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\begin{center}
\vskip 1cm{\LARGE\bf Some Classes of Numbers and Derivatives \\
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}
\vskip 1cm
\large {
Milan Janji\'c \\
Department of Mathematics and Informatics\\
 University of Banja Luka \\
Republic of Srpska, Bosnia and Herzegovina} \\
\href{mailto:agnus@blic.net}{\tt agnus@blic.net}\\
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\begin{abstract}
We prove that three classes of numbers -- the non-central Stirling
numbers of the first kind, generalized factorial coefficients, and
Gould-Hopper numbers -- may be defined by the use of derivatives. We
derive several properties of these numbers from their definitions. We
also prove a result for harmonic numbers.  The coefficients of Hermite
and  Bessel polynomials are a particular case of generalized factorial
coefficients, The coefficients of the associated Laguerre polynomials
are a particular case of Gould-Hopper numbers. So we obtain some
properties of these polynomials. In particular,
we derive an orthogonality  relation for the coefficients of Hermite
and Bessel polynomials.
\end{abstract}

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\section{Introduction}
The purpose of this paper is to investigate properties of the
non-central Stirling numbers of the first kind,  the generalized
factorial coefficients, and  Gould-Hopper numbers by the use of
derivatives.

We use the following notation throughout the paper:

\begin{itemize}

\item $(a)_n$ denotes the falling factorial of $a,$ that
 is, $(a)_n=a(a-1)\cdots(a-n+1),$ $(a)_0=1;$

\item $s(n,k)$ and $\mathbf s(n,k)$ denote the signed and the unsigned  Stirling numbers of the first kind respectively;

\item $H_n$ denotes the harmonic number $\sum_{1 \leq i \leq n} 1/i$;

\item $s(n,k,a)$  denotes the non-central Stirling number of the first kind;

\item $C(n,k,a)$ denotes the generalized factorial coefficient;

\item $C(n,k,b,a)$ denotes the non-central generalized factorial coefficient  or
Gould-Hopper number.

\end{itemize}

The notation and the terminology are taken from Charalambides' book \cite{char}.
Sloane \cite{Slo} calls the $s(n,k,a)$ the generalized Stirling numbers.
Further, 

\begin{itemize}
\item $H_n(x)$ denotes the Hermite polynomial;
\item $p_n(x)$ denotes the (reverse) Bessel polynomial, and
\item $L^k_n(x)$ denotes the associated Laguerre polynomial, where $L^0_n(x)=L_n(x)$ is a Laguerre polynomial.
\end{itemize}

The paper is organized as follows. The first section is an introduction.

In the second section we prove that the  non-central Stirling numbers
$s(n,k,a)$ of the first kind  naturally appear in the expansion of
derivatives of the function $x^{-a}\ln^b x,$ where $a$ and $b$ are
arbitrary real numbers.  We first obtain a recurrence relation for
$s(n,k,a)$ and then, using Leibnitz rule, we obtain an explicit
formula.  We then consider a particular formula for  $s(n,1,a)$
and derive some combinatorial identities. The results are related to a
number of sequences from Sloane's {\it Encyclopedia} \cite{Slo}.

In the third section we first prove that the generalized
factorial coefficients appear as coefficients in the expansion of
the $n$th derivative of the function $f(x^a),$ where $a$ is arbitrary real number, and $f\in
C^\infty(0,+\infty)$ is arbitrary function.  Choosing
suitable functions $f$ we derive some  properties of generalized
factorial coefficients.
We are particularly concerned with some properties of coefficients of Hermite and Bessel polynomials.
The results of this section are also  related to a number of sequences from \cite{Slo}.

In the fourth  section we first show that Gould-Hopper numbers
are coefficients in the expansion of the $n$th derivative of
the function $x^af(x^b),$ where $a,b$ are arbitrary real numbers, and  $f\in
C^\infty(0,+\infty)$ is arbitrary function. The coefficients of associated  Laguerre polynomials are particular case of Gould-Hopper numbers.
Using similar methods as in the third
section we prove a number of properties which describe
connections between  Gould-Hopper numbers, generalized factorial coefficients, powers, factorials, binomial coefficients, and Stirling
numbers. The results are also concerned with some sequences from \cite{Slo}.

