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\markboth{\hfil Symbolic computation of Appell polynomials using Maple \hfil}%
{\hfil H. Alkahby, G. Ansong, P. Frempong-Mireku, \& A. Jalbout \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc 16th Conference on Applied Mathematics, Univ. of Central Oklahoma},
\newline
Electronic Journal of Differential Equations, Conf. 07, 2001, pp. 1--13.
\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\vspace{\bigskipamount} \\
%
Symbolic computation of Appell polynomials using Maple
%
\thanks{ {\em Mathematics Subject Classifications:} 26B30.
\hfil\break\indent
{\em Key words:} Appell polynomials, generating function, bounded variation.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Published July 20, 2001} }
\date{}
\author{ H. Alkahby, G. Ansong, P. Frempong-Mireku, \& A. Jalbout }
\maketitle
\begin{abstract}
This work focuses on the symbolic computation of Appell polynomials using the
computer algebra system Maple. After describing the traditional approach of
constructing Appell polynomials, the paper examines the operator method of
constructing the same Appell polynomials. The operator approach enables us to
express the Appell polynomial as Bessel function whose coefficients are Euler
and Bernuolli numbers. We have also constructed algorithms using Maple to
compute Appell polynomials based on the methods we have described. The
achievement is the construction of Appell polynomials for any function of
bounded variation.
\end{abstract}
\newtheorem{thm}{Theorem}
\section{Introduction}
The aim of this work is to give a new method of constructing and studying the
properties of a definite class of polynomials called Appell polynomials.
This sequence of polynomials $P_n(x)$, of degree $n$, is defined by the
recurrence relation
\begin{equation}
\frac{d}{dx}P_n(x)=P_{n-1}(x), \label{cc}
\end{equation}
or equivalently,
\[
\exists A(t)=\sum_{n=0}^{\infty}a_{n}t^{n}, (a_0\neq 0):
A(t)e^{tx}=\sum_{n=0}^{\infty}P_n(x)t^{n}.
\]
Each $A(t)$ is called a generating function and will generate a set of Appell
polynomials. In the interval $[-1,1]$, even polynomials of the type defined by
(\ref {cc}) have two zero points at $-1$ and $1$. Two explicit methods of
construction of Appell polynomials may be obtained by \\
1.) \textbf{the traditional method} and by \\
2.) \textbf{the operator method}.
The explicit expression derived for $P_n(x)$ using the generating function $f(x)=({e^{kx}+e^{-kx}})/({e^{k}+e^{-k}})$, could be expressed in terms of
Bessel powers. The coefficients of this Appell polynomial are the product of Euler and Bernoulli numbers. The behavior of this Appell polynomials in the interval $(-1, 1)$ and outside the interval $(-1, 1)$ could also be studied. Hence, enabling one to obtain the zeros of the polynomial $P_n(x)$.
The importance of this construction is to provide us with a powerful tool for
solving boundary value problems in hydrodynamics and many other areas of
application. In particular, this model could be viewed as a fluid flow with
nodes at $-1$ and $1$. It also provides a solution to the linear differential
equation of the form:
\begin{equation} \label{e1.2}
\sum_{r=0}^n L_r(x)\frac{d^ry(x)}{dx^r}=\lambda y(x)
\end{equation}
where $L_r(x)$ is a polynomial in $x$ of degree $r$, $\lambda$ is a
parameter and under the condition that the generating function is
$A(t)=e^{Q(t)}$, where $Q(t)$ is a polynomial in $t$.
\section{Operators in a linear space}
Let $L=\{v,u,\dots\}$ be a given linear vector space of possibly infinite
dimension. Suppose that an algebra of operators $D, I, P, \dots$ is defined
in the vector space $L$. If the operator $D$ on the vector $v$ is given by
\[
Dv=0,
\]
where the vector $v\in L$, then the set of vectors $\{v\}$ is called the
zero-space of the operator $D$. The members of this space will be denoted by
$c$. Hence,
\[
Dc=0.
\]
Also, $c$ will be called the formal constant of the operator $D$. Next, let
us introduce the operator $I$ with the property
\begin{equation}
DI=1. \label{e1}
\end{equation}
From (\ref{e1.2}) it is clear that the operator $I$ is the right inverse for
the $D$ operator. If the operator $P$ is defined to be
\[
P=1-ID,
\]
then clearly, $P$ is the projection operator for the vector $v$ on the
zero-space of the operator $D$, since
\begin{equation}
D(Pv)=Dv-Dv=0. \label{e2}
\end{equation}
The relation given by (\ref{e1}) shows that
$Pv=c$.
