ChebyshevPoynomials

The key differences between Chebyshev polynomials of the first and second kind are:

Definition and Form

First Kind (Tn):

• Defined by the relation $$T_n(\cos \theta) = \cos(n\theta)$$
• Initial terms: $$T_0(x) = 1, T_1(x) = x$$
• Recurrence relation: $$T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$$

Second Kind (Un):

• Defined by $$U_n(\cos \theta)\sin \theta = \sin((n+1)\theta)$$
• Initial terms: $$U_0(x) = 1, U_1(x) = 2x$$
• Recurrence relation: $$U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)$$

Trigonometric Connection

• First kind polynomials are related to cosine functions
• Second kind polynomials are connected to sine functions

Orthogonality

A significant difference lies in their weight functions:

• First kind: orthogonal with respect to $$\frac{1}{\sqrt{1-x^2}}$$ on [-1,1]
• Second kind: orthogonal with respect to $$\sqrt{1-x^2}$$ on [-1,1][1]