Note that these considerations are related with Bell polynomials which naturally appear in derivatives of composition  functions \cite{bell, nat, xu} .
\section{Non-central Stirling numbers of the first kind}
We shall first derive a formula for the $n$th  derivative of the function   \[f(x)=x^{-a}\ln^b x,\;(a,b\in \mathbb R).\]
\begin{theorem}\label{t1} Let $a$ be a real number, and let $n$ be a nonnegative integer. Then
\begin{equation}\label{e1}\frac{d^n}{dx^n}f(x)=x^{-a-n}\sum_{i=0}^np(n,i,a)(b)_i\ln^{b-i} x.\end{equation}
where $p(n,i,a),\;(0\leq i\leq n)$ are polynomials of $a$ with integer coefficients.
\end{theorem}
\begin{proof}
Theorem \ref{t1} is true  for $n=0,$ if we define
$p(0,0,a)=1.$

If we define \[p(1,0,a)=-a,\;p(1,1,a)=1\]
then Theorem \ref{t1} is also  true for  $n=1.$

Assume Theorem \ref{t1} is valid for $n\geq 1.$

Taking derivative in (\ref{e1}) we find that
\[\frac{d^{n+1}}{dx^{n+1}}f(x)=x^{-a-n-1}\left[(-a-n)\sum_{i=0}^np(n,i,a)(b)_i\ln^{b-i}x+\sum_{i=0}^{n} p(n,i,a)(b)_{i+1}\ln^{b-i-1}x\right].\]
Replacing $i+1$ by $i$ in the second sum on the right side yields \[\frac{d^{n+1}}{dx^{n+1}}f(x)=x^{a-n-1}(-a-n)p(n,0,a)+s(n,n,a)(b)_{n+1}\ln^{b-n-1}x+\]
\[+x^{a-n-1}\sum_{i=1}^n\left[(-a-n)p(n,i,a)+s(n,i-1,a)\ln^{b-i}x\right](b)_i.\]

It follows that Theorem \ref{t1}  is true if we define
\[p(n+1,0,a)=-(a+n)p(n,0,a),\;p(n+1,n+1,a)=p(n,n,a),\]
\[p(n+1,i,a)=-(a+n)p(n,i,a)+p(n,i-1,a),\;(i=1,\ldots,n).\]
\end{proof}

The preceding equations are the recurrence relations for non-central Stirling numbers of the first kind  $s(n,i,a)$, \cite[p.\ 316]{char}.
In  what follows we shall denote $p(n,i,a)$ by
$s(n,i,a).$

It is easy to see that the following equations hold
\[s(n,0,a)=(-a)_n,\;(n=0,1,2,\ldots),\]
and
\[s(n,n,a)=1,\;(n=0,1,2,\ldots).\]
By Leibnitz rule we get
\begin{equation}\label{e3}\frac{d^n}{dx^n}f(x)=\sum_{k=0}^n{n\choose k}\frac{d^k}{dx^k}x^{-a}\frac{d^{n-k}}{dx^{n-k}}\ln^b x.\end{equation}
From the well-known formulas
\[\frac{d^k}{dx^k}x^{-a}=(-a)_kx^{a-k},\]
and
\[\frac{d^{n-k}}{dx^{n-k}}\ln^b x=x^{-n+k}\sum_{i=1}^{n-k}s(n-k,i)(b)_i\ln^{b-i}x,\] by comparing (\ref{e1}) and (\ref{e3})
we obtain the following:
\begin{proposition}\label{t2} Let $a$ be a real number, and let $n,$  $i,\;(i\leq n)$ be nonnegative integers. Then
\begin{equation}\label{e44}s(n,i,a)=\sum_{k=0}^{n-i}{n\choose k}(-a)_ks(n-k,i).\end{equation}
\end{proposition}
\begin{remark} Proposition \ref{t2} is true in the case $a=0$ with the convention that $(0)_0=1.$ \end{remark}
Taking $i=1$ in (\ref{e44}) we have the following:
\begin{proposition}\label{t4}
Let $a$ be a real number, and $n$ be a  positive  integer. Then
\[s(n,1,a)=n!\sum_{k=0}^{n-1}(-1)^{n-k-1}\frac{{-a\choose k}}{n-k}.\]
\end{proposition}
For $s(n,1,a)$ we have the following recurrence relation:
\begin{equation}\label{e5} s(1,1,a)=1,\;
s(n,1,a)=(-a-n+1)s(n-1,1,a)+(-a)_{n-1},\;(n\geq 2).\end{equation}
We shall now prove that polynomials $r(n,a),\;(n=1,2,\ldots)$ defined by
\[r(n,a)=\sum_{k=0}^{n-1}(k+1)s(n,k+1)(-a)^k\]
 satisfy (\ref{e5}). For $n=1$ this is obviously true.

 Using the two terms recurrence relation for  Stirling numbers of the first kind, for $n&gt;1$  we have
\[r(n,a)=\sum_{k=0}^{n-1}(k+1)s(n-1,k)(-a)^k-(n-1)\sum_{k=0}^{n-2}(k+1)s(n-1,k+1)(-a)^k.\]
Since $s(n-1,0)=0,$ by
replacing $k+1$ instead of $k$ in the first sum on the right side  we obtain
\[r(n,a)=(-a-n+1)r(n-1,a)+\sum_{k=0}^{n-2}s(n-1,k+1)(-a)^{k+1}.\]
Furthermore, a well known property of Stirling numbers of the first kind implies
\[ \sum_{k=0}^{n-2}s(n-1,k+1)(-a)^{k+1}=(-a)_{n-1},\]
which means that $r(n,a)$ satisfies (\ref{e5}).
We have proved the following:
\begin{proposition}\label{t5} Let $a$ be a real number, and let  $n\geq 1$ be an integer. Then
\begin{equation}\label{e6}n!\sum_{k=0}^{n-1}(-1)^{k}\frac{{-a\choose k}}{n-k}=\sum_{k=0}^{n-1}(k+1)\mathbf s(n,k+1)a^k.\end{equation}
\end{proposition}
\begin{remark} Proposition \ref{t5} is true for $a=0$ with the convention that  $0^0=1.$\end{remark}


