Binomial Theorem/Abel's Generalisation

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Theorem

$\ds \paren {x + y}^n = \sum_k \binom n k x \paren {x - k z}^{k - 1} \paren {y + k z}^{n - k}$

for $n \in \Z_{\ge 0}$ and $x \in \R_{\ne 0}$.


Special Case: $x + y = 0$

Consider Abel's Generalisation of Binomial Theorem:

$\ds \paren {x + y}^n = \sum_k \binom n k x \paren {x - k z}^{k - 1} \paren {y + k z}^{n - k}$


This holds in the special case where $x + y = 0$.


Negative $n$

Abel's Generalisation of Binomial Theorem:

$\ds \paren {x + y}^n = \sum_k \binom n k x \paren {x - k z}^{k - 1} \paren {y + k z}^{n - k}$

does not hold for $n \in \Z_{< 0}$.


Proof 1

By admitting $y = \paren {x + y} - x$, we have that:

$\paren {x + y}^n = \paren {x + \paren {x + y} - x}^n$


Expanding the right hand side in powers of $\paren {x + y}$:

\(\ds \) \(\) \(\ds \sum_k \binom n k x \paren {x - k z}^{k - 1} \paren {y + k z}^{n - k}\)
\(\ds \) \(=\) \(\ds \sum_k \binom n k x \paren {x - k z}^{k - 1} \paren {x + \paren {x + y} + k z}^{n - k}\)
\(\ds \) \(=\) \(\ds \sum_k \binom n k x \paren {x - k z}^{k - 1} \sum_j \paren {x + y}^j \paren {-x + k z}^{n - k - j} \binom {n - k} j\)
\(\ds \) \(=\) \(\ds \sum_j \binom n j \paren {x + y})^j \sum_k \binom {n - j} {n - j - k} x \paren {x - k z}^{k - 1} \paren {-x + k z}^{n - k - j}\)
\(\ds \) \(=\) \(\ds \sum_{j \mathop \le n} \binom n j \paren {x + y}^j 0^{n - j}\) Abel's Generalisation of Binomial Theorem: Special Case $x + y = 0$
\(\ds \) \(=\) \(\ds \paren {x + y}^n\) Binomial Theorem

$\blacksquare$


Proof 2

From this formula:

$(1): \quad \ds \sum_{k \mathop \in \Z} \binom n k x \paren {x + k z}^{k - 1} y \paren {y + \paren {n - k} z}^{n - k - 1} = \paren {x + y} \paren {x + y + n z}^{n - 1}$

The given formula:

$\ds \paren {x + y}^n = \sum_k \binom n k x \paren {x - k z}^{k - 1} \paren {y + k z}^{n - k}$

can then be transformed into $(1)$ by


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Proof 3

From Hurwitz's Generalisation of Binomial Theorem:

$(1): \quad \paren {x + y}^n = \ds \sum x \paren {x + \epsilon_1 z_1 + \cdots + \epsilon_n z_n}^{\epsilon_1 + \cdots + \epsilon_n - 1} \paren {y - \epsilon_1 z_1 - \cdots - \epsilon_n z_n}^{n - \epsilon_1 - \cdots - \epsilon_n}$

Setting $z = z_1 = z_2 = \cdots z_n$ we have:

\(\ds \) \(\) \(\ds \sum x \paren {x + \epsilon_1 z + \cdots + \epsilon_n z}^{\epsilon_1 + \cdots + \epsilon_n - 1} \paren {y - \epsilon_1 z - \cdots - \epsilon_n z}^{n - \epsilon_1 - \cdots - \epsilon_n}\)
\(\ds \) \(=\) \(\ds \sum \binom n k x \paren {x + k z}^{k - 1} \paren {y - k z}^{n - k}\) where $\epsilon_1 + \cdots + \epsilon_n = k$
\(\ds \) \(=\) \(\ds \sum \binom n {n - k} x \paren {x + k z}^{k - 1} \paren {y - k z}^{n - k}\) Symmetry Rule for Binomial Coefficients
\(\ds \) \(=\) \(\ds \sum \binom n k x \paren {x - k z}^{k - 1} \paren {y + k z}^{n - k}\)

Hence the result.


Source of Name

This entry was named for Niels Henrik Abel.


Historical Note

Niels Henrik Abel presented this generalisation of the Binomial Theorem in $1826$ in the first volume of August Leopold Crelle's Journal für die reine und angewandte Mathematik.


Sources

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