\documentclass[a4paper]{article}
\usepackage{amssymb}
\begin{document}
\begin{eqnarray*}
\mbox{(1)} \qquad A
 & = & \sum_{i = 0}^{i = \infty} a_{i} x^{i} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(2)} \qquad B
 & = & \sum_{j = 0}^{j = \infty} b_{j} x^{j} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(3)} \qquad C
 & = & \sum_{k = 0}^{k = \infty} c_{k} x^{k} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(4)} \qquad A + B
 & = & \left ( \sum_{i = 0}^{i = \infty} a_{i} x^{i} \right ) + \left ( \sum_{j = 0}^{j = \infty} b_{j} x^{j} \right ) \\
& = & \left ( \sum_{n = 0}^{n = \infty} a_{n} x^{n} \right ) + \left ( \sum_{n = 0}^{n = \infty} b_{n} x^{n} \right ) \\
& = & \sum_{n = 0}^{n = \infty} a_{n} x^{n} + b_{n} x^{n} \\
& = & a_{0} x^{0} + b_{0} x^{0} + a_{1} x^{1} + b_{1} x^{1} + a_{2} x^{2} + b_{2} x^{2} + a_{3} x^{3} + b_{3} x^{3}
 + a_{4} x^{4} + b_{4} x^{4} \\
 & & \qquad + \, a_{5} x^{5} + b_{5} x^{5} + \ldots \\
& = & (a_{0} + b_{0}) x^{0} + (a_{1} + b_{1}) x^{1} + (a_{2} + b_{2}) x^{2} + (a_{3} + b_{3}) x^{3}
 + (a_{4} + b_{4}) x^{4} + (a_{5} + b_{5}) x^{5} + \ldots \\
& = & \sum_{n = 0}^{n = \infty} (a_{n} + b_{n}) x^{n} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(5)} \qquad c_{n}
 & = & a_{n} + b_{n} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(6)} \qquad A - B
 & = & \left ( \sum_{i = 0}^{i = \infty} a_{i} x^{i} \right ) - \left ( \sum_{j = 0}^{j = \infty} b_{j} x^{j} \right ) \\
& = & \left ( \sum_{n = 0}^{n = \infty} a_{n} x^{n} \right ) - \left ( \sum_{n = 0}^{n = \infty} b_{n} x^{n} \right ) \\
& = & \sum_{n = 0}^{n = \infty} a_{n} x^{n} - b_{n} x^{n} \\
& = & a_{0} x^{0} - b_{0} x^{0} + a_{1} x^{1} - b_{1} x^{1} + a_{2} x^{2} - b_{2} x^{2} + a_{3} x^{3} - b_{3} x^{3}
 + a_{4} x^{4} - b_{4} x^{4} \\
 & & \qquad + \, a_{5} x^{5} - b_{5} x^{5} + \ldots \\
& = & (a_{0} - b_{0}) x^{0} + (a_{1} - b_{1}) x^{1} + (a_{2} - b_{2}) x^{2} + (a_{3} - b_{3}) x^{3}
 + (a_{4} - b_{4}) x^{4} + (a_{5} - b_{5}) x^{5} + \ldots \\
& = & \sum_{n = 0}^{n = \infty} (a_{n} - b_{n}) x^{n} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(7)} \qquad c_{n}
 & = & a_{n} - b_{n} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(8)} \qquad A \times B
 & = & \left ( \sum_{i = 0}^{i = \infty} a_{i} x^{i} \right ) \times \left ( \sum_{j = 0}^{j = \infty} b_{j} x^{j} \right ) \\
& = & \left ( \sum_{n = 0}^{n = \infty} a_{n} x^{n} \right ) \times \left ( \sum_{n = 0}^{n = \infty} b_{n} x^{n} \right ) \\
& = & (a_{0} x^{0} + a_{1} x^{1} + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + a_{5} x^{5} + \ldots) \\
 & & \qquad \times \, (b_{0} x^{0} + b_{1} x^{1} + b_{2} x^{2} + b_{3} x^{3} + b_{4} x^{4} + b_{5} x^{5} + \ldots) \\
& = & a_{0} x^{0} (b_{0} x^{0} + b_{1} x^{1} + b_{2} x^{2} + b_{3} x^{3} + b_{4} x^{4} + b_{5} x^{5} + \ldots) \\
 & & \quad + \, a_{1} x^{1} (b_{0} x^{0} + b_{1} x^{1} + b_{2} x^{2} + b_{3} x^{3} + b_{4} x^{4} + b_{5} x^{5} + \ldots) \\
 & & \quad \quad + \, a_{2} x^{2} (b_{0} x^{0} + b_{1} x^{1} + b_{2} x^{2} + b_{3} x^{3} + b_{4} x^{4} + b_{5} x^{5} + \ldots) \\
 & & \quad \quad \quad + \, a_{3} x^{3} (b_{0} x^{0} + b_{1} x^{1} + b_{2} x^{2} + b_{3} x^{3} + b_{4} x^{4} + b_{5} x^{5} + \ldots) \\
 & & \quad \quad \quad \quad + \, a_{4} x^{4} (b_{0} x^{0} + b_{1} x^{1} + b_{2} x^{2} + b_{3} x^{3} + b_{4} x^{4} + b_{5} x^{5} + \ldots) \\
 & & \quad \quad \quad \quad \quad + \, \ldots \\
& = & a_{0} b_{0} x^{0} + a_{0} b_{1} x^{1} + a_{0} b_{2} x^{2} + a_{0} b_{3} x^{3} + a_{0} b_{4} x^{4} + a_{0} b_{5} x^{5} + \ldots \\
 & & \quad + \, a_{1} b_{0} x^{1} + a_{1} b_{1} x^{2} + a_{1} b_{2} x^{3} + a_{1} b_{3} x^{4} + a_{1} b_{4} x^{5} + a_{1} b_{5} x^{6} + \ldots \\
 & & \quad \quad + \, a_{2} b_{0} x^{2} + a_{2} b_{1} x^{3} + a_{2} b_{2} x^{4} + a_{2} b_{3} x^{5} + a_{2} b_{4} x^{6} + a_{2} b_{5} x^{7} + \ldots \\
