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The probability distribution of a discrete scheme maximizing the entropy of degree $\beta$, the average value of a random variable being fixed, is proved to converge as $\beta \to 1$ to the Boltzmann distribution which maximizes, under the same conditions, the Shannon entropy.

The solution is pointed out of the generalized associativity equation $$\phi (p_1+p_2,p_3,\phi (p_1,p_2,u_1,u_2),u_3)=\phi (p_1,p_2+p_3,u_1,\phi (p_2,p_3,u_2,u_3))$$ within suitable assumptions of continuity, monotonicity and idempotence.

The probability distribution of a discrete scheme is determined which maximizes the entropy of degree $\beta$, the average value of a given random variable being fixed.