Editing Chebyshev polynomials (section)

===Generating functions===

The [[generating function|ordinary generating function]] for {{mvar|T<sub>n</sub>}} is
<math display="block">\sum_{n=0}^\infty T_n(x)\,t^n = \frac{1 - tx}{1 - 2tx + t^2}.</math>

There are several other [[generating function]]s for the Chebyshev polynomials; the [[exponential generating function]] is
<math display="block">\begin{align}
\sum_{n=0}^\infty T_n(x) \frac{t^n}{n!}
  &= {\tfrac{1}{2}} \Bigl({\exp}\Bigl({\textstyle t\bigl(x - \sqrt{x^2 - 1}~\!\bigr)}\Bigr)
                      + {\exp}\Bigl({\textstyle t\bigl(x + \sqrt{x^2 - 1}~\!\bigr)}\Bigr)\Bigr) \\
  &= e^{tx} \cosh\left({\textstyle t\sqrt{x^2 - 1} }~\! \right).
\end{align}</math>

The generating function relevant for 2-dimensional [[potential theory]] and [[Cylindrical multipole moments|multipole expansion]] is
<math display="block">\sum\limits_{n=1}^\infty T_{n}(x)\,\frac{t^n}{n} = \ln\left(\frac{1}{\sqrt{1 - 2tx + t^2 }}\right).</math>

The ordinary generating function for {{mvar|U<sub>n</sub>}} is
<math display="block">\sum_{n=0}^\infty U_n(x)\,t^n = \frac{1}{1 - 2tx + t^2},</math>
and the exponential generating function is
<math display="block">
 \sum_{n=0}^\infty U_n(x) \frac{t^n}{n!} = e^{tx} \biggl(\!\cosh\left(t\sqrt{x^2 - 1}\right) + \frac{x}{\sqrt{x^2 - 1}} \sinh\left(t\sqrt{x^2 - 1}\right)\biggr).
</math>