The power of the operator $I$ will be defined in the usual form as
$I^n=II^{n-1}$, and $\;\; I^0=1$, from which one can obtain the relation
\begin{equation}
DI^n=I^{n-1}. \label{e3}
\end{equation}
The $n$th order of the power of the formal variable $x$ generated by the
operator $I$ with coefficients $c$ is the expression given by
\[
n!I^n c=cx^n.
\]
Clearly, the following expression holds
\[
Dx^nc=nx^{n-1}c.
\]
Now, consider the expression of the form
\[
v=c_0+Ic_1+\dots+I^nc_n,
\]
which is called a polynomial operator. The formula for computing the
coefficients is:
\[
c_i=PD^iv.
\]
In terms of the variable $x$, one can write
\[
v=c_0+c_1x +\dots+c_nx^n,
\]
which is a polynomial of order $n$ with coefficients $c_0, c_1,\dots,c_n$.
Thus, the formula for the coefficients could be written as
\[
c_i=\frac{1}{i!}PD^iv.
\]
It must be noted that the powers of this special kind of variable
generate a differential operator. Now, consider a linear vector space $L$,
the vectors of which consist of matrices with dimension $2x2$, and components
$u_{11}$, $u_{12}$, $u_{21}$, $u_{22}$, which are functions of one variable
$x$. Thus, if $u$ is such a vector then
\[
u=\left (
\begin{array}{cc}
u_{11}(x) & u_{12}(x) \\
u_{21}(x) & u_{22}(x)
\end{array}
\right ).
\]
The differential operator $D$ takes the form
\[
D=\left (
\begin{array}{cc}
0 & \frac{d}{dx} \\
\frac{d}{dx} & 0
\end{array}
\right ).
\]
It is a straight forward computation to obtain the formula
\[
D^n u =\left\{
\begin{array}{ll}
\left(
\begin{array}{cc}
\frac{d^{2i}u_{11}}{dx^{2i}} & \frac{d^{2i}u_{12}}{dx^{2i}} \\[3pt]
\frac{d^{2i}u_{21}}{dx^{2i}} & \frac{d^{2i}u_{22}}{dx^{2i}}
\end{array}
\right ),& n=2i \\[16pt]
\left(
\begin{array}{cc}
\frac{d^{2i+1}u_{11}}{dx^{2i+1}} & \frac{d^{2i+1}u_{12}}{dx^{2i+1}} \\[3pt]
\frac{d^{2i+1}u_{21}}{dx^{2i+1}} & \frac{d^{2i+1}u_{22}}{dx^{2i+1}}
\end{array}
\right ), &n=2i+1,
\end{array}
\right.
\]
and
\[
PD^n u =\left\{
\begin{array}{ll}
\left(
\begin{array}{cc}
\frac{d^{2i}u_{11}}{dx^{2i}}|_{x_{1}} & \frac{d^{2i}u_{12}}{dx^{2i}}|_{x_{1}}
\\[3pt]
\frac{d^{2i}u_{21}}{dx^{2i}}|_{x_{2}} & \frac{d^{2i}u_{22}}{dx^{2i}}|_{x_{2}}
\end{array}
\right ), &n=2i \\[16pt]
\left(
\begin{array}{cc}
\frac{d^{2i+1}u_{11}}{dx^{2i+1}}|_{x_{1}} & \frac{d^{2i+1}u_{12}}{dx^{2i+1}}
|_{x_{1}} \\[3pt]
\frac{d^{2i+1}u_{21}}{dx^{2i+1}}|_{x_{2}} & \frac{d^{2i+1}u_{22}}{dx^{2i+1}}
|_{x_{2}}
\end{array}
\right ), &n=2i+1 .
\end{array}
\right.
\]
To perform the expansion for an arbitrary function $u$ by the constructed
formal powers, it is necessary to use the form of the $n$-th derivative of
\[
u=\left (
\begin{array}{cc}
u_{11}(x) & u_{12}(x) \\
u_{21}(x) & u_{22}(x)
\end{array}
\right ).
