Special Polynomial Families

Classical orthogonal polynomial families including Legendre, Chebyshev (1st and 2nd kind), Hermite, and Laguerre polynomials with both symbolic expansion and numerical evaluation capabilities.

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Mathematical Definition

Orthogonal Polynomials: A sequence $\{P_n(x)\}$ satisfying orthogonality relation:

\int_a^b P_n(x) P_m(x) w(x) \, dx = 0 \quad \text{for } n \neq m

where $w(x)$ is the weight function on interval $[a, b]$ .

Three-Term Recurrence: All orthogonal polynomials satisfy:

P_{n+1}(x) = (a_n x + b_n) P_n(x) - c_n P_{n-1}(x)

Family Definitions:

1. Legendre: Interval $[-1, 1]$ , $w(x) = 1$ - Differential equation: $(1-x^2)P_n'' - 2xP_n' + n(n+1)P_n = 0$ - Recurrence: $P_{n+1} = \frac{(2n+1)xP_n - nP_{n-1}}{n+1}$

2. Chebyshev (1st): Interval $[-1, 1]$ , $w(x) = \frac{1}{\sqrt{1-x^2}}$ - Definition: $T_n(\cos\theta) = \cos(n\theta)$ - Recurrence: $T_{n+1} = 2xT_n - T_{n-1}$

3. Chebyshev (2nd): Interval $[-1, 1]$ , $w(x) = \sqrt{1-x^2}$ - Definition: $U_n(\cos\theta) = \frac{\sin((n+1)\theta)}{\sin\theta}$ - Recurrence: $U_{n+1} = 2xU_n - U_{n-1}$

4. Hermite: Interval $(-\infty, \infty)$ , $w(x) = e^{-x^2}$ - Differential equation: $H_n'' - 2xH_n' + 2nH_n = 0$ - Rodriguez formula: $H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n}(e^{-x^2})$ - Recurrence: $H_{n+1} = 2xH_n - 2nH_{n-1}$

5. Laguerre: Interval $[0, \infty)$ , $w(x) = e^{-x}$ - Differential equation: $xL_n'' + (1-x)L_n' + nL_n = 0$ - Recurrence: $L_{n+1} = \frac{(2n+1-x)L_n - nL_{n-1}}{n+1}$

Code Examples

Legendre Polynomials

Solutions to Legendre's differential equation

use mathhook_core::core::polynomial::special_families::Legendre;
use mathhook_core::core::polynomial::special_families::OrthogonalPolynomial;
use mathhook_core::symbol;

let x = symbol!(x);

// Symbolic expansion
let p0 = Legendre::polynomial(0, &x);  // 1
let p1 = Legendre::polynomial(1, &x);  // x
let p2 = Legendre::polynomial(2, &x);  // (3x^2 - 1)/2

// Numerical evaluation
let val = Legendre::evaluate(2, 0.5);  // P_2(0.5) = -0.125

// Recurrence: P_{n+1} = ((2n+1)x*P_n - n*P_{n-1}) / (n+1)
let (a, b, c) = Legendre::recurrence_coefficients(2);

Chebyshev Polynomials (First Kind)

Defined by T_n(cos(theta)) = cos(n*theta)

use mathhook_core::core::polynomial::special_families::ChebyshevT;
use mathhook_core::core::polynomial::special_families::OrthogonalPolynomial;
use mathhook_core::symbol;

let x = symbol!(x);

// Symbolic
let t0 = ChebyshevT::polynomial(0, &x);  // 1
let t1 = ChebyshevT::polynomial(1, &x);  // x
let t2 = ChebyshevT::polynomial(2, &x);  // 2x^2 - 1

// Numerical
let val = ChebyshevT::evaluate(2, 0.5);  // T_2(0.5) = -0.5

// Recurrence: T_{n+1} = 2x*T_n - T_{n-1}

Hermite Polynomials

Solutions to Hermite's equation (physicist's convention)

use mathhook_core::core::polynomial::special_families::Hermite;
use mathhook_core::core::polynomial::special_families::OrthogonalPolynomial;
use mathhook_core::symbol;

let x = symbol!(x);

// Symbolic
let h0 = Hermite::polynomial(0, &x);  // 1
let h1 = Hermite::polynomial(1, &x);  // 2x
let h2 = Hermite::polynomial(2, &x);  // 4x^2 - 2

// Numerical
let val = Hermite::evaluate(1, 0.5);  // H_1(0.5) = 1

// Recurrence: H_{n+1} = 2x*H_n - 2n*H_{n-1}

Laguerre Polynomials

Solutions to Laguerre's equation

use mathhook_core::core::polynomial::special_families::Laguerre;
use mathhook_core::core::polynomial::special_families::OrthogonalPolynomial;
use mathhook_core::symbol;

let x = symbol!(x);

// Symbolic
let l0 = Laguerre::polynomial(0, &x);  // 1
let l1 = Laguerre::polynomial(1, &x);  // 1 - x
let l2 = Laguerre::polynomial(2, &x);  // (x^2 - 4x + 2)/2

// Numerical
let val = Laguerre::evaluate(1, 0.5);  // L_1(0.5) = 0.5

// Recurrence: L_{n+1} = ((2n+1-x)*L_n - n*L_{n-1}) / (n+1)

Variable Substitution

Use any variable symbol in polynomial generation

use mathhook_core::core::polynomial::special_families::Legendre;
use mathhook_core::core::polynomial::special_families::OrthogonalPolynomial;
use mathhook_core::symbol;

// Use variable t instead of x
let t = symbol!(t);
let p2_t = Legendre::polynomial(2, &t);
// Result uses t: (3t^2 - 1)/2

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