 & & \quad \quad \quad + \, a_{3} b_{0} x^{3} + a_{3} b_{1} x^{4} + a_{3} b_{2} x^{5} + a_{3} b_{3} x^{6} + a_{3} b_{4} x^{7} + a_{3} b_{5} x^{8} + \ldots \\
 & & \quad \quad \quad \quad + \, a_{4} b_{0} x^{4} + a_{4} b_{1} x^{5} + a_{4} b_{2} x^{6} + a_{4} b_{3} x^{7} + a_{4} b_{4} x^{8} + a_{4} b_{5} x^{9}
 + \ldots \\
 & & \quad \quad \quad \quad \quad + \, \ldots \\
& = & a_{0} b_{0} x^{0}
 + (a_{0} b_{1} + a_{1} b_{0}) x^{1}
 + (a_{0} b_{2} + a_{1} b_{1} + a_{2} b_{0}) x^{2}
 + (a_{0} b_{3} + a_{1} b_{2} + a_{2} b_{1} + a_{3} b_{0}) x^{3} \\
 & & \qquad + \, (a_{0} b_{4} + a_{1} b_{3} + a_{2} b_{2} + a_{3} b_{1} + a_{4} b_{0}) x^{4}
 + \ldots \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(9a)} \qquad c_{0}
 & = & a_{0} b_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(9b)} \qquad c_{1}
 & = & a_{0} b_{1} + a_{1} b_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(9c)} \qquad c_{2}
 & = & a_{0} b_{2} + a_{1} b_{1} + a_{2} b_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(9d)} \qquad c_{3}
 & = & a_{0} b_{3} + a_{1} b_{2} + a_{2} b_{1} + a_{3} b_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(9e)} \qquad c_{4}
 & = & a_{0} b_{4} + a_{1} b_{3} + a_{2} b_{2} + a_{3} b_{1} + a_{4} b_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(10)} \qquad c_{n}
 & = & \sum_{m = 0}^{m = n} a_{m} b_{n - m} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(11)} \qquad \frac{A}{B}
 & = & \frac{\sum_{i = 0}^{i = \infty} a_{i} x^{i}}{\sum_{j = 0}^{j = \infty} b_{j} x^{j}} \\
& = & \frac{a_{0} x^{0} + a_{1} x^{1} + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + a_{5} x^{5} + \ldots}
{b_{0} x^{0} + b_{1} x^{1} + b_{2} x^{2} + b_{3} x^{3} + b_{4} x^{4} + b_{5} x^{5} + \ldots} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(12)} \qquad & & \sum_{j = 0}^{j = \infty} b_{j} x^{j} \Biggl / \ \sum_{i = 0}^{i = \infty} a_{i} x^{i} \ \Biggl \backslash \ ? \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(13)} \qquad & & b_{0} x^{0} + b_{1} x^{1} + b_{2} x^{2} + b_{3} x^{3} \Biggl / \ a_{0} x^{0} + a_{1} x^{1} + a_{2} x^{2} + a_{3} x^{3} \
 \Biggl \backslash \ \frac{a_{0}}{b_{0}} x^{0}
 + \left ( \frac{a_{1}}{b_{0}} - \frac{a_{0} b_{1}}{b_{0}^{2}} \right ) x^{1} \\
& & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad
 + \left ( \frac{a_{2}}{b_{0}} - \frac{a_{0} b_{2}}{b_{0}^{2}} - \frac{a_{1} b_{1}}{b_{0}^{2}} + \frac{a_{0} b_{1}^{2}}{b_{0}^{3}} \right ) x^{2} \\
& & \qquad \qquad \qquad \qquad \qquad \qquad \, \, \, \, \,
 a_{0} x^{0} + \frac{a_{0} b_{1}}{b_{0}} x^{1} + \frac{a_{0} b_{2}}{b_{0}} x^{2} + \frac{a_{0} b_{3}}{b_{0}} x^{3} \\
& & \qquad \qquad
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \, - \\
& & \qquad \qquad \qquad \qquad
 \left ( a_{1} - \frac{a_{0} b_{1}}{b_{0}} \right ) x^{1}
 + \left ( a_{2} - \frac{a_{0} b_{2}}{b_{0}} \right ) x^{2}
 + \left ( a_{3} - \frac{a_{0} b_{3}}{b_{0}} \right ) x^{3} \\
& & \qquad \qquad \qquad \qquad
 \left ( a_{1} - \frac{a_{0} b_{1}}{b_{0}} \right ) x^{1}
 + \left ( \frac{a_{1} b_{1}}{b_{0}} - \frac{a_{0} b_{1}^{2}}{b_{0}^{2}} \right ) x^{2}
 + \left ( \frac{a_{1} b_{2}}{b_{0}} - \frac{a_{0} b_{1} b_{2}}{b_{0}^{2}} \right ) x^{3} \\
& & \qquad \qquad
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \, - \\
& & \qquad \qquad \qquad \qquad
 \left ( a_{2} - \frac{a_{0} b_{2}}{b_{0}} - \frac{a_{1} b_{1}}{b_{0}} + \frac{a_{0} b_{1}^{2}}{b_{0}^{2}} \right ) x^{2}
 + \left ( a_{3} - \frac{a_{0} b_{3}}{b_{0}} - \frac{a_{1} b_{2}}{b_{0}} + \frac{a_{0} b_{1} b_{2}}{b_{0}^{2}} \right ) x^{3} \\
& & \qquad \qquad \qquad \qquad
 \left ( a_{2} - \frac{a_{0} b_{2}}{b_{0}} - \frac{a_{1} b_{1}}{b_{0}} + \frac{a_{0} b_{1}^{2}}{b_{0}^{2}} \right ) x^{2}
 + \left ( \frac{a_{2} b_{1}}{b_{0}} - \frac{a_{0} b_{1} b_{2}}{b_{0}^{2}} - \frac{a_{1} b_{1}^{2}}{b_{0}^{2}} + \frac{a_{0} b_{1}^{3}}{b_{0}^{3}}
 \right ) x^{3} \\
& & \qquad \qquad
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \! - \! \! \!
