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Finite Difference Question

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You just say G(x) = f(x+1) - f(x)

G(x) is a polynomial, not a set.   And for specific values of x, G(x) is a number.

So G(x) = f(x+1)-f(x) represents both the statement that the polynomial G(x) is the same as the polynomial f(x+1)-f(x) and the statement that the number G(x) is the same as the number f(x+1)-f(x) for every number x.

You can also use the delta operator ∆ to denote the same thing.   For any function f(x), not just polynomial,  ∆f(x) means f(x+1)-f(x).   And then you can use that to denote the finite difference of a finite difference.  The thing you wrote above as G'(x) could be written as ∆^(2)f(x)

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