When one writes the fractional heat equation in self-similar variables a drift term appears. We study the associated eigenvalue problem for this equation, which has a fractional Laplacian and a first order term under competition. Our main contribution is to give explicit Euclidean formulae of the fractional analogue of Hermite polynomials. A crucial tool is the Mellin transform, which is essentially the Fourier transform in logarithmic variable and which turns the gradient into multiplication. This is joint work with Hardy Chan, Marco Fontelos and Juncheng Wei.