• \[ e_k(X_1, X_2, \dots, X_n) = \sum_{1\le i_1 < i_2 <\cdots < i_k\le n}X_{i_1}\cdots X_{i_k} \]

• \[ m_\alpha(X_1, X_2, \dots, X_n) = \sum_{\alpha\subset \beta, |\alpha| = |\beta|}X_{1}^{\beta_1}X_{2}^{\beta_2}\cdots X_{n}^{\beta_k} \]
• where \(\alpha\) is the multi-index with nonzero indices.

• \[ p_k(X_1, X_2,\dots, X_n) = \sum_{i=1}^n X_i^k \]

• \[ h_k(X_1, X_2, \dots, X_n) = \sum_{|\alpha| = k}X_1^{\alpha_1}X_2^{\alpha_2}\cdots X_n^{\alpha_n} \]

• Jacobi's Bialternant Formula
• \[ s_\lambda(X_1, X_2, \dots, X_n) = \frac{ \det \left[ \begin{matrix} X_1^{\lambda_1+n-1} & X_2^{\lambda_1+n-1} & \dots & X_n^{\lambda_1+n-1} \\ X_1^{\lambda_2+n-2} & X_2^{\lambda_2+n-2} & \dots & X_n^{\lambda_2+n-2} \\ \vdots & \vdots & \ddots & \vdots \\ X_1^{\lambda_n} & X_2^{\lambda_n} & \dots & X_n^{\lambda_n} \end{matrix} \right]}{ \det \left[ \begin{matrix} X_1^{n-1} & X_2^{n-1} & \dots & X_n^{n-1} \\ X_1^{n-2} & X_2^{n-2} & \dots & X_n^{n-2} \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \dots & 1 \end{matrix} \right] } \]
• where \(\lambda\) is the with \(\lambda_i \ge \lambda_{i+1}\ge 0\).
• The denominator is called the Vandermonde determinant or the Vandermonde polynomial: \[ V_n = \prod_{1 \leq i < j \leq n} (X_i-X_j). \]

• \[ f(X_1, X_2, \dots, X_n) = \operatorname{sgn}(\sigma)f(X_{\sigma(1)}, X_{\sigma(2)}, \dots, X_{\sigma(n)}) \]

• Product of two alternating polynomials is an symmetric polynomial.

• Any symmetric polynomial can be written in terms of elementary symmetric polynomials.

• \[ ke_k = \sum_{i=1}^k(-1)^{i-1}e_{k-i}p_i \]
• Equivalently: \[ 0 = \sum_{i=0}^k (-1)^{k-i}e_{k-i}p_i \]
• And further: \[ 0 = \sum_{i=k-n}^k(-1)^{i-1}e_{k-i}p_i \]
  • for all \(k > n\ge 1\).
  • Multiply both sides of the \(k=n\) case above by \(x_i^{k-n}\).

• \(T_n(x)\)

• \(U_n(x)\)

• Airfoil Polynomials
• \(V_n(x)\)
• \[ V_n(\cos\theta) = \frac{\cos((n+1/2)\theta)}{\cos(\theta/2)}. \]
• \[ V_n(x) = \sqrt{\frac{2}{1+x}}T_{2n+1}\left(\sqrt{\frac{x+1}{2}}\right). \]

• Airfoil Polynomials
• \(W_n(x)\)
• \[ W_n(\cos\theta) = \frac{\sin((n+1/2)\theta)}{\sin(\theta/2)}. \]
• \[ W_n(x) = U_{2n}\left(\sqrt{\frac{x+1}{2}}\right). \]
• It is related to the \(D_n(x)\) by: \[ W_n(\cos x) = D_n(x). \]

• The chebyshev polynomials can be defined as the solutions to the in a ./algebra/Ring Theory.html#org1fc0d16 \(\mathbb{R}[x]\): \[ T_n(x)^2 - (x^2-1)U_{n-1}(x)^2 = 1. \]
• This can easily be shown by substituting \(x\) with \(\cos\theta\).

\[ \frac{1}{\sqrt{1-2xt + t^2}} = \sum_{n=0}^\infty P_n(x)t^n. \]

It appears at the multipole expansion in electrostatics, as \( 1/r'' \) where \( r'' \) is the distance between the charge and a point in space. Using this analogy, \( x \) can be mapped to \( \cos\theta \) and \( t \) to \( r'/r \), therefore the denominator on the left hand side is the length obtained by the law of cosine with \( \mathbf{r} \) and \( \mathbf{r'} \).

