Chebyshev Expansions | M.E. Irizarry-Gelpí

Fri 16 September 2016
Maths
#Chebyshev

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.