 - \! \! \! - \! \! \! - \, - \\
& & \qquad \qquad \qquad \qquad
 \left ( a_{3} - \frac{a_{0} b_{3}}{b_{0}} - \frac{a_{1} b_{2}}{b_{0}} + \frac{a_{0} b_{1} b_{2}}{b_{0}^{2}}
 - \frac{a_{2} b_{1}}{b_{0}} + \frac{a_{0} b_{1} b_{2}}{b_{0}^{2}} + \frac{a_{1} b_{1}^{2}}{b_{0}^{2}} - \frac{a_{0} b_{1}^{3}}{b_{0}^{3}} \right ) x^{3} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(14a)} \qquad c_{0}
 & = & \frac{a_{0}}{b_{0}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(14b)} \qquad c_{1}
 & = & \frac{a_{1}}{b_{0}} - \frac{a_{0} b_{1}}{b_{0}^{2}} \\
& = & \frac{a_{1}}{b_{0}} - \frac{b_{1} c_{0}}{b_{0}} \\
& = & \frac{1}{b_{0}} (a_{1} - b_{1} c_{0}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(14c)} \qquad c_{2}
 & = & \frac{a_{2}}{b_{0}} - \frac{a_{0} b_{2}}{b_{0}^{2}} - \frac{a_{1} b_{1}}{b_{0}^{2}} + \frac{a_{0} b_{1}^{2}}{b_{0}^{3}} \\
& = & \frac{a_{2}}{b_{0}} - \frac{b_{2} c_{0}}{b_{0}} - \frac{a_{1} b_{1}}{b_{0}^{2}} + \frac{b_{1}^{2} c_{0}}{b_{0}^{2}} \\
& = & \frac{1}{b_{0}} \left ( a_{2} - b_{2} c_{0} - \frac{a_{1} b_{1}}{b_{0}} + \frac{b_{1}^{2} c_{0}}{b_{0}} \right ) \\
& = & \frac{1}{b_{0}} \left ( a_{2} - b_{2} c_{0} - \frac{b_{1}}{b_{0}} (a_{1} - b_{1} c_{0}) \right ) \\
& = & \frac{1}{b_{0}} (a_{2} - b_{2} c_{0} - b_{1} c_{1}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(14d)} \qquad c_{3}
 & = & \frac{a_{3}}{b_{0}} - \frac{a_{0} b_{3}}{b_{0}^{2}} - \frac{a_{1} b_{2}}{b_{0}^{2}} + \frac{a_{0} b_{1} b_{2}}{b_{0}^{3}}
 - \frac{a_{2} b_{1}}{b_{0}^{2}} + \frac{a_{0} b_{1} b_{2}}{b_{0}^{3}} + \frac{a_{1} b_{1}^{2}}{b_{0}^{3}} - \frac{a_{0} b_{1}^{3}}{b_{0}^{4}} \\
& = & \frac{a_{3}}{b_{0}} - \frac{b_{3} c_{0}}{b_{0}} - \frac{a_{1} b_{2}}{b_{0}^{2}} + \frac{b_{1} b_{2} c_{0}}{b_{0}^{2}}
 - \frac{a_{2} b_{1}}{b_{0}^{2}} + \frac{b_{1} b_{2} c_{0}}{b_{0}^{2}} + \frac{a_{1} b_{1}^{2}}{b_{0}^{3}} - \frac{b_{1}^{3} c_{0}}{b_{0}^{3}} \\
& = & \frac{1}{b_{0}} \left ( a_{3} - b_{3} c_{0} - \frac{a_{1} b_{2}}{b_{0}} + \frac{b_{1} b_{2} c_{0}}{b_{0}}
 - \frac{a_{2} b_{1}}{b_{0}} + \frac{b_{1} b_{2} c_{0}}{b_{0}} + \frac{a_{1} b_{1}^{2}}{b_{0}^{2}} - \frac{b_{1}^{3} c_{0}}{b_{0}^{2}} \right ) \\
& = & \frac{1}{b_{0}} \left ( a_{3} - b_{3} c_{0} - \frac{b_{2}}{b_{0}} (a_{1} - b_{1} c_{0})
 - \frac{a_{2} b_{1}}{b_{0}} + \frac{b_{1} b_{2} c_{0}}{b_{0}} + \frac{b_{1}^{2}}{b_{0}^{2}} (a_{1} - b_{1} c_{0}) \right ) \\
& = & \frac{1}{b_{0}} \left ( a_{3} - b_{3} c_{0} - b_{2} c_{1}
 - \frac{a_{2} b_{1}}{b_{0}} + \frac{b_{1} b_{2} c_{0}}{b_{0}} + \frac{b_{1}^{2} c_{1}}{b_{0}} \right ) \\
& = & \frac{1}{b_{0}} \left ( a_{3} - b_{3} c_{0} - b_{2} c_{1} - \frac{b_{1}}{b_{0}} (a_{2} - b_{2} c_{0} - b_{1} c_{1}) \right ) \\
& = & \frac{1}{b_{0}} (a_{3} - b_{3} c_{0} - b_{2} c_{1} - b_{1} c_{2}) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15)} \qquad c_{n}
 & = & \frac{1}{b_{0}} \left ( a_{n} - \sum_{m = 1}^{m = n} b_{m} c_{n - m} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(16)} \qquad A^{p}
 & = & \underbrace{A \times A \times \ldots \times A \times A}_{p \mbox{ termen}} \qquad p \, \in \, \mathbb{N} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(17)} \qquad A^{1}
 & = & A \\
& = & \sum_{i = 0}^{i = \infty} a_{i} x^{i} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(18)} \qquad A^{2}
 & = & A \times A \\
& = & \left ( \sum_{i = 0}^{i = \infty} a_{i} x^{i} \right ) \times \left ( \sum_{i = 0}^{i = \infty} a_{i} x^{i} \right ) \\
& = & (a_{0} x^{0} + a_{1} x^{1} + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + a_{5} x^{5} + \ldots) \\
 & & \qquad \times \, (a_{0} x^{0} + a_{1} x^{1} + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + a_{5} x^{5} + \ldots) \\