\]
The integral operators $I_{1}$ and $I_{2}$ are defined by
$I_{1} =\int_{x_{1}=0}^x d\xi$ and $I_{2}=\int_{x_{2}=1}^x d\xi$.
The following notation will be used:
\[
y^{2i}(x)=y^{2i}*1=(2i)!(I_1I_2)^i*1,
\]
and
\[
y^{2i+1}(x)=y^{2i+1}*1=(2i+1)!I_2(I_1I_2)^i*1.
\]
Similarly, the expression $(I_2I_1)^i*1$ and $I_1(I_2I_1)^i*1$ are denoted
by
\[
\hat{y}^{2i}(x)=\hat{y}^{2i}*1=(I_2I_1)^i*1,
\]
and
\[
\hat{y}^{2i+1}(x)=\hat{y}^{2i+1}*1=I_1(I_2I_1)^i*1
\]
respectively. The expressions $y^{n}$ and $\hat{y}^{n}$ are called the
general Bessel powers. The computation of the Bessel's powers to the order
$n=16$ for $x_1=0$ and $x_2=1$ will be shown later.
The integral takes the following form
\[
I=\left (
\begin{array}{cc}
0 & \int_{x_1}^x d\xi \\
\int_{x_2}^x d\xi & 0
\end{array}
\right )\; =\left (
\begin{array}{cc}
0 & I_1 \\
I_2 & 0
\end{array}
\right ),
\]
where the lower limit of the integrals can differ. Without loss of
generalityone can take $x_1=0$, and $x_2=1$
For the projection operator, it has the form
\[
P=1-ID=\left(
\begin{array}{cc}
\dots|_{x_1=0} & 0 \\
0 & \dots|_{x_2=0}
\end{array}
\right ).
\]
The formal powers are also given by the expression
\[
x^nc =n!\left\{
\begin{array}{ll}
\left(
\begin{array}{cc}
(I_1I_2)^iC_{11} & (I_1I_2)^iC_{12} \\[3pt]
(I_2I_1)^iC_{21} & (I_2I_1)^iC_{22}
\end{array}
\right ), &n=2i \\[15pt]
\left(
\begin{array}{cc}
I_1(I_2I_1)^iC_{21} & I_1(I_2I_1)^iC_{22} \\[3pt]
I_2(I_1I_2)^iC_{11} & I_2(I_1I_2)^iC_{12}
\end{array}
\right ), &n=2i+1,
\end{array}
\right.
\]
where $C$ is the formal constant for the operator $D$ and
\[
C=\left (
\begin{array}{cc}
C_{11} & C_{12} \\
C_{21} & C_{22}
\end{array}
\right).
\]
\section{\textbf{\ A Class of Appell Polynomials}}
Let $P_n$ denote a polynomial of degree $n$ of a variable $x$.
Consider a class of polynomials which satisfy the conditions
\begin{equation}
\frac{dP_0}{dx}=0, \quad \frac{dP_n}{dx}=P_{n-1} \label{eq1}
\end{equation}
The class of polynomials defined below is called Appell polynomials. In fact, there are large numbers of varying forms of this kind of Appells polynomials. An arbitrary constant is
obtained each time (\ref{eq1}) is integrated sequentially.
For example,
\begin{eqnarray*}
y^1&=&\int dx = x+C_0 \\
y^2&=&2\int y^1 dx = 2\int(x+C_0)dx= x^2+2xC_0 +C_1 \\
y^3&=&3\int y^2dx=3\int(x^2+2xC_0+C_1)dx \\
&=& x^3+3x^2C_0+3xC_1+C_2 \\
&\vdots& \\
y^n&=&n\int y^{n-1}dx=x^n+nx^{n-1}C_0+\dots+C_{n-1}.
\end{eqnarray*}
The above can be represented in a compact form as an Abel-Goncarov
integral
\[
P_n(x)=\int_{\alpha_0}^x dx_1\int_{\alpha_1}^{x_1}dx_2
\int_{\alpha_2}^{x_2}dx_3\dots\int_{\alpha_{n-1}}^{x_{n-1}}dx_n,
\]
where
\[
\alpha_i =\left\{
\begin{array}{ll}
1, & i= \lambda \mathop{\rm mod}(\lambda+1), \\
0, & i\neq \lambda \mathop{\rm mod}(\lambda+1).