In the  case that $a$ is a negative integer and  $n\leq -a,$ the identity (\ref{e6}) is  related to the harmonic numbers.
\begin{proposition}\label{t6}
Define  $h(n,m)$ such that
\[h(n,m)=(H_{m}-H_{m-n})\frac{m!}{(m-n)!},\;(m=1,2,\ldots;n=1,2,\ldots,m).\]
Then  $h(n,m)$ satisfies (\ref{e5}).
\end{proposition}
\begin{proof}
The proof goes by induction with respect to  $n.$ For $n=1$ we have
\[h(1,m)=\frac{(m)!}{(m-1)!}(H_{m}-H_{m-1})=1.\]
Furthermore, for $n&gt;1$  we have
\[(m-n+1)h(n-1,m)+(m)_{n-1}=\frac{(m)!}{(m-n)!}(H_{m}-H_{m-n+1})+(m)_{n-1}=\]\[=\frac{(m)!}{(m-n)!}(H_{m}-H_{m-n}),\]
since $\frac{(m)!}{(m-n)!(m-n+1)}=(m)_{n-1}.$
It follows that
\[h(n,m)=(m-n+1)h(n-1,m)+(m)_{n-1},\] and the result is proved.
\end{proof}
As an immediate consequence of Proposition \ref{t6} we obtain
\begin{proposition} Let $m$ be a positive integer and let  $n,\;(1\leq n\leq m)$ be any integers. Then
\[H_m-H_{m-n}=\frac{(-1)^{n+1}}{{m\choose n}}\sum_{k=0}^{n-1}\frac{(-1)^k{m\choose k}}{n-k}.\]
\end{proposition}
\begin{remark} The results of this section are concerned with the following sequences in \cite{Slo}:
\seqnum{A001701},
\seqnum{A001702},
\seqnum{A001705},
\seqnum{A001706},
\seqnum{A001707},
\seqnum{A001708},
\seqnum{A001709},
\seqnum{A001711},
\seqnum{A001712},
\seqnum{A001713},
\seqnum{A001716},
\seqnum{A001717},
\seqnum{A001718},
\seqnum{A001722},
\seqnum{A001723},
\seqnum{A001724},
\seqnum{A049444},
\seqnum{A049458},
\seqnum{A049459},
\seqnum{A049600},
\seqnum{A051338},
\seqnum{A051339},
\seqnum{A051379},
\seqnum{A051523},
\seqnum{A051524},
\seqnum{A051525},
\seqnum{A051545},
\seqnum{A051546},
\seqnum{A051560},
\seqnum{A051561},
\seqnum{A051562},
\seqnum{A051563},
\seqnum{A051564},
\seqnum{A051565}.
\end{remark}
\section{Generalized factorial coefficients}
The first result in this section is a closed formula for the
$n$th derivative of the function $f(x^a),$ where $f\in
C^\infty(0,+\infty),$ and $a$ is a real number. Such one formula may be obtained as a particular case of  Fa\'{a} di Bruno's formula. We obtain here
the formula which is easily proved by induction. In addition, we obtain a
recurrence relation for coefficients.
\begin{theorem} Let $n&gt;0$ be  an integer, and let $a$ be a real number. Then
\begin{equation}\label{xnas}\frac{d^n}{dx^n}f(x^a)=\sum_{k=1}^nq(n,k,a)x^{ak-n}\frac{d^k}{dx^k}f(t),\end{equation}
where $t=x^a,$ and $q(n,k,a)$ is a polynomials of $a$ with integer coefficients. The degree of $q(n,k,a)$ is  $n$, and  it does not depend on $f.$
\end{theorem}
\begin{proof}
The result is true for $n=1$ if we take $q(1,1,a)=a.$
Assume that the result is true for $n\geq 1.$
Taking derivative in (\ref{xnas}) we obtain
\[\frac{d^{n+1}}{dx^{n+1}}f(x^a)=\sum_{k=1}^n(ka-n)q(n,k,a)x^{ka-n-1}\frac{d^k}{dx^k}f(t)+\]\[+
a\sum_{k=1}^nq(n,k,a) x^{ka-n+a-1}\frac{d^{k+1}}{dx^{k+1}}f(t)=\]\[=
\sum_{k=2}^n\big[(ka-n)q(n,k,a)+a q(n,k-1,a)]x^{ka-n-1}\frac{d^k}{dx^k}f(t)+\]\[+
(a-n)q(n,1,a)x^{a-n-1}\frac{d}{dx}f(t)+a q(n,n,a)x^{(n+1)(a-1)}\frac{d^{n+1}}{dx^{n+1}}f(t).\]
Define
\[q(n,0,a)=0,\;q(n,k,a)=0,\;(k&gt;n),\] and
\begin{equation}\label{e4}q(n+1,k,a)=(ka-n)q(n,k,a)+a q(n,k-1,a),\;(k=1,\ldots,n+1),\end{equation}
to obtain
\[x^{n+1}\frac{d^{n+1}}{dx^{n+1}}f(x^a)=\sum_{k=1}^{n+1}q(n+1,k,a)t^k\frac{d^k}{dx^k}f(t),\]
and  the result is proved.
\end{proof}
If, additionally, we define $q(0,0,a)=1$ then the formula (\ref{xnas}) may be written in the form
\begin{equation}\label{ee4}\frac{d^n}{dx^n}f(x^a)=\sum_{k=0}^nq(n,k,a)\frac{d^k}{dx^k}f(t)x^{ak-n},\;(n=0,1,\ldots).\end{equation}
The equations (\ref{e4})  shows that the  polynomials $q(n,k,a)$ are in fact the generalized factorial coefficients  $C(n,k,a)$ (\cite[p.\ 309]{char}).
\begin{proposition} Generalized factorial coefficients satisfy the following equations:
\[C(n,1,a)=(a)_n,\;C(n,n,a)=a^n,\;C(n,k,1)=0,\;(k&lt;n)(n=1,2,\ldots).\]
\end{proposition}
\begin{proof}
For the first equation it is enough to take $f(t)\equiv t$ in
(\ref{ee4}).