& = & a_{0} x^{0} (a_{0} x^{0} + a_{1} x^{1} + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + a_{5} x^{5} + \ldots) \\
 & & \quad + \, a_{1} x^{1} (a_{0} x^{0} + a_{1} x^{1} + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + a_{5} x^{5} + \ldots) \\
 & & \quad \quad + \, a_{2} x^{2} (a_{0} x^{0} + a_{1} x^{1} + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + a_{5} x^{5} + \ldots) \\
 & & \quad \quad \quad + \, a_{3} x^{3} (a_{0} x^{0} + a_{1} x^{1} + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + a_{5} x^{5} + \ldots) \\
 & & \quad \quad \quad \quad + \, a_{4} x^{4} (a_{0} x^{0} + a_{1} x^{1} + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + a_{5} x^{5} + \ldots) \\
 & & \quad \quad \quad \quad \quad + \, \ldots \\
& = & a_{0}^{2} x^{0} + a_{0} a_{1} x^{1} + a_{0} a_{2} x^{2} + a_{0} a_{3} x^{3} + a_{0} a_{4} x^{4} + a_{0} a_{5} x^{5} + \ldots \\
 & & \quad + \, a_{1} a_{0} x^{1} + a_{1}^{2} x^{2} + a_{1} a_{2} x^{3} + a_{1} a_{3} x^{4} + a_{1} a_{4} x^{5} + a_{1} a_{5} x^{6} + \ldots \\
 & & \quad \quad + \, a_{2} a_{0} x^{2} + a_{2} a_{1} x^{3} + a_{2}^{2} x^{4} + a_{2} a_{3} x^{5} + a_{2} a_{4} x^{6} + a_{2} a_{5} x^{7} + \ldots \\
 & & \quad \quad \quad + \, a_{3} a_{0} x^{3} + a_{3} a_{1} x^{4} + a_{3} a_{2} x^{5} + a_{3}^{2} x^{6} + a_{3} a_{4} x^{7} + a_{3} a_{5} x^{8} + \ldots \\
 & & \quad \quad \quad \quad + \, a_{4} a_{0} x^{4} + a_{4} a_{1} x^{5} + a_{4} a_{2} x^{6} + a_{4} a_{3} x^{7} + a_{4}^{2} x^{8} + a_{4} a_{5} x^{9}
 + \ldots) \\
 & & \quad \quad \quad \quad \quad + \, \ldots \\
& = & a_{0}^{2} x^{0}
 + (a_{0} a_{1} + a_{1} a_{0}) x^{1}
 + (a_{0} a_{2} + a_{1}^{2} + a_{2} a_{0}) x^{2}
 + (a_{0} a_{3} + a_{1} a_{2} + a_{2} a_{1} + a_{3} a_{0}) x^{3} \\
 & & \qquad + \, (a_{0} a_{4} + a_{1} a_{3} + a_{2}^{2} + a_{3} a_{1} + a_{4} a_{0}) x^{4}
 + \ldots \\
& = & a_{0}^{2} x^{0} + 2 a_{0} a_{1} x^{1} + (2 a_{0} a_{2} + a_{1}^{2}) x^{2} + (2 a_{0} a_{3} + 2 a_{1} a_{2}) x^{3} \\
 & & \qquad + \, (2 a_{0} a_{4} + 2 a_{1} a_{3} + a_{2}^{2}) x^{4} + \ldots \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(19)} \qquad A^{3}
 & = & A \times A^{2} \\
& = & (a_{0} x^{0} + a_{1} x^{1} + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + a_{5} x^{5} + \ldots) \\
 & & \qquad \times \, (a_{0}^{2} x^{0}
 + 2 a_{0} a_{1} x^{1}
 + (2 a_{0} a_{2} + a_{1}^{2}) x^{2}
 + (2 a_{0} a_{3} + 2 a_{1} a_{2}) x^{3} \\
 & & \qquad \qquad + \, (2 a_{0} a_{4} + 2 a_{1} a_{3} + a_{2}^{2}) x^{4}
 + \ldots) \\
& = & a_{0} x^{0} (a_{0}^{2} x^{0} + 2 a_{0} a_{1} x^{1} + (2 a_{0} a_{2} + a_{1}^{2}) x^{2} + (2 a_{0} a_{3} + 2 a_{1} a_{2}) x^{3} \\
 & & \qquad \qquad + \, (2 a_{0} a_{4} + 2 a_{1} a_{3} + a_{2}^{2}) x^{4} + \ldots) \\
& & \quad + \, a_{1} x^{1} (a_{0}^{2} x^{0} + 2 a_{0} a_{1} x^{1} + (2 a_{0} a_{2} + a_{1}^{2}) x^{2} + (2 a_{0} a_{3} + 2 a_{1} a_{2}) x^{3} \\
 & & \quad \qquad \qquad + \, (2 a_{0} a_{4} + 2 a_{1} a_{3} + a_{2}^{2}) x^{4} + \ldots) \\
& & \quad \quad + \, a_{2} x^{2} (a_{0}^{2} x^{0} + 2 a_{0} a_{1} x^{1} + (2 a_{0} a_{2} + a_{1}^{2}) x^{2} + (2 a_{0} a_{3} + 2 a_{1} a_{2}) x^{3} \\
 & & \quad \quad \qquad \qquad + \, (2 a_{0} a_{4} + 2 a_{1} a_{3} + a_{2}^{2}) x^{4} + \ldots) \\
& & \quad \quad \quad + \, a_{3} x^{3} (a_{0}^{2} x^{0} + 2 a_{0} a_{1} x^{1} + (2 a_{0} a_{2} + a_{1}^{2}) x^{2} + (2 a_{0} a_{3} + 2 a_{1} a_{2}) x^{3} \\
 & & \quad \quad \quad \qquad \qquad + \, (2 a_{0} a_{4} + 2 a_{1} a_{3} + a_{2}^{2}) x^{4} + \ldots) \\
& & \quad \quad \quad \quad
 + \, a_{4} x^{4} (a_{0}^{2} x^{0} + 2 a_{0} a_{1} x^{1} + (2 a_{0} a_{2} + a_{1}^{2}) x^{2} + (2 a_{0} a_{3} + 2 a_{1} a_{2}) x^{3} \\
 & & \quad \quad \quad \quad \qquad \qquad + \, (2 a_{0} a_{4} + 2 a_{1} a_{3} + a_{2}^{2}) x^{4} + \ldots) \\
 & & \quad \quad \quad \quad \quad + \, \ldots \\
