Poisson distribution
Some transforms of this law

We give values of some important transforms of the free Poisson law; the computation can be found in e.g. in the book Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher

The R-transform of the free Poisson law is given by R (z) = λ α 1 − α z. {\displaystyle R(z)={\frac {\lambda \alpha }{1-\alpha z}}.}

The Cauchy transform (which is the negative of the Stieltjes transformation) is given by G (z) = z + α − λ α − (z − α (1 + λ)) 2 − 4 λ α 2 2 α z {\displaystyle G(z)={\frac {z+\alpha -\lambda \alpha -{\sqrt {(z-\alpha (1+\lambda))^{2}-4\lambda \alpha ^{2}}}}{2\alpha z}}}

The S-transform is given by S (z) = 1 z + λ {\displaystyle S(z)={\frac {1}{z+\lambda }}} in the case that α = 1. {\displaystyle \alpha =1.}
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