The second equation follows immediately from (\ref{e4}).

If $a=1$ then (\ref{ee4}) takes the form
\[x^n\frac{d^n}{dx^n}f(x)=\sum_{k=0}^nC(n,k,1)x^k\frac{d^k}{dx^k}f(t).\]
and since $f$ is arbitrary function  the third equation is also true.
\end{proof}

The generalized factorial coefficients are related with Hermite and Bessel polynomials.

Taking $f(t)=e^{bt}$  in (\ref{ee4}) we obtain
\begin{equation}\label{abx}\frac{d^n}{dx^n}e^{bx^a}=e^{bx^a}\sum_{k=1}^nC(n,k,a)b^kx^{ak-n}.\end{equation}
\begin{proposition}\label{t12}
Let $n$ and $m$  be positive integers. Then
 \begin{equation}\label{uhp}\frac{d^n}{dx^n}e^{-x^m}=e^{-x^m}\sum_{k=\lceil\frac{n}{m}\rceil}^n(-1)^kC(n,k,m)x^{mk-n}.\end{equation}
\end{proposition}
 \begin{proof} Take $b=-1$ in (\ref{abx}), hence
 \[\frac{d^n}{dx^n}e^{-x^m}=e^{-x^m}\sum_{k=0}^nC(n,k,m)x^{mk-n},\;(n\geq 0).\]
It is clear that taking derivatives on the left-hand  side of this
equation can not produce negative powers of $x.$ This means that
$C(n,k,m)=0$ if $km-n&lt;0,$ and Proposition \ref{t12} is proved.
\end{proof}
\begin{remark} The equation (\ref{uhp}) defines generalized Hermite polynomials, \cite{Ber}.\end{remark}
\begin{proposition} If $H_n(x),\;n=1,\ldots$ are Hermite polynomials then
\begin{equation}\label{her}H_n(x)=\sum_{k=\lceil\frac{n}{2}\rceil}^n(-1)^{n+k}C(n,k,2)x^{2k-n}.\end{equation}
\end{proposition}
It is easy to check that functions $f(n,k),\;(n=1,\ldots;\;k=1,\ldots,n)$ defined by
\[f(n,k)=\frac{(-1)^{n-k}(2n-k-1)!}{2^{2n-k}(n-k)!(k-1)!}\]
fulfill the recurrence relation (\ref{e4}) for $a=\frac 12.$
We thus obtain the following:
\begin{proposition}
Bessel polynomials $p_n(x)$ satisfy the following equation:
\begin{equation}\label{bes}p_n(x)=2^n\sum_{k=1}^n(-1)^{n-k}C\bigg(n,k,\frac 12\bigg)x^k.\end{equation}
\end{proposition}
The next result shows that generalized factorial coefficients are coefficients in the expansion of falling factorials of $b$ in terms
of falling factorials of $a,$ where $a$ and $b$ are arbitrary real  numbers.
\begin{proposition} Let $n$ be a nonnegative integer, and   let $a,b$  be arbitrary real numbers. Then
\begin{equation}\label{bnan}(b)_n=\sum_{k=0}^nC\bigg(n,k,\frac ba\bigg)(a)_k.\end{equation}
\end{proposition}
\begin{proof}
Replacing $a$ by $\frac{b}{a}$  in (\ref{ee4}) we have
\[x^n\frac{d^n}{dx^n}f(x^{\frac ba})=\sum_{k=1}^nC\bigg(n,k,\frac{b}{a}\bigg)t^k \frac{d^k}{dx^k}f(t),\]
where $t=x^{\frac ba}.$