& = & a_{0}^{3} x^{0} + 2 a_{0}^{2} a_{1} x^{1} + (2 a_{0}^{2} a_{2} + a_{0} a_{1}^{2}) x^{2} + (2 a_{0}^{2} a_{3} + 2 a_{0} a_{1} a_{2}) x^{3} \\
 & & \qquad \qquad + \, (2 a_{0}^{2} a_{4} + 2 a_{0} a_{1} a_{3} + a_{0} a_{2}^{2}) x^{4} + \ldots \\
& & \quad + \, a_{0}^{2} a_{1} x^{1} + 2 a_{0} a_{1}^{2} x^{2} + (2 a_{0} a_{1} a_{2} + a_{1}^{3}) x^{3} + (2 a_{0} a_{1} a_{3} + 2 a_{1}^{2} a_{2}) x^{4} \\
 & & \quad \qquad \qquad + \, (2 a_{0} a_{1} a_{4} + 2 a_{1}^{2} a_{3} + a_{1} a_{2}^{2}) x^{5} + \ldots \\
& & \quad \quad + \, a_{0}^{2} a_{2} x^{2} + 2 a_{0} a_{1} a_{2} x^{3} + (2 a_{0} a_{2}^{2} + a_{1}^{2} a_{2}) x^{4}
 + (2 a_{0} a_{2} a_{3} + 2 a_{1} a_{2}^{2}) x^{5} \\
 & & \quad \quad \qquad \qquad + \, (2 a_{0} a_{2} a_{4} + 2 a_{1} a_{2} a_{3} + a_{2}^{3}) x^{6} + \ldots \\
& & \quad \quad \quad + \, a_{0}^{2} a_{3} x^{3} + 2 a_{0} a_{1} a_{3} x^{4} + (2 a_{0} a_{2} a_{3} + a_{1}^{2} a_{3}) x^{5}
 + (2 a_{0} a_{3}^{2} + 2 a_{1} a_{2} a_{3}) x^{6} \\
 & & \quad \quad \quad \qquad \qquad + \, (2 a_{0} a_{3} a_{4} + 2 a_{1} a_{3}^{2} + a_{2}^{2} a_{3}) x^{7} + \ldots \\
& & \quad \quad \quad \quad
 + \, a_{0}^{2} a_{4} x^{4} + 2 a_{0} a_{1} a_{4} x^{5} + (2 a_{0} a_{2} a_{4} + a_{1}^{2} a_{4}) x^{6} + (2 a_{0} a_{3} a_{4} + 2 a_{1} a_{2} a_{4}) x^{7} \\
 & & \quad \quad \quad \quad \qquad \qquad + \, (2 a_{0} a_{4}^{2} + 2 a_{1} a_{3} a_{4} + a_{2}^{2} a_{4}) x^{8} + \ldots \\
 & & \quad \quad \quad \quad \quad + \, \ldots \\
& = & a_{0}^{3} x^{0} + 3 a_{0}^{2} a_{1} x^{1} + (3 a_{0}^{2} a_{2} + 3 a_{0} a_{1}^{2}) x^{2}
 + (3 a_{0}^{2} a_{3} + 6 a_{0} a_{1} a_{2} + a_{1}^{3}) x^{3} \\
 & & \qquad + \, (3 a_{0}^{2} a_{4} + 6 a_{0} a_{1} a_{3} + 3 a_{0} a_{2}^{2} + 3 a_{1}^{2} a_{2}) x^{4} + \ldots \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(20)} \qquad A^{4}
 & = & A \times A^{3} \\
& = & (a_{0} x^{0} + a_{1} x^{1} + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4} + a_{5} x^{5} + \ldots) \\
 & & \qquad \times \, (a_{0}^{3} x^{0} + 3 a_{0}^{2} a_{1} x^{1} + (3 a_{0}^{2} a_{2} + 3 a_{0} a_{1}^{2}) x^{2}
 + (3 a_{0}^{2} a_{3} + 6 a_{0} a_{1} a_{2} + a_{1}^{3}) x^{3} \\
 & & \qquad \qquad + \, (3 a_{0}^{2} a_{4} + 6 a_{0} a_{1} a_{3} + 3 a_{0} a_{2}^{2} + 3 a_{1}^{2} a_{2}) x^{4} + \ldots) \\
& = & a_{0} x^{0} (a_{0}^{3} x^{0} + 3 a_{0}^{2} a_{1} x^{1} + (3 a_{0}^{2} a_{2} + 3 a_{0} a_{1}^{2}) x^{2}
 + (3 a_{0}^{2} a_{3} + 6 a_{0} a_{1} a_{2} + a_{1}^{3}) x^{3} \\
 & & \qquad \qquad + \, (3 a_{0}^{2} a_{4} + 6 a_{0} a_{1} a_{3} + 3 a_{0} a_{2}^{2} + 3 a_{1}^{2} a_{2}) x^{4} + \ldots) \\
 & & \quad + \, a_{1} x^{1} (a_{0}^{3} x^{0} + 3 a_{0}^{2} a_{1} x^{1} + (3 a_{0}^{2} a_{2} + 3 a_{0} a_{1}^{2}) x^{2}
 + (3 a_{0}^{2} a_{3} + 6 a_{0} a_{1} a_{2} + a_{1}^{3}) x^{3} \\
 & & \quad \qquad \qquad + \, (3 a_{0}^{2} a_{4} + 6 a_{0} a_{1} a_{3} + 3 a_{0} a_{2}^{2} + 3 a_{1}^{2} a_{2}) x^{4} + \ldots) \\
 & & \quad \quad + \, a_{2} x^{2} (a_{0}^{3} x^{0} + 3 a_{0}^{2} a_{1} x^{1} + (3 a_{0}^{2} a_{2} + 3 a_{0} a_{1}^{2}) x^{2}
 + (3 a_{0}^{2} a_{3} + 6 a_{0} a_{1} a_{2} + a_{1}^{3}) x^{3} \\
 & & \quad \quad \qquad \qquad + \, (3 a_{0}^{2} a_{4} + 6 a_{0} a_{1} a_{3} + 3 a_{0} a_{2}^{2} + 3 a_{1}^{2} a_{2}) x^{4} + \ldots) \\
 & & \quad \quad \quad + \, a_{3} x^{3} (a_{0}^{3} x^{0} + 3 a_{0}^{2} a_{1} x^{1} + (3 a_{0}^{2} a_{2} + 3 a_{0} a_{1}^{2}) x^{2}
 + (3 a_{0}^{2} a_{3} + 6 a_{0} a_{1} a_{2} + a_{1}^{3}) x^{3} \\
 & & \quad \quad \quad \qquad \qquad + \, (3 a_{0}^{2} a_{4} + 6 a_{0} a_{1} a_{3} + 3 a_{0} a_{2}^{2} + 3 a_{1}^{2} a_{2}) x^{4} + \ldots) \\
 & & \quad \quad \quad \quad + \, a_{4} x^{4} (a_{0}^{3} x^{0} + 3 a_{0}^{2} a_{1} x^{1} + (3 a_{0}^{2} a_{2} + 3 a_{0} a_{1}^{2}) x^{2}
 + (3 a_{0}^{2} a_{3} + 6 a_{0} a_{1} a_{2} + a_{1}^{3}) x^{3} \\
