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The probability generating function isFor independent random variables, the bounded factors $z^X$ and $z^Y$ have factorizing expectation, soThis includes complex $z$ in the indicated disk by absolute convergence.

For the dice construction, encode the labels of an ordinary die by $F(z)=z+z^2+\cdots +z^6$. Its probability generating function is $F(z)/6$. A useful factorization into polynomial factors isUse redistribution of polynomial factors for dice to defineBoth polynomials have nonnegative integer coefficients, no constant term, and coefficient sum six. They therefore describe fair six-sided dice with strictly positive face labelsWhen these dice are thrown with independent outcomes, the sum has probability generating function $A(z)B(z)/36=F(z{)}^2/36$. Equality of coefficients proves equality of every sum probability, including all totals from two to twelve, with zero probability outside that range. Both dice are nonstandard, completing the construction.