\end{array}
\right.
\]
for a regular monotonic polynomials on $[0, 1]$ with type numbers,
$\lambda$.
When defining Appell polynomials, it is important to add an additional condition, due to the constants which arise in the equations. This condition may cause the polynomial to have
zeros at definite points on the real-axis. Let all the polynomials of even
degree $P_{2i}$ have zeros at the points $-1,\;1$. Then by performing the
integration and sequentially determining the constants, one gets the
following sequence of equations
\[ \displaylines{
y^2=x^2-1, \cr
y^4=x^4-6x^2+5, \cr
y^6= x^6-15x^4+75x^2-61, \cr
y^8=x^8-28x^6+350x^4-1708x^2+1385, \cr
y^{10}=x^{10}-45x^8-1050x^6-12810x^4+62325x^2-50521.
}
\]
\section{Using Generating function}
When constructing Appell polynomials with two zero-points, the
method of generating functions may the used. It may be shown that there
exists a functional sequence, the differential of which gives the Appell
polynomials with two zero-points. Consider the function
\begin{equation}
f(x)=\frac{e^{kx}+e^{-kx}}{e^{k}+e^{-k}}. \label{eq2}
\end{equation}
The Taylor expansion to the degree of $k$ can be easily computed. The
coefficients $C_{2i}$, of the expansion are found by the formula
\begin{equation}
C_{2i}=\frac{d^{2i}}{dk^{2i}}\left(\frac{e^{kx}+e^{-kx}}{e^{k}+e^{-k}}%
\right) \Big|_{k=0}. \label{eq3}
\end{equation}
From the Taylor expansion, it is clear that $C_{2i}$ is a function of one
variable, say $x$. Hence, $f(x)$ can be written in terms of $C_{2i}$ as
\begin{equation}
f(x)=\frac{e^{kx}+e^{-kx}}{e^{k}+e^{-k}}=\sum_{i=0}^{\infty} C_{2i}(x)
k^{2i}. \label{eq4}
\end{equation}
It can also be demonstrated that $C_{2i}$ is related to the Bessel powers
\[
y^{2i}(x;-1,+1),
\]
with two zero-points. The connection is given by the relation
\begin{equation}
C_{2i}=\frac{1}{(2i)!} y^{2i}(x;-1,+1). \label{eq5}
\end{equation}
Thus, the expansion for $f(x)$ which is given by the sum
\begin{equation}
f(x)=\sum_{i=0}^{\infty}C_{2i}(x) k^{2i}, \label{eq6}
\end{equation}
satisfies the following properties
\begin{equation}
\frac{d^{2}}{dx^{2}}f(x)= k^2f(x). \label{eq7}
\end{equation}
Equation (\ref{eq7}) could be generalized to take the following form
\begin{equation}
\frac{d^{2i}}{dx^{2i}}f(x)= k^{2i}f(x). \label{eq8}
\end{equation}
The series given in (\ref{eq6}) is called an Appell polynomial
with two zero-points. Therefore, the set of polynomials obtained by the
traditional method in section $3$, converges in even powers $x^{2i}C.$
These polynomials are called Appell polynomials. Thus, a sequence of Appell polynomials,
if the values of the zero-points are related to $x_1=0$, and $x_2=1$, can
also be easily computed. An example of this kind of Appell polynomial is
shown below:
\[ \displaylines{
y^2=x^2-x, \cr
y^4=x^4-2x^3+x, \cr
y^6= x^6-3x^5+5x^3-3x, \cr
y^8=x^8-4x^7+14x^5-28x^2+17x, \cr
y^{10}=x^{10}-5\,x^{9}+30\,x^{7}-126\,x^{5}+255\,x^{3}-155\,x \cr
y^{12}=x^{12}-6\,x^{11}+55\,x^{9}-396\,x^{7}+1683\,x^{5}-3410\,x^{\ 3}+
2073\,x\,. \cr
}\]
Substitute the sum given in (\ref{eq6}) into (\ref{eq7}) to
obtain
\begin{equation}
\sum_{i=0}^{\infty}k^{2i}(C_{2i})''=
\sum_{i=0}^{\infty}k^{2i}(C_{2i}) \label{eq9}
\end{equation}
where $C_2''=1$ and $C_0''=0$. Generalizing
the obtained relation, one gets
\[
(2i)!C_{2i}^{''}=2i(2i-1)C_{2i-2}(2i-2)!,
\]
or
\[
{y^{2i}}^{''}=(2i-1)y^{2i-2}.