 & & \quad \quad \quad \quad \qquad \qquad + \, (3 a_{0}^{2} a_{4} + 6 a_{0} a_{1} a_{3} + 3 a_{0} a_{2}^{2} + 3 a_{1}^{2} a_{2}) x^{4} + \ldots) \\
 & & \quad \quad \quad \quad \quad + \, \ldots \\
& = & a_{0}^{4} x^{0} + 3 a_{0}^{3} a_{1} x^{1} + (3 a_{0}^{3} a_{2} + 3 a_{0}^{2} a_{1}^{2}) x^{2}
 + (3 a_{0}^{3} a_{3} + 6 a_{0}^{2} a_{1} a_{2} + a_{0} a_{1}^{3}) x^{3} \\
 & & \qquad \qquad + \, (3 a_{0}^{3} a_{4} + 6 a_{0}^{2} a_{1} a_{3} + 3 a_{0}^{2} a_{2}^{2} + 3 a_{0} a_{1}^{2} a_{2}) x^{4} + \ldots \\
 & & \quad + \, a_{0}^{3} a_{1} x^{1} + 3 a_{0}^{2} a_{1}^{2} x^{2} + (3 a_{0}^{2} a_{1} a_{2} + 3 a_{0} a_{1}^{3}) x^{3}
 + (3 a_{0}^{2} a_{1} a_{3} + 6 a_{0} a_{1}^{2} a_{2} + a_{1}^{4}) x^{4} \\
 & & \quad \qquad \qquad + \, (3 a_{0}^{2} a_{1} a_{4} + 6 a_{0} a_{1}^{2} a_{3} + 3 a_{0} a_{1} a_{2}^{2} + 3 a_{1}^{3} a_{2}) x^{5} + \ldots \\
 & & \quad \quad + \, a_{0}^{3} a_{2} x^{2} + 3 a_{0}^{2} a_{1} a_{2} x^{3} + (3 a_{0}^{2} a_{2}^{2} + 3 a_{0} a_{1}^{2} a_{2}) x^{4}
 + (3 a_{0}^{2} a_{2} a_{3} + 6 a_{0} a_{1} a_{2}^{2} + a_{1}^{3} a_{2}) x^{5} \\
 & & \quad \quad \qquad \qquad + \, (3 a_{0}^{2} a_{2} a_{4} + 6 a_{0} a_{1} a_{2} a_{3} + 3 a_{0} a_{2}^{3} + 3 a_{1}^{2} a_{2}^{2}) x^{6} + \ldots \\
 & & \quad \quad \quad + \, a_{0}^{3} a_{3} x^{3} + 3 a_{0}^{2} a_{1} a_{3} x^{4} + (3 a_{0}^{2} a_{2} a_{3} + 3 a_{0} a_{1}^{2} a_{3}) x^{5} \\
 & & \quad \quad \quad \qquad \qquad + \, (3 a_{0}^{2} a_{3}^{2} + 6 a_{0} a_{1} a_{2} a_{3} + a_{1}^{3} a_{3}) x^{6} \\
 & & \quad \quad \quad \quad \qquad \qquad + \, (3 a_{0}^{2} a_{3} a_{4} + 6 a_{0} a_{1} a_{3}^{2} + 3 a_{0} a_{2}^{2} a_{3}
 + 3 a_{1}^{2} a_{2} a_{3}) x^{7} + \ldots \\
 & & \quad \quad \quad \quad + \, a_{0}^{3} a_{4} x^{4} + 3 a_{0}^{2} a_{1} a_{4} x^{5} + (3 a_{0}^{2} a_{2} a_{4} + 3 a_{0} a_{1}^{2} a_{4}) x^{6} \\
 & & \quad \quad \quad \quad \quad \qquad \qquad + \, (3 a_{0}^{2} a_{3} a_{4} + 6 a_{0} a_{1} a_{2} a_{4} + a_{1}^{3} a_{4}) x^{7} \\
 & & \quad \quad \quad \quad \quad \quad \qquad \qquad + \, (3 a_{0}^{2} a_{4}^{2} + 6 a_{0} a_{1} a_{3} a_{4} + 3 a_{0} a_{2}^{2} a_{4}
 + 3 a_{1}^{2} a_{2} a_{4}) x^{8} + \ldots \\
 & & \quad \quad \quad \quad \quad + \, \ldots \\
& = & a_{0}^{4} x^{0} + 4 a_{0}^{3} a_{1} x^{1} + (4 a_{0}^{3} a_{2} + 6 a_{0}^{2} a_{1}^{2}) x^{2}
 + (4 a_{0}^{3} a_{3} + 12 a_{0}^{2} a_{1} a_{2} + 4 a_{0} a_{1}^{3}) x^{3} \\
 & & \qquad + \, (4 a_{0}^{3} a_{4} + 12 a_{0}^{2} a_{1} a_{3} + 6 a_{0}^{2} a_{2}^{2} + 12 a_{0} a_{1}^{2} a_{2}) x^{4} + \ldots \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(21a)} \qquad p \, \, = \, \, 1: \qquad c_{0}
 & = & a_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(21b)} \qquad p \, \, = \, \, 2: \qquad c_{0}
 & = & a_{0}^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(21c)} \qquad p \, \, = \, \, 3: \qquad c_{0}
 & = & a_{0}^{3} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(21d)} \qquad p \, \, = \, \, 4: \qquad c_{0}
 & = & a_{0}^{4} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22a)} \qquad p \, \, = \, \, 1: \qquad c_{1}
 & = & a_{1} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22b)} \qquad p \, \, = \, \, 2: \qquad c_{1}
 & = & 2 a_{0} a_{1} \\
& = & a_{0} a_{1} + a_{1} a_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22c)} \qquad p \, \, = \, \, 3: \qquad c_{1}
 & = & 3 a_{0}^{2} a_{1} \\
& = & a_{0} a_{0} a_{1} + a_{0} a_{1} a_{0} + a_{1} a_{0} a_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(22d)} \qquad p \, \, = \, \, 4: \qquad c_{1}
 & = & 4 a_{0}^{3} a_{1} \\
& = & a_{0} a_{0} a_{0} a_{1} + a_{0} a_{0} a_{1} a_{0} + a_{0} a_{1} a_{0} a_{0} + a_{1} a_{0} a_{0} a_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(23a)} \qquad p \, \, = \, \, 1: \qquad c_{2}