Choosing $f(t)=t^a$ implies $f(x^{\frac{b}{a}})=x^b,$ hence
\[x^n\frac{d^n}{dx^n}x^{b}=\sum_{k=1}^nC\bigg(n,k,\frac{b}{a}\bigg)t^k \frac{d^k}{dx^k}t^a,\] that is,
\[(b)_{n}x^{b}=\sum_{k=1}^nC\bigg(n,k,\frac ba\bigg)(a)_kt^{a}.\]
Since $x^b=t^a$ the result follows.
\end{proof}
\begin{remark}The equation (\ref{bnan}) serves as the definition of generalized
factorial coefficients in \cite[Definition 8.2]{char}.
\end{remark}
Choosing $b=-a$ implies $(-a)_n=(-1)^na(a+1)\cdots(a+n-1).$ We thus obtain the expression in which  rising factorials
 are given in terms of falling factorials.
\begin{proposition} Let $a$ be a real number. Then
\[a(a+1)\cdots(a+n-1)=(-1)^n\sum_{k=1}^nC(n,k,-1)(a)_k.\]
\end{proposition}
\begin{remark}
The preceding equation means that $C(n,k,-1)$ are Lah
numbers.
\end{remark}
From the equation (\ref{bnan}) we shall derive some properties of coefficients of Hermite and Bessel polynomials.
Denote by $b(n,k)$ the coefficient by $x^k$ in the expansion of
$P_n(x)$ in
 (\ref{bes}). Then
 \[C\bigg(n,k,\frac 12\bigg)=(-1)^{n-k}2^{-n}b(n,k),\;(n=1,2,\ldots,\;k=1,2,\ldots,n).\]
 Next, denote by $h(n,k)$ the coefficient by $x^k$ in the expansion of $H_n(x)$ in (\ref{her}).  It follows that
 \[C(n,k,2)=(-1)^{n+k}h(n,2k-n),\] where $h(n,2k-n)=0$ if $2k-n&lt;0.$ We have thus proved the following:
\begin{proposition} Let $a$ be a real number, and let $n$ be a positive integer. Then the following equations hold
\[(2a)_n=\sum_{k=1}^n(-1)^{n+k}h(n,2k-n)(a)_k,\]
and
\[(a)_n=\sum_{k=1}^n(-1)^{n-k}2^{-n}b(n,k)(2a)_k.\]
\end{proposition}
The following proposition gives a known property of
generalized factorial coefficients, \cite[Theorem 8.18]{char}.
\begin{proposition}\label{t20}
Let $n\geq k$ be integers, and let  $a,\;b$ be real numbers. Then
\begin{equation}\label{konv}C(n,k,a_1a_2)=\sum_{j=k}^nC(n,j,a_2)C(j,k,a_1).\end{equation}
\end{proposition}
\begin{proof}
Take $f_1(t)=t^{a_1}$ and $f_2(t)=t^{a_2},$  hence  \[f(x^{a_1a_2})=(f\circ f_1)(x^{a_2}).\]
 Firstly, it follows from (\ref{xnas}) that
\begin{equation}\label{uk1}x^n\frac{d^n}{dx^n}f(x^{a_1a_2})=\sum_{k=1}^nC(n,k,a_1a_2)x^{a_1a_2k}\frac{d^k}{dx^k}f(t),\;(t=x^{a_1a_2}).\end{equation}
On the other hand,  (\ref{xnas}) also implies
\[x^n\frac{d^n}{dx^n}(f\circ f_1)(x^{a_2})=\sum_{j=1}^nC(n,j,a_2)x^{a_2j}\frac{d^j}{dx^j}(f\circ f_1)(u),\;(u=x^{a_2}).\]
Applying (\ref{xnas}) once more   yields
\[x^n\frac{d^n}{dx^n}(f\circ f_1)(x^{a_2})=\sum_{j=1}^n\sum_{k=1}^jC(n,j,a_2)C(j,k,a_1)x^{a_2j}u^{-j}v^k\frac{d^k}{dx^k}f(v),\;(v=u^{a_1}).\]
Changing the order of summation and taking into account that $v=u^{a_2}=x^{a_1a_2}=t$ we obtain
\[x^n\frac{d^n}{dx^n}(f\circ f_1)(x^{a_2})=\sum_{k=1}^n\left(\sum_{j=k}^nC(n,j,a_2)C(j,k,a_1)\right)x^{a_1a_2k}\frac{d^k}{dx^k}f(t).\]
Comparing (\ref{uk1}) and the preceding  equation shows that Proposition \ref{t20} is true.
\end{proof}
From Proposition \ref{t20} we derive an orthogonality  relation between coefficients of Hermite and Bessel
polynomials.
 \begin{proposition}If $h(n,k)$ and $b(n,k)$ are the coefficients of Hermite and Bessel polynomials respectively,  then
\[\sum_{k=1}^{n}b(n,k)h(n,2k-n)=0.\]
\end{proposition}
\begin{proof}
Since $C(n,k,1)=0$ for $k&lt;n$ the result follows from (\ref{her}) and (\ref{bes}).
\end{proof}
\begin{remark} The results of this section are related to the following sequences in \cite{Slo}:
\seqnum{A000369},
\seqnum{A001497},
\seqnum{A001801},
\seqnum{A004747},
\seqnum{A008297},
\seqnum{A013988},
\seqnum{A035342},
\seqnum{A035469},
\seqnum{A049029},
\seqnum{A049385},
\seqnum{A059343},
\seqnum{A092082},
\seqnum{A105278},
\seqnum{A111596},
\seqnum{A122850},
\seqnum{A132056},
\seqnum{A132062},
\seqnum{A136656}.
\end{remark}

