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We propose a Stein characterization of the Kummer distribution on (0,$\infty$). This result follows from our observation that the density of theKummer distribution satisfies a certain differential equation, leading to asolution of the related Stein equation. A bound is derived for the solution,under a condition on the parameters. The derivation of this bound is carriedout using the same framework as in Gaunt 2017 [A Stein characterisation of thegeneralized hyper-bolic distribution. ESAIM: Probability and Statistics, 21,303--316] in the case of the generalized inverse Gaussian distribution, whichwe revisit by correcting a minor error in the latter paper.

About the Stein equation for the generalized inverse Gaussian and Kummer distributions