choose k } d_{n-k}^{(\nu)} x^k
 = \sum_{k=0}^n { \nu+n-k \choose n-k } \frac{n!}{k!}\, (x-1)^k
$$
and have exponential generating series
\begin{equation}\label{series-Dnx}
 D^{(\nu)}(x;t) = \sum_{n\geq0} D^{(\nu)}_n(x) \frac{t^n}{n!} = \frac{\ee^{(x-1)t}}{(1-t)^{\nu+1}} \, .
\end{equation}
In particular, we have $ D^{(\nu)}_n(0) = d^{(\nu)}_n $,
$ D^{(\nu)}_n(1) = { \nu+n \choose n } n! $ and $ D^{(\nu)}_n(2) = a^{(\nu)}_n $.

The generalized derangement polynomials and the Tricomi polynomials
are related in the following way\footnote{Notice that
the numbers $ d^{(\nu)}_n $ and $ a^{(\nu)}_n $,
and the polynomials $ D_n^{(\nu)}(x) $ considered here
are very similar to those considered in some recent papers
\cite{CFMunariniZ,FerrariMunarini,MunariniD,FerrariMunariniZagaglia}
and that all the results we obtain here can be easily adapted to these variants.
}.
\begin{theorem}
 For every $ n \in \NN $, we have the identity
 \begin{equation}\label{TriD}
  D^{(\nu)}_n(x) = \Lambda^{(\nu+x)}_n(x-1)\, .
 \end{equation}
 In particular, for $ x = 0 $, $ x = 1 $ and $ x = 2 $, we have the identities
 $$
  \Lambda^{(\nu)}_n(-1) = d^{(\nu)}_n \, , \qquad
  \Lambda^{(\nu+1)}_n(0) = { \nu + n \choose n } n!
  \qquad\text{and}\qquad
  \Lambda^{(\nu+2)}_n(1) = a^{(\nu)}_n \, .
 $$
\end{theorem}
\begin{proof}
 By series (\ref{series-TT}) and (\ref{Id-TriD}), we have
 $$ \Lambda^{(\nu+x)}(x-1;t) = (1-t)^{x-1-\nu-x} \ee^{(x-1)t} = \frac{\ee^{(x-1)t}}{(1-t)^{\nu+1}} = D^{(\nu)}(x;t)\, . $$
 This relation implies identity (\ref{TriD}) at once.
\end{proof}

Moreover, we have the following result.
\begin{theorem}
 For every $ n \in \NN $, we have the identity
 \begin{equation}\label{Id-TriD}
  \sum_{k=0}^n { n \choose k }  D^{(\alpha)}_k(x)\, \Lambda^{(\beta)}_{n-k}(y) =
  \sum_{k=0}^n { n \choose k } (x)_k\, \Lambda^{(\alpha+\beta)}_{n-k}(x+y-1)\, .
 \end{equation}
 In particular, for $ x = 0, 2 $ and $ y = 0 $, we have the identities
 \begin{equation*}
  \sum_{k=0}^n { n \choose k }  d^{(\alpha)}_k\, (\beta)_{n-k} = d^{(\alpha+\beta)}_n 
 \end{equation*}
 and
 \begin{equation*}
  \sum_{k=0}^n { n \choose k }  a^{(\alpha)}_k\, (\beta)_{n-k} =
  \sum_{k=0}^n { n \choose k }  (k+1)!\, a^{(\alpha+\beta-2)}_{n-k}\, .
 \end{equation*}
\end{theorem}
\begin{proof}
 By series (\ref{Id-TriD}) and (\ref{series-TT}), we have
 \begin{align*}
  D^{(\alpha)}(x;t)\, \Lambda^{(\beta)}(x;t)
  &= \frac{\ee^{(x-1)t}}{(1-t)^{\alpha+1}}\cdot (1-t)^{y-\beta} \ee^{yt} \\
  &= \frac{1}{(1-t)^x}\cdot (1-t)^{x+y-1-\alpha-\beta} \ee^{(x+y-1)t} \\
  &= \frac{1}{(1-t)^x}\cdot \Lambda^{(\alpha+\beta)}(x+y-1;t)\, .
 \end{align*}
 This relation is equivalent to identity (\ref{Id-TriD}).
\end{proof}

