maximizes a certain free entropy functional. An easy application of Gibbs’ inequality shows that the classical GIG maximizes the same functional of classical entropy.

The paper is organized as follows: In Section 2 we shortly recall basics of free probability and next we study some properties of fGIG distributions. Section 3 is devoted to the study of free infinite divisibility, free regularity, free selfdecomposability and unimodality of the fGIG distribution. In Section 4 we show that the free counterpart of the characterization of GIG distribution by holds true, and we discuss entropy analogies between GIG and fGIG.

Free GIG distributions
======================

In this section we recall the definition of free GIG distribution and study basic properties of this distribution. In particular we study in detail the $R$-transform of fGIG distribution. Some of the properties established in this section will be crucial in the subsequent sections where we study free infinite divisibility of the free GIG distribution and characterization of the free GIG distribution. The free GIG distribution appeared for the first time (not under the name free GIG) as the almost sure weak limit of empirical spectral distribution of GIG matrices (see [@Fer06]).

Basics of free probability
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This paper 