\section{Gould-Hopper numbers}
In the first result of this section we prove  that Gould-Hopper numbers are coefficients in the expansion of the $n$th derivative of $x^af(x^b),$
 where $a,b$ are arbitrary real numbers, and $f\in C^\infty(0,+\infty)$ is arbitrary function.
\begin{theorem}\label{t24}
Let $n$ be a positive integer, and let $a,b$ be real numbers. Then
\begin{equation}\label{gh}\frac{d^n}{dx^n}[x^af(x^b)]=x^{a-n}\sum_{k=0}^np(n,k,b,a)x^{bk}\frac{d^k}{dx^k}f(t),\end{equation}
where $t=x^b,$ and $p(n,k,b,a)$ are polynomials of $a$ and $b$ with integer coefficients, which do not depend on $f.$
\end{theorem}
\begin{proof}
Using Leibnitz rule and (\ref{xnas}) we easily obtain
\[\frac{d^n}{dx^n}[x^af(x^b)]=
x^{a-n}\left[(a)_nf(x^b)+\sum_{j=1}^n\sum_{k=1}^j{n\choose j}C(j,k,b)\frac{d^k}{dx^k}f(t)(a)_{n-j}x^{bk}\right].\]
Changing the order of summation implies
\[\frac{d^n}{dx^n}[x^af(x^b)]=x^{a-n}\left[(a)_nf(x^b)+\sum_{k=1}^n\left[\sum_{j=k}^n{n\choose j}C(j,k,b)(a)_{n-j}\right]\frac{d^k}{dx^k}f(t)x^{bk}\right].\]
Theorem \ref{t24}  is true if we define
\[p(n,0,b,a)=(a)_n,\;p(n,k,b,a)=\sum_{j=k}^n{n\choose j}C(j,k,b)(a)_{n-j},\;(k=1,\ldots,n).\]
\end{proof}
\begin{remark}
According to \cite[p.\ 318]{char} we see that
$p(n,k,b,a)$ are Gould-Hopper numbers or non-central
 generalized factorial coefficients and will be dented by $C(n,k;b,a).$
\end{remark}
Gould-Hopper numbers generalize coefficients of associated Laguerre polynomials $L^k_n(x),$ \cite[p.\ 726]{arh}.

Namely, $L^k_n(x)$ are defined to be
\[L^k_n(x)=\frac{e^xx^{-k}}{n!}\frac{d^n}{dx^n}(e^{-x}x^{n+k}),\;L^0_n(x)=L_n(x).\]