\]
An explicit expression for $y^{2i}$ in the form of a polynomial can be
obtained. For this, one uses the Leibnitz formula for finding the differential
of order $2i$ from a differentiable function. Thus, one obtains
\begin{equation}
y^{2i}(x;-1,1) =\left.\sum_{j=0}^{i}C(2i,2j)x^{2i-2j}\frac{d^{2j}}{dk^{2j}}
\left(\frac{1}{e^k+e^{-k}}\right) \right|_{k=0}.
\end{equation}
The expression
\begin{equation}
2\frac{d^{2j}}{dk^{2j}}(e^k+e^{-k})^{-1}=E_{2j} \label{eq10}
\end{equation}
where the $E_{2j}$ are the Euler numbers. Therefore, the formula simplifies to
\begin{equation}
y^{2i} (x;-1,1)=\sum_{j=0}^{i}x^{2i-2j}C(2i,2j)E_{2j}. \label{eq11}
\end{equation}
The Euler numbers could be computed from Eq. (\ref{eq10}). An example of
computing Euler numbers $E_{2j}$ for $2j=0,2,4,..$. by this method is
shown in Table 1.
\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
2j & $0$ & $2 $ & $4$ & $6$ & $8$ & $10$ & $12$ & $14$ \\ \hline
E & 1 & -1 & 5 & -61 & 1385 & -50521 & 270276 & 19936098 \\ \hline
\end{tabular}
\end{center}
\caption{Euler numbers}
\end{table}
\section{ Computation of Appell Polynomials Using Stieltjes Integral}
As pointed out by Thone [1] Appell polynomials can be given in terms of
Stieltjes integrals. This enables us to give a new characterization to Appell
polynomials that expands its applications to functional methods.
The definition of bounded variation is important to the construction.
\paragraph{Definition}
A real valued function $f$ is of bounded variation on a closed
interval $\lbrack a,b\rbrack $ if for any partition $\pi $,
$\sum_{j=1}^{n}\left| \Delta f_{j}\right| \leq M,$for $M>0$ and $\Delta
f_{j}=f(x_{j})-f(x_{j-1})$.
\begin{thm}
If $\alpha (x)$ is a function of bounded variation on $\lbrack a,b\rbrack $
\ and the integrals
\[
\mu _{n}=\int_{a}^{b}x^{n}d\alpha (x),\: n=0,1,2,\dots,\mu _{0}\neq 0
\]
all exist then $\exists \{\phi _{n}(x)\},n=0,1,\dots$, $\ \phi _{n}(x)$ is of
degree $n$ such that
\begin{equation}
\int_{a}^{b}\phi _{n}^{\lbrack r\rbrack }(x)d\alpha (x)=\delta
_{n}^{r}=\left\{
\begin{array}{ll}
0 & n\neq r\\
1 & n=r
\end{array}
\right. \label{eq12}
\end{equation}
\end{thm}
\paragraph{Sketch of Proof:} If
\begin{equation}
\phi _{n}(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots+a_{0} \label{eq13}
\end{equation}
Applying (\ref{eq12}) to (\ref{eq13}) yields a system of equation which has
a unique solution if $n!(n-1)!(n-2)!\dots..1\mu _{0}^{n+1}\neq 0$.
Since $u_0\neq 0$, the solution is as follows
\[
a_{n+1-r}=\frac{\left|
\begin{array}{cccccc}
\mu _{0} & 0 & 0 & \dots & 0 & 1 \\
\frac{\mu _{0}}{1!} & \mu _{0} & 0 & \dots & 0 & 0 \\
\frac{\mu _{2}}{2!} & \frac{\mu _{1}}{1!} & \mu _{0} & \dots & 0 & 0 \\
\vdots &\vdots & \vdots& & \vdots &\vdots \\
\frac{\mu _{r-1}}{(r-1)!!} & \frac{\mu _{r-2}}{(r-2)!!} & \frac{\mu _{r-3}}{%
(r-3)!!} & \dots & \frac{\mu _{0}}{1!} & 0
\end{array}
\right| }{(n+1-r)!\mu _{0}^{r}}
\]
This solution is used to find the Appel polynomials for any function of
bounded variation.
