\documentclass[12pt]{article}

\pagestyle{empty}

\begin{document}

\section{$E_k(x)$ : Eulerian Polynomials}

\begin{eqnarray}
E_0(x) &=& 1 \\
E_1(x) &=& x \\
E_2(x) &=& x^2 + x \\
E_3(x) &=& x^3 + 4x^2 + x \\
E_4(x) &=& x^4 + 11x^3 + 11x^2 + x \\
E_5(x) &=& x^5 + 26x^4 + 66x^3 + 26x^2 + x
\end{eqnarray}

\begin{equation}
E_k(x) = (x-x^2)E_{k-1}'(x)+xkE_{k-1}(x)
\end{equation}

\begin{equation}
E_k(1) = k!
\end{equation}

\section{$T_k(x)$ : Chebyshev Polynomials, 1st Kind}

\begin{eqnarray}
T_0(x) &=& 1 \\
T_1(x) &=& x \\
T_2(x) &=& 2x^2 - 1 \\
T_3(x) &=& 4x^3 - 3x \\
T_4(x) &=& 8x^4 - 8x^2 + 1 \\
T_5(x) &=& 16x^5 - 20x^3 + 5x
\end{eqnarray}

\begin{equation}
T_k(x) = 2xT_{k-1}(x)-T_{k-2}(x)
\end{equation}

\begin{equation}
(1-x^2) \frac{d^2}{dx^2} T_k(x) - x \frac{d}{dx} T_k(x) + k^2 T_k(x) = 0
\end{equation}

\begin{equation}
\sum_{k \geq 0} t^k T_k(x) = \frac{1-tx}{1-2tx+t^2}
\end{equation}

\begin{equation}
T_0(x) + 2 \sum_{k \geq 1} t^k T_k(x) = \frac{1-t^2}{1-2tx+t^2}
\end{equation}

\section{$P_k(x)$ : Legendre Polynomials}

\begin{eqnarray}
P_0(x) &=& 1 \\
P_1(x) &=& x \\
P_2(x) &=& \frac{1}{2} (3x^2 - 1) \\
P_3(x) &=& \frac{1}{2} (5x^3 - 3x) \\
P_4(x) &=& \frac{1}{8} (35x^4 - 30x^2 + 3) \\
P_5(x) &=& \frac{1}{8} (63x^5 - 70x^3 + 15x)
\end{eqnarray}

\begin{equation}
P_k(x) = 2x(1-\frac{1}{2k})P_{k-1}(x)-(1-\frac{1}{k})P_{k-2}(x)
\end{equation}

\begin{equation}
(1-x^2) \frac{d^2}{dx^2} P_k(x) - 2x \frac{d}{dx} P_k(x) + k(k+1) P_k(x) = 0
\end{equation}

\begin{equation}
\frac{d}{dx} \left( (1-x^2) \frac{d}{dx} P_k(x) \right) + k(k+1) P_k(x) = 0
\end{equation}

\begin{equation}
\sum_{k \geq 0} t^k P_k(x) = (1-2tx+t^2)^{-1/2}
\end{equation}

\end{document}
