Is there a general form of Chebyshev expansion coefficients for Gaussian distribution

Question

$\newcommand{\chebyshevt}{\text{chebyshevt}}$ $\newcommand{\Norm}{\text{Norm}}$ I tried to calculate the coefficient for distribution $Norm(x, \mu, \sigma)$ via $$\int_{-1}^{1} \chebyshevt(x, t) \Norm(x, \mu, \sigma)/\sqrt{1-x^2}dx$$ in Wolfram but time exceeded. (t is an integral number). But if I ignore the parameters $$\int_{-1}^{1} \chebyshevt(x, t) \Norm(x, 0, 1)/\sqrt{1-x^2}dx$$ it is the expression including modified Bessel function of the first kind. So I wonder if a general form exists (with (modified) Bessel function).