Generalized binomial

The formula of the binomial generalized makes it possible to develop a real or complex power of a sum of two terms in the form of a sum of series and generalizes the Formule of the binomial theorem.

We have for all realities or complex R , X and there ( there ≠ 0) such as | X / there |<1,

$(x+y)\; ^r=\; \backslash \; sum\_\; \{k=0\}\; ^\; \backslash \; infty\; \{R\; \backslash \; choose\; K\}\; x^k\; y^\; \{r-k\}$

where $\{R\; \backslash \; choose\; K\}\; =\; \backslash \; frac\; \{R\; (r-1)\; (r-2)\; \backslash \; ldots\; (r-k+1)\}\{K!\}$ is a binomial Coefficient.

(which if K = 0 is a Produit vacuum and thus equal to 1, and if K = 1 is equal to R , additional factors (R - 1), etc, not appearing in this case).

The corresponding series is convergent and the equality remains valid all the times that the real numbers or complex X and are there in a report/ratio of module strictly lower than 1.

The nap of a geometrical series is a particular case of the formula obtained by taking there = 1 and R = -1.

The formula remains also valid for elements X and of a Algèbre of Banach, which commutates ( xy = yx ), such as is invertible there there and || X .y^-1||< 1.

See too

• Formula of the binomial theorem
• Formula of the negative binomial

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