Variance function of boolean additive convolution

Suppose
is the pseudo-variance function of the Cauchy-Stieltjes Kernel (CSK) family
generated by a non degenerate probability measure
with support bounded from above. We determine the formula for pseudo-variance function (or variance function
in case of existence) under boolean additive convolution power. This formula is used to identify the relation between variance functions under Boolean Bercovici–Pata Bijection between probability measures. We also give the connection between boolean cumulants and variance function and we relate boolean cumulants of some probability measures to Catalan numbers and Fuss Catalan numbers.