  We investigate a family of distributions having a property of
stability-under-addition, provided that the number $\nu$ of added-up random
variables in the random sum is also a random variable. We call the
corresponding property a \,$\nu$-stability and investigate the situation with
the semigroup generated by the generating function of $\nu$ is commutative.
Using results from the theory of iterations of analytic functions, we show that
the characteristic function of such a $\nu$-stable distribution can be
represented in terms of Chebyshev polynomials, and for the case of $\nu$-normal
distribution, the resulting characteristic function corresponds to the
hyperbolic secant distribution. We discuss some specific properties of the
class and present particular examples.
