Let's recall two things.
1.
Weierstrass transform W(ξ) is defined by
$$W(ξ) = \frac{1}{\sqrt{4π}}\int_{-\infty}^{\infty} e^{-\frac{(ξ - x)^2}{4}}f(x) \,dx.$$, that is, the convolution of f with the Gaussian function.
2.
The probability density function of Student's t distribution with n degress of freedom is $$f_n(x) = c(1 + \frac{x^2}{n})^\frac{-(n+1)}{2}.$$ where c is a normalization constant $c = \frac{1}{\sqrt{n}B(\frac{n}{2},\frac{1}{2})}$ and B is the beta function.
Considering 1 and 2, we get $$W(ξ) = \frac{c}{\sqrt{4π}}\int_{-\infty}^{\infty} e^{-\frac{(ξ - x)^2}{4}}(1 + \frac{x^2}{n})^\frac{-(n+1)}{2}\,dx.$$
Please solve this. I’m very curious of what this will be. But it’s over my capabilities.