 & = & a_{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(23b)} \qquad p \, \, = \, \, 2: \qquad c_{2}
 & = & 2 a_{0} a_{2} + a_{1}^{2} \\
& = & a_{0} a_{2} + a_{1} a_{1} + a_{2} a_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(23c)} \qquad p \, \, = \, \, 3: \qquad c_{2}
 & = & 3 a_{0}^{2} a_{2} + 3 a_{0} a_{1}^{2} \\
& = & a_{0} a_{0} a_{2} + a_{0} a_{1} a_{1} + a_{0} a_{2} a_{0} + a_{1} a_{0} a_{1} + a_{1} a_{1} a_{0} + a_{2} a_{0} a_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(23d)} \qquad p \, \, = \, \, 4: \qquad c_{2}
 & = & 4 a_{0}^{3} a_{2} + 6 a_{0}^{2} a_{1}^{2} \\
& = & a_{0} a_{0} a_{0} a_{2} + a_{0} a_{0} a_{1} a_{1} + a_{0} a_{0} a_{2} a_{0} + a_{0} a_{1} a_{0} a_{1} + a_{0} a_{1} a_{1} a_{0}
 + a_{0} a_{2} a_{0} a_{0} \\
 & & \qquad + \, a_{1} a_{0} a_{0} a_{1} + a_{1} a_{0} a_{1} a_{0} + a_{1} a_{1} a_{0} a_{0} + a_{2} a_{0} a_{0} a_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(24a)} \qquad p \, \, = \, \, 1: \qquad c_{3}
 & = & a_{3} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(24b)} \qquad p \, \, = \, \, 2: \qquad c_{3}
 & = & 2 a_{0} a_{3} + 2 a_{1} a_{2} \\
& = & a_{0} a_{3} + a_{1} a_{2} + a_{2} a_{1} + a_{3} a_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(24c)} \qquad p \, \, = \, \, 3: \qquad c_{3}
 & = & 3 a_{0}^{2} a_{3} + 6 a_{0} a_{1} a_{2} + a_{1}^{3} \\
& = & a_{0} a_{0} a_{3} + a_{0} a_{1} a_{2} + a_{0} a_{2} a_{1} + a_{0} a_{3} a_{0} + a_{1} a_{0} a_{2} + a_{1} a_{1} a_{1}
 + a_{1} a_{2} a_{0} + a_{2} a_{0} a_{1} \\
 & & \qquad + \, a_{2} a_{1} a_{0} + a_{3} a_{0} a_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(24d)} \qquad p \, \, = \, \, 4: \qquad c_{3}
 & = & 4 a_{0}^{3} a_{3} + 12 a_{0}^{2} a_{1} a_{2} + 4 a_{0} a_{1}^{3} \\
& = & a_{0} a_{0} a_{0} a_{3} + a_{0} a_{0} a_{1} a_{2} + a_{0} a_{0} a_{2} a_{1} + a_{0} a_{0} a_{3} a_{0} + a_{0} a_{1} a_{0} a_{2}
 + a_{0} a_{1} a_{1} a_{1} \\
 & & \qquad + \, a_{0} a_{1} a_{2} a_{0} + a_{0} a_{2} a_{0} a_{1} + a_{0} a_{2} a_{1} a_{0} + a_{0} a_{3} a_{0} a_{0}
 + a_{1} a_{0} a_{0} a_{2} + a_{1} a_{0} a_{1} a_{1} \\
 & & \qquad \qquad + \, a_{1} a_{0} a_{2} a_{0} + a_{1} a_{1} a_{0} a_{1} + a_{1} a_{1} a_{1} a_{0}
 + a_{1} a_{2} a_{0} a_{0} + a_{2} a_{0} a_{0} a_{1} \\
 & & \qquad \qquad \qquad + \, a_{2} a_{0} a_{1} a_{0} + a_{2} a_{1} a_{0} a_{0} + a_{3} a_{0} a_{0} a_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(25a)} \qquad p \, \, = \, \, 1: \qquad c_{4}
 & = & a_{4} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(25b)} \qquad p \, \, = \, \, 2: \qquad c_{4}
 & = & 2 a_{0} a_{4} + 2 a_{1} a_{3} + a_{2}^{2} \\
& = & a_{0} a_{4} + a_{1} a_{3} + a_{2} a_{2} + a_{3} a_{1} + a_{4} a_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(25c)} \qquad p \, \, = \, \, 3: \qquad c_{4}
 & = & 3 a_{0}^{2} a_{4} + 6 a_{0} a_{1} a_{3} + 3 a_{0} a_{2}^{2} + 3 a_{1}^{2} a_{2} \\
& = & a_{0} a_{0} a_{4} + a_{0} a_{1} a_{3} + a_{0} a_{2} a_{2} + a_{0} a_{3} a_{1} + a_{0} a_{4} a_{0} + a_{1} a_{0} a_{3}
 + a_{1} a_{1} a_{2} + a_{1} a_{2} a_{1} \\
 & & \qquad + \, a_{1} a_{3} a_{0} + a_{2} a_{0} a_{2} + a_{2} a_{1} a_{1} + a_{2} a_{2} a_{0} + a_{3} a_{0} a_{1} + a_{3} a_{1} a_{0}
 + a_{4} a_{0} a_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(25d)} \qquad p \, \, = \, \, 4: \qquad c_{4}
 & = & 4 a_{0}^{3} a_{4} + 12 a_{0}^{2} a_{1} a_{3} + 6 a_{0}^{2} a_{2}^{2} + 12 a_{0} a_{1}^{2} a_{2} \\
& = & a_{0} a_{0} a_{0} a_{4} + a_{0} a_{0} a_{1} a_{3} + a_{0} a_{0} a_{2} a_{2} + a_{0} a_{0} a_{3} a_{1} + a_{0} a_{0} a_{4} a_{0} \\
 & & \quad + \, a_{0} a_{1} a_{0} a_{3} + a_{0} a_{1} a_{1} a_{2} + a_{0} a_{1} a_{2} a_{1} + a_{0} a_{1} a_{3} a_{0} + a_{0} a_{2} a_{0} a_{2} \\