It is the solution to the differential equation: \[ (1-x^2)P_n''(x) - 2xP_n'(x) + n(n+1)P_n(x) = 0. \]

It appears when applying separation of variables to the Laplace equation to the spherical coordinates as the equations in \( \theta \). The form in this problem is: \[ \frac{1}{\sin\theta} \frac{d}{d\theta} \left( \sin\theta \frac{d\Theta(\theta)}{d\theta}\right) = -\ell(\ell+1)\Theta(\theta). \] The differentiation can be transformed into using variable \( x := \cos\theta \), by setting \( \Theta(\theta) = P(\cos\theta) \) and using the relation: \[ \frac{d}{d\theta} = \frac{dx}{d\theta}\frac{d}{dx} = -\sin\theta\frac{d}{dx}. \]

When the equation is not fully expanded, the Sturm-Liouville form can be found: \[ \frac{d}{dx}\left((1-x^2)\frac{d}{dx}\right)P(x) = -\lambda P(x). \]

Rodrigues' Formula \[ P_n(x) = \frac{1}{2^nn!}\frac{d^n}{dx^n}(x^2 - 1)^n. \]

Taking derivatives of the generating function with respect to \( t \) yields the recursion formula: \[ (n+1)P_{n+1}(x) = (2n+1)xP_n(x) - nP_{n-1}(x). \]

\[ \int_{-1}^1P_m(x)P_n(x)\,dx = \frac{2}{2n+1}\delta_{mn}. \]

Canonical solutions of the general Legendre equation: \[ (1-x^2)\frac{d^2}{dx^2}P_\ell^m(x) - 2x\frac{d}{dx}P_\ell^m(x) + \left[ \ell(\ell+1) - \frac{m^2}{1-x^2}\right] P_\ell^m(x) = 0. \]

This appears in the eigenvalue equation of the Laplacian in spherical coordinates, in which the \( \phi \) dependence is not vanished.

For \( m\ge 0 \), \[ P_\ell^m(x) = (-1)^m(1-x^2)^{m/2}\frac{d^m}{dx^m}(P_\ell(x)). \]

For lower indices: \[ \int_{-1}^1 P_k^mP_\ell^m\,dx = \frac{2(\ell+m)!}{(2\ell + 1)(\ell - m)!}\delta_{k\ell}. \]

For upper indices: \[ \int_{-1}^1 \frac{P_\ell^mP_\ell^n}{1-x^2}\,dx = \begin{cases} 0 & \text{if $m\neq n$} \\ \frac{(\ell +m)!}{m(\ell - m)!} & \text{if $m=n\neq 0$} \\ \infty & \text{if $m=n=0$}.\end{cases} \]

\[ P_\ell^m(-x) = (-1)^{\ell - m}P_\ell^m(x). \]

Physicist's Hermite Equation: \[ H_n''(x) - 2xH_n'(x) + 2nH_n(x) = 0. \]

The differential equation for the harmonic oscillator: \[ y'' + (m-x^2)y = 0 \] contains the Hermite equation when the solution takes the form \( H_n(x)e^{-x^2/2} \), with \( m \) being \( 2n-1 \).

\[ H_n(x) = (-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}. \]

Rodrigues' Formula \[ H_n(x) = \left(2x - \frac{d}{dx}\right)^n1. \]

\[ H_n(-x) = (-1)^nH_n(x). \]

They are orthogonal with respect to the weight function \( w(x) = e^{-x^2} \): \[ \int_{-\infty}^\infty H_n(x)H_m(x)w(x)\,dx = \sqrt{\pi}2^nn!\delta_{nm}. \]

• \[ L_n(x) = \sum_{k=0}^n\binom{n}{k}\frac{(-1)^k}{k!}x^k. \]
• \[ L_n(x) = \frac{1}{n!}e^x\frac{d^n}{dx^n}(x^ne^{-x}). \]

\[ \begin{align*} L_0(x) &= 1\\ L_1(x) &= 1-x\\ L_{k+1}(x) &= \frac{(2k+1-x)L_k(x) - kL_{k-1}(x)}{k+1}. \end{align*} \]

• \[ \sum_{n=0}^\infty L_n(x)t^n = \frac{1}{1-t}e^{-tx/(1-t)}. \]