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The Fractional Calculus is currently a very important research field in several different areas: physics (including classical and quantum mechanics and thermodynamics), chemistry, biology, economics and control theory ([5], [6], [7]). In 1989, K.S.Miller and Ross [8] introduced the discrete analogue of the Riemann-Liouville fractional derivative and proved some properties of the fractional difference operator. The main definition of fractional difference equation (as done in [8]) is the ν fractional sum of f(t) by

BERNOULLI'S n-FORMULA AND n${\_}$MULTI-SERIES BY THE GENERALIZED DIFFERENCE EQUATION