 & & \quad \quad + \, a_{0} a_{2} a_{1} a_{1} + a_{0} a_{2} a_{2} a_{0} + a_{0} a_{3} a_{0} a_{1} + a_{0} a_{3} a_{1} a_{0} + a_{0} a_{4} a_{0} a_{0} \\
 & & \quad \quad \quad + \, a_{1} a_{0} a_{0} a_{3} + a_{1} a_{0} a_{1} a_{2} + a_{1} a_{0} a_{2} a_{1} + a_{1} a_{0} a_{3} a_{0}
 + a_{1} a_{1} a_{0} a_{2} \\
 & & \quad \quad \quad \quad + \, a_{1} a_{1} a_{2} a_{0} + a_{1} a_{2} a_{0} a_{1} + a_{1} a_{2} a_{1} a_{0} + a_{1} a_{3} a_{0} a_{0}
 + a_{2} a_{0} a_{0} a_{2} \\
 & & \quad \quad \quad \quad \quad + \, a_{2} a_{0} a_{1} a_{1} + a_{2} a_{0} a_{2} a_{0} + a_{2} a_{1} a_{0} a_{1} + a_{2} a_{1} a_{1} a_{0}
 + a_{2} a_{2} a_{0} a_{0} \\
 & & \quad \quad \quad \quad \quad \quad + \, a_{3} a_{0} a_{0} a_{1} + a_{3} a_{0} a_{1} a_{0} + a_{3} a_{1} a_{0} a_{0} + a_{4} a_{0} a_{0} a_{0} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(26a)} \qquad \qquad c_{0}
 & = & a_{0}^{p} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(26b)} \qquad c_{n}
 & = & \frac{1}{a_{0}} \sum_{m = 1}^{m = n} \left ( \frac{m}{n} (p + 1) - 1 \right ) a_{m} c_{n - m} \qquad n \, \, \, \ge \, \, \, 1 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(27a)} \qquad C
 & = & A + B \\
& = & \sum_{n = 0}^{n = \infty} (a_{n} + b_{n}) x^{n} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(27b)} \qquad C
 & = & A - B \\
& = & \sum_{n = 0}^{n = \infty} (a_{n} - b_{n}) x^{n} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(27c)} \qquad C
 & = & A \times B \\
& = & \sum_{n = 0}^{n = \infty} \left ( \sum_{m = 0}^{m = n} a_{m} b_{n - m} \right ) x^{n} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(27d)} \qquad C
 & = & \frac{A}{B} \\
& = & \frac{1}{b_{0}} \sum_{n = 0}^{n = \infty} \left ( a_{n} - \sum_{m = 1}^{m = n} b_{m} c_{n - m} \right ) x^{n} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(27e)} \qquad C
 & = & A^{p} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad p \, \in \, \mathbb{N} \\
& = & a_{0}^{p} + \frac{1}{a_{0}} \sum_{n = 1}^{n = \infty} \sum_{m = 1}^{m = n} \left ( \frac{m}{n} (p + 1) - 1 \right ) a_{m} c_{n - m} x^{n} \\
\end{eqnarray*}
\end{document}

p = 2:

a_{0} c_{1} = \left ( \frac{1}{1} (2 + 1) - 1 \right ) a_{1} c_{0} \\
 = 2 a_{1} c_{0} \\
 = 2 a_{1} a_{0}^{2} \\
c_{1} = 2 a_{1} a_{0} \\ klopt!

a_{0} c_{2} = \left ( \frac{1}{2} (2 + 1) - 1 \right ) a_{1} c_{1}
 + \left ( \frac{2}{2} (2 + 1) - 1 \right ) a_{2} c_{0} \\
 = \frac{1}{2} a_{1} c_{1} + 2 a_{2} c_{0} \\
 = \frac{1}{2} a_{1} 2 a_{1} a_{0} + 2 a_{2} a_{0}^{2} \\
 = a_{1}^{2} a_{0} + 2 a_{2} a_{0}^{2} \\
c_{2} = a_{1}^{2} + 2 a_{2} a_{0} \\ klopt!

a_{0} c_{3} = \left ( \frac{1}{3} (2 + 1) - 1 \right ) a_{1} c_{2}
 + \left ( \frac{2}{3} (2 + 1) - 1 \right ) a_{2} c_{1}
 + \left ( \frac{3}{3} (2 + 1) - 1 \right ) a_{3} c_{0} \\
 = a_{2} c_{1} + 2 a_{3} c_{0} \\
 = a_{2} 2 a_{1} a_{0} + 2 a_{3} a_{0}^{2} \\
c_{3} = 2 a_{2} a_{1} + 2 a_{3} a_{0} \\ klopt!

a_{0} c_{4} = \left ( \frac{1}{4} (2 + 1) - 1 \right ) a_{1} c_{3}
 + \left ( \frac{2}{4} (2 + 1) - 1 \right ) a_{2} c_{2}
 + \left ( \frac{3}{4} (2 + 1) - 1 \right ) a_{3} c_{1}
 + \left ( \frac{4}{4} (2 + 1) - 1 \right ) a_{4} c_{0} \\
 = - \frac{1}{4} a_{1} c_{3} + \frac{1}{2} a_{2} c_{2} + \frac{5}{4} a_{3} c_{1} + 2 a_{4} c_{0} \\
 = - \frac{1}{4} a_{1} (a_{0} a_{3} + a_{1} a_{2} + a_{2} a_{1} + a_{3} a_{0}) + \frac{1}{2} a_{2} (2 a_{0} a_{2} + a_{1}^{2})
 + \frac{5}{4} a_{3} 2 a_{0} a_{1} + 2 a_{4} a_{0}^{2} \\
 = - \frac{1}{4} a_{1} a_{0} a_{3} - \frac{1}{4} a_{1} a_{1} a_{2} - \frac{1}{4} a_{1} a_{2} a_{1} - \frac{1}{4} a_{1} a_{3} a_{0}
 + \frac{1}{2} a_{2} 2 a_{0} a_{2} + \frac{1}{2} a_{2} a_{1}^{2}
 + \frac{5}{4} a_{3} 2 a_{0} a_{1} + 2 a_{4} a_{0}^{2} \\
 = 2 a_{3} a_{0} a_{1} + a_{2}^{2} a_{0} + 2 a_{4} a_{0}^{2} \\
c_{4} = 2 a_{3} a_{1} + a_{2}^{2} + 2 a_{4} a_{0} \\ klopt!