Take $a=n+k, b=1, f(x)=e^{-x}$ in (\ref{gh}) to obtain
\begin{proposition} Let $L^k_n(x),\;(n=0,1,\ldots,\;k=0,1,\ldots)$ be associated  Laguerre polynomials. Then
\[L^k_n(x)=\frac{1}{n!}\sum_{i=0}^n(-1)^iC(n,i,1,n+k)x^i.\]
\end{proposition}
\begin{proposition} Let $n$ be a nonnegative integer, and let $a,b,c$ be nonzero real numbers. Then
\[(a+bc)_n=\sum_{k=0}^nC(n,k;b,a)(c)_k.\]
\end{proposition}
\begin{proof}
Take $f(t)=t^{c}$ in (\ref{gh}) to obtain $x^af(x^b)=x^{a+bc},$ and the resul follows.
\end{proof}
\begin{remark}
The equation from the preceding proposition serves as the definition of Gould-Hopper numbers in \cite[p.\ 317]{char}.
\end{remark}
The following result shows that  Gould-Hopper numbers, with a suitable chosen sign,  are coefficients in the expression of falling factorial of $a$ in terms rising
factorial of $b.$
\begin{proposition} Let $n$ be a positive integer, and let $a,b$ be nonzero  real numbers. Then
\[(a)_n=\sum_{k=1}^n(-1)^{k-1}C\big(n,k;\frac ab,a\big)\cdot b\cdot(b+1)\cdots(b+k-1).\]
\end{proposition}
\begin{proof}
Take $f(t)=t^{-\frac ac},$ where $c\not=0.$ Then $x^af(x^c)=1,$ hence $\frac{d^n}{dx^n}(x^af(x^c))=0,\;(n&gt;0).$
Applying (\ref{gh}) we obtain
\[\sum_{k=0}^nC\bigg(n,k,\frac ab,a\bigg)(-b)_k=0,\] where $b=\frac ac,$  and the result holds.
\end{proof}
The next result is an explicit formula  for  $C(n,k;b,a)$ in terms of generalized factorial coefficients.
\begin{proposition}\label{t30}
Let $m\leq n$ be  nonnegative integers,  and let $a,b$ be  nonzero real numbers. Then
\[C(n,m;b,a)=\sum_{k=m}^nC\bigg(k,m,\frac ba\bigg)\bigg[C(n,k,a)+(k+1)C(n,k+1,a)\bigg].\]
\end{proposition}
\begin{proof}
Let us choose $f_1(t)=t,\;f_2(t)=f(t^{\frac ba}),$ where $f$ is arbitrary function. Then \[f_1(x^a)f_2(x^a)=x^af(x^b).\]
Using (\ref{ee4}) and  Leibnitz rule we obtain
\[\frac{d^n}{dx^n}[x^af(x^b)]=x^{-n}\sum_{j=0}^n\sum_{k=0}^jC(n,k,a){m\choose j}\frac{d^j}{dt^j}t\frac{d^{k-j}}{dt^{k-j}}\big[f(t^{\frac ba})\big]x^{ak},\]
where $t=x^a.$ On the right side of this equation only terms obtained for $j=0$ and $j=1$ remain.
It follows that
\[\frac{d^n}{dx^n}[x^af(x^b)]=x^{-n}
\sum_{k=0}^nC(n,k,a)\frac{d^k}{dt^k}\big[f(t^{\frac ba})\big]x^{(k+1)a}+\sum_{k=1}^nkC(n,k,a)\frac{d^{k-1}}{dt^{k-1}}\big[f(t^{\frac ba})\big]x^{ak}=\]
\[=\sum_{k=0}^n[C(n,k,a)+(k+1)C(n,k+1,a)]\frac{d^k}{dt^k}\big[f(t^{\frac ba})\big]x^{(k+1)a}.\]
According to (\ref{ee4}) we have
\[\frac{d^n}{dx^n}[x^af(x^b)]=x^{a-n}\sum_{k=0}^n\sum_{m=0}^kC\bigg(k,m,\frac ba\bigg)\big[C(n,k,a)+(k+1)C(n,k+1,a)\big]f^{(m)}(u)x^{bm},\]
where $u=t^{\frac ba}=x^b.$

Interchanging the order of summation gives
\[\left[x^af(x^b)\right]^{(n)}=x^{a-n}\sum_{m=0}^n\left[\sum_{k=m}^nC\bigg(k,m,\frac ba\bigg)\big[C(n,k,a)+(k+1)C(n,k+1,a)\big]\right]f^{(m)}(t)x^{bm}.\]
Comparing this equation with (\ref{gh}) implies
\[C(n,m,b,a)=\sum_{k=m}^nC\bigg(k,m,\frac ba\bigg)\big[C(n,k,a)+(k+1)C(n,k+1,a)\big],\;(m=0,1,\ldots,n)],\]
 and  Proposition \ref{t30} is proved.
\end{proof}
We finish with a  result connecting  Gould-Hopper numbers, Stirling numbers of the first kind, powers, binomial coefficients, and falling factorials.
\begin{proposition} Let $j\leq n$ be  nonnegative integers  and let $a,b$ be nonzero real numbers. Then
\[\sum_{k=j}^n C(n,k,b,a)s(k,j)=b^{j}\sum_{k=j}^n{n\choose k}(a)_{n-k}s(k,j).\]
\end{proposition}
\begin{proof}
Take $f(t)=\ln^c t,$ where $c$ is a real number such that $(c)_i\not=0,\;(i=1,2,\ldots).$  It follows that  $x^af(x^b)=x^ab^c\ln^c x.$ From (\ref{e1}) and (\ref{e44}) we
  conclude that
\[b^c\frac{d^n}{dx^n}(x^a\ln^c x)=b^c(a)_nx^{a-n}\ln x+bx^{a-n}\sum_{k=1}^n\sum_{j=1}^k{n\choose k}(a)_{n-k}s(k,j)(c)_j\ln^{c-j}x.\]
Using (\ref{gh}) yields
\[b^c\left[x^a\ln^c x\right]^{(n)}=b^c(a)_nx^{a-n}\ln^cx+x^{a-n}b^c\sum_{k=1}^n\sum_{j=1}^kC(n,k,b,a)s(k,j)b^{-j}(c)_j\ln^{c-j}x .\]
Changing the order of summation in both sums leads to  the following equation:
  \[\sum_{j=1}^n\left[\sum_{k=j}^n{n\choose k}(a)_{n-k}s(k,j)\right](c)_j\ln^{c-j}x=\]\[=\sum_{j=1}^n\left[\sum_{k=j}^nC(n,k,b,a)s(k,j)b^{-j}\right](c)_j\ln^{c-j}x.\]
Comparing terms by the same $\ln^{c-j} x$, and then dividing by $(c)_j\not=0$ proves the result.
\end{proof}
\begin{remark}
The results of this section are concerned with the following sequences in \cite{Slo}:
\seqnum{A000522},
\seqnum{A021009},
\seqnum{A035342},
\seqnum{A035469},
\seqnum{A049029},
\seqnum{A049385},
\seqnum{A072019},
\seqnum{A072020},
\seqnum{A084358},
\seqnum{A092082},
\seqnum{A094587},
\seqnum{A105278},
\seqnum{A111596},
\seqnum{A132013},
\seqnum{A132014},
\seqnum{A132056},
\seqnum{A132159},
\seqnum{A132681},
\seqnum{A132710},
\seqnum{A132792},
\seqnum{A136215},
\seqnum{A136656}.
\end{remark}


