Abstract
We express the Jacobi sequences of the square of a real valued random variable with all moments, not necessarily symmetric, as functions of the corresponding sequences of the random variable itself. In the symmetric case, the result is known and, we give a short, purely algebraic proof of it. We apply our result to the square of the Gamma distribution, i.e. the 4th power of the standard Gaussian. The result confirms the conjecture that
belongs to the polynomial class, but its principal Jacobi sequence grows like
, not
as expected.