More generally, we have the following formulas.
\begin{theorem}
 For every $ m,n \in \NN $, we have the identities
 \begin{align}
  & \sum_{k=0}^n { m+k \choose k } \frac{d^{(\nu)}_{n-k}}{(n-k)!}\, \ell^{(\alpha)}_{m+k}(x) =
    \sum_{k=0}^m { x-\alpha \choose k } (-1)^k \frac{x^{m-k}}{(m-k)!}\, \ell^{(\alpha+\nu+k)}_n(x-1)
    \label{id-derT} \\
  & \sum_{k=0}^n { m+k \choose k } \frac{a^{(\nu)}_{n-k}}{(n-k)!}\, \ell^{(\alpha)}_{m+k}(x) =
    \sum_{k=0}^m { x-\alpha \choose k } (-1)^k \frac{x^{m-k}}{(m-k)!}\, \ell^{(\alpha+\nu+k+2)}_n(x+1) \, .
    \label{id-arrT}
 \end{align}
 Equivalently, we have the identities
 \begin{align}
  & \sum_{k=0}^n { n \choose k } d^{(\nu)}_k\, \Lambda^{(\alpha)}_{m+n-k}(x) =
    \sum_{k=0}^m { m \choose k } { x-\alpha \choose k } (-1)^k k!\, x^{m-k}\, \Lambda^{(\alpha+\nu+k)}_n(x-1)
    \label{id-derTT} \\
  & \sum_{k=0}^n { n \choose k } a^{(\nu)}_k\, \Lambda^{(\alpha)}_{m+n-k}(x) =
    \sum_{k=0}^m { m \choose k } { x-\alpha \choose k } (-1)^k k!\, x^{m-k}\, \Lambda^{(\alpha+\nu+k+2)}_n(x+1) \, .
    \label{id-arrTT}
 \end{align}
\end{theorem}
\begin{proof}
 From identity (\ref{id-DmT}) and series (\ref{series-dn}), we have
 \begin{align*}
  d^{(\nu)}(t)\, \frac{1}{m!} D^m_t \ell^{(\alpha)}(x;t)
  &= \sum_{k=0}^m { x-\alpha \choose k } (-1)^k \frac{x^{m-k}}{(m-k)!}\; d^{(\nu)}(t)\, \ell^{(\alpha+k)}(x;t) \\
  &= \sum_{k=0}^m { x-\alpha \choose k } (-1)^k \frac{x^{m-k}}{(m-k)!}\; (1-t)^{x-1-\alpha+\nu-k}\, \ee^{(x-1)t} \\
  &= \sum_{k=0}^m { x-\alpha \choose k } (-1)^k \frac{x^{m-k}}{(m-k)!}\; \ell^{(\alpha+\nu+k)}(x-1;t)
 \end{align*}
 from which we obtain identity (\ref{id-derT}).
 In a similar way, we also obtain identity (\ref{id-arrT}).
\end{proof}

Consider the \emph{Touchard polynomials} $ T_n(x) $, \cite{RotaBell,Touchard},
and the associated  polynomials $ U_n(x) $ defined by
\begin{align*}
 & T_n(x) = \sum_{k=0}^n { n \choose k } (-1)^{n-k} \falling{x}{k} \\
 & U_n(x) = \sum_{k=0}^n { n \choose k } (-1)^{n-k} (x)_k
\end{align*}
and having exponential generating series
\begin{align}
 & T(x;t) = \sum_{n\geq0} T_n(x) \frac{t^n}{n!} = \ee^{-t} (1+t)^x \label{series-Touchard} \\
 & U(x;t) = \sum_{n\geq0} U_n(x) \frac{t^n}{n!} = \frac{\ee^{-t}}{(1-t)^x} \label{series-TouchardU} \, .
\end{align}
The following identities relate the Tricomi polynomials and the generalized derangement polynomials
by means of the Touchard polynomials.
\begin{theorem}
 We have the identities
 \begin{align}
  & \Lambda_n^{(\alpha)}(x) = \sum_{k=0}^n { n \choose k } (-1)^k\, T_k(x+1)\, D_{n-k}^{(\alpha)}(x) \label{id-TriTouDer} \\
  & D_n^{(\alpha)}(x) = \sum_{k=0}^n { n \choose k } U_k(x+1)\, \Lambda_{n-k}^{(\alpha)}(x) \label{id-TriTouUDer} \, .
 \end{align}
\end{theorem}
\begin{proof}
 By series (\ref{series-TT}), (\ref{series-Dnx}) and (\ref{series-Touchard}), we have
 $$
  \Lambda^{(\alpha)}(x;t)
  = (1-t)^{x-\alpha} \ee^{xt}
  = \ee^t (1-t)^{x+1} \cdot \frac{\ee^{(x-1)t}}{(1-t)^{\alpha+1}}
  = T(x+1;-t)\cdot D^{(\alpha)}(x;t)
 $$
 from which we get identity (\ref{id-TriTouDer}) at once.
 Similarly, by series (\ref{series-TT}), (\ref{series-Dnx}) and (\ref{series-TouchardU}), we have
 $$
  D^{(\alpha)}(x;t)
  = \frac{\ee^{(x-1)t}}{(1-t)^{\alpha+1}}
  = \frac{\ee^{-t}}{(1-t)^{x+1}}\cdot(1-t)^{x-\alpha} \ee^{xt}
  = U(x+1;t)\cdot\Lambda^{(\alpha)}(x;t)
 $$
 from which we get identity (\ref{id-TriTouUDer}) at once.
\end{proof}

