Special function expression for the (truncated) moment of the generalized Gamma distribution

Question

The, kind of, truncated moment of the generalized Gamma distribution for positive $a,d$ and $p$ is $$\int_0^\infty (x-k)_+^a x^{d-1}e^{-x^p}dx=\frac1p\int_0^\infty (y^{\frac1p}-k)_+^a\,y^{\frac dp-1}e^{-y}\,dy. \tag1$$ The truncated moment of the Gamma distribution can be transformed into the confluent hypergeometric function of the second kind defined as such. Is there a similar special function transform for integral $(1)$?