Some examples are demonstrated in the next section.
\section{\textbf{Symbolic Algebra and Appell Polynomials}}
Now, we demonstrate how a symbolic algebra system can aid us in the construction of Appell polynomials. This would enable us to generate Appell polynomials of any order say ($k$). The name of the algorithms are \emph{apPoly(n1,n2,n3)} and \emph{apPolyeb(n1,n2,n3)} where $n1$ is the degree of the polynomials, $(n2, n3)$ are the two zeros of the polynomial which could be $0,\;1$ or $-1,\;1$. The Maple function needed when constructing Appell polynomials by the traditional method is \emph{int.} The program to do this is given in appendix 1. The algorithm \emph{appelPoly(f,x,a,b,n),} where $(f,x)$ is a function of bounded variation and the variables $a$ and $b$ are the lower and upper limits of integration, and $n$ is the degree of the polynomials, computes the general Appell polynomials for a given function of bounded variation. The first example has the two zeros at $-1,\;1$ and the second has the two zeros at $0,\;1$.
\begin{verbatim}
for i to 4 do apPoly(2*i,-1,1) od;
\end{verbatim}
\vspace{-.5cm}
\[\displaylines{
y^{2}=x^{2}-1 \cr
y^{4}=x^{4}-6\,x^{2}+5 \cr
y^{6}=x^{6}-15\,x^{4}+75\,x^{2}-61 \cr
y^{8}=x^{8}-28\,x^{6}+350\,x^{4}-1708\,x^{2}+1385 \cr
y^{10}=x^{10}-45\,x^{8}+1050\,x^{6}-12810\,x^{4}+62325\,x^{2}-50521
}
\]
\begin{verbatim}
for i to 4 do apPoly(2*i,0,1) od;
\end{verbatim}
\vspace*{-0.5cm}
\[\displaylines{
y^{2}=x^{2}-x \cr
y^{4}=x^{4}-2\,x^{3}+x \cr
y^{6}=x^{6}-3\,x^{5}+5\,x^{3}-3\,x \cr
y^{8}=x^{8}-4\,x^{7}+14\,x^{5}-28\,x^{3}+17\,x \cr
y^{10}=x^{10}-5\,x^{9}+30\,x^{7}-126\,x^{5}+255\,x^{3}-155\,x
}
\]
The operator method began with the function given in (\ref{eq2}) whose
Taylor expansion in $k$ to the order $12$ is given by the Maple command
\begin{verbatim}
taylor(f(x), k=0, 12);
\end{verbatim}
$$\displaylines{
1+\left(\frac{x^2}{2}-{\frac {1}{2}}\right){k}^{2}+\left(\frac{x^24}{24}+{\frac {5x^2}{24}}-\frac{x^2}{4}\right){k}^{4}+\left ({\frac {5}{48}}
-{\frac {61}{720}}+{\frac {x^6}{720}}\,-\frac{x^4}{48}
\right ){k}^{6}\cr
+\left (-{\frac {61x^2}{1440}}+{\frac {277}{8064}}+{\frac {x^8}{40320}}
+{\frac {5x^4}{576}}-{\frac {x^6}{1440}}\right){k^8} \cr
+\left ({\frac {277x^2}{16128}}-{\frac {50521}{3628800}}
+{\frac {x^10}{3628800}}\,-{\frac {61x^4}{17280}}+{\frac {x^6}{3456}}
-{\frac {x^8}{80640}}\right){k^{10}}
}$$
All the coefficients could be extracted by the following Maple command:
\begin{verbatim}
for i to 5 do coeff(tt,k,2*i) od;
\end{verbatim}
\[\displaylines{
{\frac {x^{2}}{2}}-\frac{1}{2} \cr
{\frac {x^{4}}{24}}-{\frac {x^{2}}{4}}+{\frac {5}{24}} \cr
{{\frac {x^{6}}{720}} -{\frac {x^{4}}{48}}+\frac {5\,x^{2}}{48}}-{\frac
{61}{720}} \cr
{\frac {x^{8}}{40320}} -{\frac {x^{6}}{1440}}+ {\frac {5\,x^{4}}{576}}-{%
\frac {61\,x^{2}}{1440}}+{\frac {277}{8064}} \cr
{\frac {x^{10}}{3628800}}-{\frac {x^{8}}{80640}}+{\frac {x^{6}}{3456}}-{%
\frac {61\,x^{4}}{17280}} +{\frac {277\,x^{2}}{16128}}-{\frac
{50521}{3628800}}
}\]
Equation (\ref{eq10}) is used to build the symbolic computation algorithm
for computing Appell polynomials. The coefficients are expressed in terms
of Bernuolli and Euler numbers. The algorithm is given in appendix $2$.
