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4. Binomial Formula
 
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In the expression (1.2) the exponentian function allows the substitution of the binomial formula as done in (2.1), but this is not possible for any given function. For applying this substitution we require the following displacement operator,

(4.1)

whose iteration yields

(4.2)

what allows the application of the binomial formula (1.6) for natural numbers and the generalized binomial formula (1.7) for complex numbers,

(4.3)

so that for any complex number a it can be generalized

(4.4)

This sheds more light on the derivative and its generalization. Using the expression of the generalized binomial formula (1.7) for non-integer numbers,

(4.5)

In the case of integer values the summatory only extends a terms and it is equal to the ordinary derivative.

Finally, it is obvious that as h goes to 0 the last equation is equivalent to the following

(4.6)