  We study the Stein equation associated with the one-dimensional Gamma
distribution, and provide novel bounds, allowing one to effectively deal with
test functions supported by the whole real line. We apply our estimates to
derive new quantitative results involving random variables that are non-linear
functionals of random fields, namely: (i) a non-central quantitative de Jong
theorem for sequences of degenerate U-statistics satisfying minimal uniform
integrability conditions, significantly extending previous findings by de Jong
(1990), Nourdin, Peccati and Reinert (2010) and D\"obler and Peccati (2016),
(ii) a new Gamma approximation bound on the Poisson space, refining previous
estimates by Peccati and Th\"ale (2013), and (iii) new Gamma bounds on a
Gaussian space, strengthening estimates by Nourdin and Peccati (2009). As a
by-product of our analysis, we also deduce a new inequality for Gamma
approximations via exchangeable pairs, that is of independent interest.