The arguments of the function are the order of the polynomils and the two
end points. The following is an illustrative symbolic computation by the
algorithm, which computes 3 Appell polynomials.
\begin{verbatim}
for i from 2 by 2 to 6 do y^i:=apPolyed(i,-1,1) od;
\end{verbatim}
\[\displaylines{
y^2=x^2 - Eu(0) \cr
y^4=x^4C(4, 0) Eu(0) + x^2 C(4, 2) Eu(2) + C(4, 4) Eu(4) \cr
y^6 = x^6 C(6, 0) Eu(0) + x^4 C(6, 2) Eu(2) + x^2C(6, 4) Eu(4) + C(6, 6)
Eu(6) \cr
}
\]
Maple can be asked to simplify the coefficients. The command for
evaluating the coefficients of Appell polynomials of order $12$ is given
as:
\begin{verbatim}
for i from 2 by 2 to 16 do y^i:=Eval(subs(C=binomial, Eu=euler,
apPolyed(i,-1,1))); od;
\end{verbatim}
\[\displaylines{
y^2 := x^2 - 1 \cr
y^4 := x^4 - 6 x^2 + 5 \cr
y^6 := x^6 - 15 x^4 + 75 x^2 - 61 \cr
y^8 := x^8 - 28 x^6 + 350 x^4 - 1708 x^2 + 1385 \cr
y^{10} := x^{10} - 45 x^8 + 1050 x^6 - 12810 x^4 + 62325 x^2 - 50521 \cr
y^{12} := x^{12} - 66 x^{10} + 2475 x^8 - 56364 x^6 + 685575 x^4 - 3334386
x^2 + 2702765
}
\]
Using Theorem 1, a set of Appell polynomial and their graphs for the function
of bounded variation f(x)=x on [0,1]is shown below. The graphs of lower
degrees passes through the maximum and minimum points of the graphs of
higher degree. Hence, Appell polynomials always contain the optimization
function. One can easily calculate the optimization points by location the
intersection of any two successive points. This property of Appell polynomials
makes it natural candidate for optimization problems.
\[\displaylines{
f1 := -\frac{1}{2} + x \cr
f2 := \frac{1}{12} -\frac{1}{2} x + \frac{1}{2} x \cr
f3 := \frac{1}{12}x - \frac{1}{4} x^2 + \frac{1}{6} x^3 \cr
f4 := -\frac{1}{720} + \frac{1}{24}x^2 - \frac{1}{12}x^3 + \frac{1}{24}x^4 \cr
}
\]
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1.eps}
\end{center}
\caption{Appell Polynomial for $f(x) = x, [0, 1]$}
\end{figure}
\paragraph{Conclusion.}
We have demonstrated a new approach for the computation of Appell polynomials
using Maple. Both the elementary and operator methods basically yield the
same result. This fact confirms the correctness of the approach used in the
construction. The interesting part of the work is the
symbolic algebra system construction of Appell polynimials, which can be
used to generate a set of Appell polynomial for any function of bounded
variation. In the future, work will be done on using Appell polynomials to
solve linear differential equation of form defined by Equation two.
