Chebyshev Polynomials - Definition

Definition

The Chebyshev polynomials of the first kind are defined by the recurrence relation

\begin{align} T_0(x) & = 1 \\ T_1(x) & = x \\ T_{n+1}(x) & = 2xT_n(x) - T_{n-1}(x). \end{align}

The conventional generating function for Tn is

The exponential generating function is

The generating function relevant for 2-dimensional potential theory and multipole expansion is

The Chebyshev polynomials of the second kind are defined by the recurrence relation

\begin{align} U_0(x) & = 1 \\ U_1(x) & = 2x \\ U_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x). \end{align}

One example of a generating function for Un is

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