\section{Final remarks}

The rising factorials and the falling factorials
form two polynomial sequences of binomial type
and have several characterizations and combinatorial interpretations \cite{JoniRotaSagan}.
In Remark \ref{rem-main}, we noticed that these polynomials satisfy
the generalized binomial theorems (\ref{thm-bin-rising}) and (\ref{thm-bin-falling}), respectively.
More generally, we have the following characterization.
\begin{theorem}
 Let $ \{p_n(x)\}_{n\in\NN} $ be a polynomial sequence,
 where each polynomial $ p_n(x) $ has degree $ n $.
 There exists a constant $ \ll \ne 0 $ for which the binomial identity
 \begin{equation}\label{thm-bin}
  \sum_{k=0}^n { n \choose k } p_{m+k}(x)\, p_{n-k}(y) = p_m(x)\, p_n(x+y+\ll\, m)
  \qquad \forall m, n \in \NN
 \end{equation}
 holds if and only if there exists a constant $ \mu $ such that
 \begin{equation}\label{PocSym}
  p_n(x) = (\ll\mu)^n (x/\ll)_n \, .
 \end{equation}
\end{theorem}
\begin{proof}
 If identity (\ref{thm-bin}) is true for every $ m, n \in \NN $,
 then it is true also for $ m = 0 $ and $ n \in \NN $.
 This implies that $ \{p_n(x)\}_{n\in\NN} $ is a polynomial sequence of binomial type
 and, consequently, that it has exponential generating series
 $$ p(x;t) = \sum_{n\geq0} p_n(x)\,\frac{t^n}{n!} = \ee^{xf(t)} $$
 for a given exponential series $ f(t) = \sum_{n\geq0} f_n \frac{t^n}{n!} $
 with $ f_0 = 0 $ and $ f_1 \ne 0 $.
 Hence, identity (\ref{thm-bin}) turns out to be equivalent to the identity
 $$ ( D_t^m p(x;t) )\, p(y;t) = p_m(x)\,  p(x+y+\ll m;t) \qquad \forall m \in \NN $$
 that is
 $$ ( D_t^m \ee^{xf(t)} )\, \ee^{yf(t)} = p_m(x)\, \ee^{(x+y+\ll m)f(t)} \qquad \forall m \in \NN \, . $$
 that is
 $$ D_t^m \ee^{xf(t)} = p_m(x)\, \ee^{(x+\ll m)f(t)} \qquad \forall m \in \NN \, . $$
 For $ m = 1 $, this relation reduces to
 $$ x f'(t)\, \ee^{xf(t)} = p_1(x)\, \ee^{(x+\ll)f(t)} $$
 or $$ x f'(t) = p_1(x)\, \ee^{\ll f(t)} $$
 or $$ \frac{f'(t)}{\ee^{\ll f(t)}} = \frac{p_1(x)}{x} = \mu $$
 for a constant $ \mu $.
 This is equivalent to $ p_1(x) = \mu\, x $ and $ f'(t) = \mu\,\ee^{\ll f(t)} $.
 By integrating this last differential equation, we obtain
 $$ f(t) = \frac{1}{\ll} \ln\frac{1}{1-\ll\mu t} $$
 and consequently
 $$
  p(x;t) = \ee^{\frac{x}{\ll} \ln\frac{1}{1-\ll\mu t}}
         = \frac{1}{(1-\ll\mu t)^{x/\ll}}
         = \sum_{n\geq0} (\ll\mu)^n (x/\ll)_n \frac{t^n}{n!}
 $$
 from which we have identity (\ref{PocSym}).
 Vice versa, employing identity (\ref{thm-bin-rising}),
 we can say that the polynomials defined by formula (\ref{PocSym})
 satisfy the binomial identity (\ref{thm-bin}).
\end{proof}

Notice that the falling factorials can be expressed by identity (\ref{PocSym})
for $ \ll = -1 $ and $ \mu = 1 $, namely $ \falling{x}{n} = (-1)^n (-x)_n $.

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\bigskip
\hrule
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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 05A19;
Secondary 05A10, 05A15.

\noindent \emph{Keywords: }
combinatorial sum, binomial sum, Sheffer sequence, Appell sequence.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A000166} and
\seqnum{A000522}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received May 15 2020;
revised version received  August 26 2020.
Published in {\it Journal of Integer Sequences}, October 15 2020.

\bigskip
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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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