{\large Appendix 1}
\begin{verbatim}
This algorithm computes Appell polynomials
apPoly:=proc(n,n1,n2) local a1, a2, a3, y, i;
%n order of the polynomial
%n1, n2 the two zero points
if not ((n1=-1 and n2=1) or (n1=1 and n2=-1) or (n1=0 and n2=1)
or (n1=1 and n2=0)) then ERROR(` n1, n2= 0, -1 or 1`) fi;
if n1=0 and n2=1 then aa1:=n1; aa2:=n2;
elif n1=1 and n2=0 then aa1:=n2; aa2:=n1;
elif n1=-1 and n2=1 then aa1:=n1; aa2:=n2;
else aa1:=n2; aa2:=n1;
fi;
if n < 1 or type(n, odd) then ERROR(` n must be even.`);
elif n = 2 and aa1=-1 and aa2=1 then y2:=x^2-1;
elif n = 2 and aa1=0 and aa2=1 then y2:=x^2-x;
elif n > 2 and aa1=-1 and aa2=1 then y2:=x^2-1;
elif n > 2 and aa1=0 and aa2=1 then y2:=x^2-x;
fi;
if n > 2 then
for i from 1 to n/2-1 do
y.(2*i+1):=(2*i+1)*(int(y.(2*i),x)+c);
y.(2*i+2):=(2*i+2)*(int(y.(2*i+1),x)+d);
a1:=subs(x= aa2, y.(2*i+2)=0);
a2:=subs(x=aa1, y.(2*i+2)=0);
a3:=solve({a1, a2}, {c,d});
y.(2*i+2):=subs(op(a3), y.(2*i+2)) ;
od;
fi;
y^n=";
end;
\end{verbatim}
\vspace{1cm}
\textbf{{\large Appendix 2} }
\begin{verbatim}
Algorithm for computing Appell Polynomial in terms of
Binomial (C) and Euler (Eu) function
apPolyeb:=proc(n,n1,n2) local aa1, aa2;
if not ((n1=-1 and n2=1) or (n1=1 and n2=-1)) then
ERROR(` n1, n2= -1 or 1`) fi;
if n1=-1 and n2=1 then aa1:=n1; aa2:=n2;
else aa1:=n2; aa2:=n1; fi;
if n < 1 or type(n,odd) then
ERROR(` n must be even and greater than 1.`);
elif n = 2 and aa1=-1 and aa2=1 then y2:=x^2-Eu(0);
elif n > 2 then
Appell:=sum(x^(2*n/2-2*j)*C(n,2*j)*Eu(2*j), j=0..n/2);
fi; end;
\end{verbatim}
\textbf{{\large Appendix 3} }
\begin{verbatim}
appelpoly := proc (f, x, a, b, n) local i, j, r, ii;
co.n := 1/(n!*int(diff(f,x),x = a .. b)); with(linalg):
for r from 2 to n+1 do
co.(n+1-r) := matrix(r,r,[]);
for i to r do for j to r do
co.(n+1-r)[1,r] := 1;
if i = j and j <> r then
co.(n+1-r)[i,j] := int(diff(f,x),x = a .. b);
co.(n+1-r)[r,r] := 0 elif j < i then
co.(n+1-r)[i,j] := int(x^(i-j)*diff(f,x),x = a .. b)/(i-j)!
elif i < j then co.(n+1-r)[i,j] := 0 fi od od;
co.(n+1-r) := det(convert(co.(n+1-r),matrix))/
((n+1-r)!*int(diff(f,x),x = a .. b)^r);
add(co.ii*x^ii,ii = 0 .. n) od end
\end{verbatim}
\begin{thebibliography}{9}
\bibitem{t1} Thorne, C. J., A Property of Appell Sets,
Amer. Math. Monthly 52 (1945)
191-193.
\bibitem{o1}Ozegov, V. B. Some Extremal Properties of Generalized Appell Polynomials,
Soviet Math. 5 (1964) 1651-1653.
\bibitem{s1} Sheffer, I.M., Some Properties of Polynomial Sets of Type Zero
Duke Math. J., 5 (1939) 590-622.
\end{thebibliography}
\noindent\textsc{H. Alkahby }\\
Department of Mathematics, Dillard University \\
New Orleans, LA 70122, USA \\
e-mail: halkahby@aol.com
\smallskip
\noindent\textsc{G. Ansong} \\
Department of Accounting, St. Mary's University \\
Canada, B3K 3C3, CA \\
e-mail: granville.ansong@stmarys.ca \smallskip
\noindent\textsc{Peter Frempong-Mireku}\\
Department of Mathematics

Dillard University \\
New Orleans, LA 70122, USA \\
e-mail:pfmireku@dillard.edu \smallskip
\noindent\textsc{A. Jalbout }\\
Department of Chemistry, University of New Orleans \\
New Orleans, LA 70148, USA
\end{document}