M.E. Irizarry-Gelpí

Physics impostor. Mathematics interloper. Husband. Father.

Chebyshev Expansions


The ordinary generating functions for Chebyshev polynomials \(T_{n}(x)\) and \(U_{n}(x)\) are

$$ \frac{1 - x r}{1 - 2xr + r^{2}} = \sum_{n = 0}^{\infty} r^{n} T_{n}(x), \qquad \vert x \vert \leq 1, \qquad \vert r \vert < 1; $$
$$ \frac{1}{1 - 2xr + r^{2}} = \sum_{n = 0}^{\infty} r^{n} U_{n}(x), \qquad \vert x \vert \leq 1, \qquad \vert r \vert < 1. $$

But these expressions can also be understood as Chebyshev expansions of the functions. If you need to know the expansion in the region \(\vert r \vert > 1\), then you need to use

$$ \frac{1 - x r}{1 - 2xr + r^{2}} = \frac{1 - x \rho}{1 - 2x\rho + \rho^{2}} + \left( \frac{1}{r^{2}} - 1 \right) \left( \frac{1}{1 - 2x\rho + \rho^{2}} \right); $$
$$ \frac{1}{1 - 2xr + r^{2}} = \frac{1}{r^{2}}\frac{1}{1 - 2x\rho + \rho^{2}}. $$

Here \(\rho = 1 / r\). Thus, you also have

$$ \frac{1 - x r}{1 - 2xr + r^{2}} = \sum_{n = 0}^{\infty} \left( \frac{1}{r} \right)^{n} T_{n}(x) + \left( \frac{1}{r^{2}} - 1 \right) \sum_{n = 0}^{\infty} \left( \frac{1}{r} \right)^{n} U_{n}(x), \qquad \vert x \vert \leq 1, \qquad \vert r \vert > 1; $$
$$ \frac{1}{1 - 2xr + r^{2}} = \frac{1}{r^{2}} \sum_{n = 0}^{\infty} \left( \frac{1}{r} \right)^{n} U_{n}(x), \qquad \vert x \vert \leq 1, \qquad \vert r \vert > 1. $$

This is similar to the trick used in the multipole expansion.