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In this paper a condition is presented on parameters (n1,n2,Λk), for arbitrary partition n={n1, n2}, (n1≤ n2) and k=1,...,n-1, which guarantees that two different interpretations of characters of fundamental modules L(Λk) for the affine Kac-Moody Lie algebra generate extended classical Gauss series-product identities

The generating condition for the extension of the classical Gauss series-product identity