\begin{thebibliography}{9999}
\bibitem{arh}
G. Arfken,  {\it Laguerre Functions, Mathematical Methods for Physicists},
3rd ed., Academic Press, 1985, 721--731.

\bibitem{bell}
E. T. Bell, Exponential polynomials, {\it Ann. Math.} {\bf 35}
(1934), 258--277.

\bibitem{Ber}
A. Bernardini and P. E. Ricci, Bell polynomials and differential equations of Freud-type polynomials,
{\it Math. Comput. Modelling,} {\bf 36} (2002), 1115--1119.

\bibitem{char}
Ch. A. Charalambides, {\it Enumerative Combinatorics,}
Chapman \&amp; Hall/CRC, 2002.

\bibitem{nat}
P. Natalini and P. E. Ricci, Bell polynomials and some of their applications,
{\it Cubo Mat. Educ.,} {\bf 5} (2003), 263--274.

\bibitem{Slo} N. J. Sloane,  The Encyclopedia of Integer Sequences,
published electronically at
\href{http://www.research.att.com/$\sim$njas/sequences/}{\tt
http://www.research.att.com/$\sim$njas/sequences/}

\bibitem{xu}
A. Xu and C. Wang, On the divided difference form of Fa\'{a} di Bruno's
formula, {\it J.  Comput. Math.} {\bf 25} (2007), 697--704.

\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2000 {\it Mathematics Subject Classification}: Primary
05A10; Secondary 11C08.

\noindent \emph{Keywords: } non-central Stirling numbers of the first kind, generalized factorial coefficients, Gould-Hopper numbers, generalized Stirling numbers.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A000369},
\seqnum{A000522},
\seqnum{A001497},
\seqnum{A001701},
\seqnum{A001702},
\seqnum{A001705},
\seqnum{A001706},
\seqnum{A001707},
\seqnum{A001708},
\seqnum{A001709},
\seqnum{A001711},
\seqnum{A001712},
\seqnum{A001713},
\seqnum{A001716},
\seqnum{A001717},
\seqnum{A001718},
\seqnum{A001722},
\seqnum{A001723},
\seqnum{A001724},
\seqnum{A001801},
\seqnum{A004747},
\seqnum{A008297},
\seqnum{A013988},
\seqnum{A021009},
\seqnum{A035342},
\seqnum{A035469},
\seqnum{A049029},
\seqnum{A049385},
\seqnum{A049444},
\seqnum{A049458},
\seqnum{A049459},
\seqnum{A049600},
\seqnum{A051338},
\seqnum{A051339},
\seqnum{A051379},
\seqnum{A051523},
\seqnum{A051524},
\seqnum{A051525},
\seqnum{A051545},
\seqnum{A051546},
\seqnum{A051560},
\seqnum{A051561},
\seqnum{A051562},
\seqnum{A051563},
\seqnum{A051564},
\seqnum{A051565},
\seqnum{A059343},
\seqnum{A072019},
\seqnum{A072020},
\seqnum{A084358},
\seqnum{A092082},
\seqnum{A094587},
\seqnum{A105278},
\seqnum{A111596},
\seqnum{A122850},
\seqnum{A132013},
\seqnum{A132014},
\seqnum{A132056},
\seqnum{A132062},
\seqnum{A132159},
\seqnum{A132681},
\seqnum{A132710},
\seqnum{A132792},
\seqnum{A136215},
\seqnum{A136656}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received  August 4 2009;
revised version received November 19 2009.
Published in {\it Journal of Integer Sequences}, November 25 2009.
Minor correction, January 29 2010.

\bigskip
\hrule
\bigskip

\noindent
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