The Mathematica code for the integration of Gaussian function with Chebyshev polynomial is 



```
sigma = 1.0;
(1./(Sqrt[2*pi]*sigma))*Integrate[Exp[-(x^2)/(2*sigma^2)]*Cos[n*ArcCos[x]],{-inf,inf}]

```

But, unfortunately, the Mathematica software was unable to integrate.


Thank You



How can I integrate a Gaussian function with a combination of Chebyshev polynomials?
$$C\_n=(2\pi)^{-1/2}\int\_{-\infty}^\infty e^{-x^2/2}\cos[n\,{\rm arccos}\,(x)]\,dx$$


$C\_n=0$ for odd $n$, and for even $n$ it equals an integer:


$C\_0=1$, $C\_2=1$, $C\_4=17$, $C\_6=353$, $C\_8=10049$, $C\_{10}=365089$, $C\_{12}=16157329$, $C\_{14}=843550273$,...


No closed form expression for all $n$ that I am aware of, but there might well be one. (I checked that the series of integers does not give a match on [OEIS](http://oeis.org).) The generating function


$$F(t)=\sum\_{n=0}^\infty C\_n t^n=\int\_{-\infty}^\infty\frac{e^{-x^2/2} (1-t x)}{t^2-2 t x+1}\,dx$$
has a closed-form answer (in terms of the error function), but this does not readily yield the $C\_n$'s.


