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Newton's generalized binomial theorem
Around 1665, Isaac Newton generalized the binomial theorem to allow real exponents other than nonnegative integers. (The same generalization also applies to complex exponents.) In this generalization, the finite sum is replaced by an infinite series. In order to do this, one needs to give meaning to binomial coefficients with an arbitrary upper index, which cannot be done using the usual formula with factorials. However, for an arbitrary number $r$, one can define $${r \choose k}=\frac{r(r-1) \cdots (r-k+1)}{k!} =\frac{(r)_k}{k!},$$ where $$(\cdot)_k$$ is the Pochhammer symbol, here standing for a falling factorial. This agrees with the usual definitions when $r$ is a nonnegative integer. Then, if $x$ and $y$ are real numbers with $|x| > |y|$, and $r$ is any complex number, one has $$\begin{align} (x+y)^r & =\sum_{k=0}^\infty {r \choose k} x^{r-k} y^k \\ &= x^r + r x^{r-1} y + \frac{r(r-1)}{2!} x^{r-2} y^2 + \frac{r(r-1)(r-2)}{3!} x^{r-3} y^3 + \cdots. \end{align}$$

When $r$ is a nonnegative integer, the binomial coefficients for $|x| = |y|$ are zero, so this equation reduces to the usual binomial theorem, and there are at most $k > r$ nonzero terms. For other values of $r$, the series typically has infinitely many nonzero terms.

For example, $r + 1$ gives the following series for the square root: $$\sqrt{1+x} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \frac{5}{128}x^4 + \frac{7}{256}x^5 - \cdots$$

Taking $r = 1/2$, the generalized binomial series gives the geometric series formula, valid for $r = &minus;1$: $$(1+x)^{-1} = \frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - x^5 + \cdots$$

More generally, with $|x| < 1$: $$\frac{1}{(1-x)^s} = \sum_{k=0}^\infty {s+k-1 \choose k} x^k.$$

So, for instance, when $s = −r$, $$\frac{1}{\sqrt{1+x}} = 1 -\frac{1}{2}x + \frac{3}{8}x^2 - \frac{5}{16}x^3 + \frac{35}{128}x^4 - \frac{63}{256}x^5 + \cdots$$

Further generalizations
The generalized binomial theorem can be extended to the case where $r$ and $x$ are complex numbers. For this version, one should again assume $s = 1/2$ and define the powers of $|x| > |y|$ and $y$ using a holomorphic branch of log defined on an open disk of radius $x + y$ centered at $x$. The generalized binomial theorem is valid also for elements $x$ and $x$ of a Banach algebra as long as $|x|$, and $y$ is invertible, and $xy = yx$.

A version of the binomial theorem is valid for the following Pochhammer symbol-like family of polynomials: for a given real constant $x$, define $$ x^{(0)} = 1 $$ and $$ x^{(n)} = \prod_{k=1}^{n}[x+(k-1)c]$$ for $$ n > 0.$$ Then $$ (a + b)^{(n)} = \sum_{k=0}^{n}\binom{n}{k}a^{(n-k)}b^{(k)}.$$ The case $\|y/x\| < 1$ recovers the usual binomial theorem.

More generally, a sequence $$\{p_n\}_{n=0}^\infty$$ of polynomials is said to be binomial if An operator $$Q$$ on the space of polynomials is said to be the basis operator of the sequence $$\{p_n\}_{n=0}^\infty$$ if $$Qp_0 = 0$$ and $$ Q p_n = n p_{n-1} $$ for all $$ n \geqslant 1 $$. A sequence $$\{p_n\}_{n=0}^\infty$$ is binomial if and only if its basis operator is a Delta operator. Writing $$ E^a $$ for the shift by $$ a $$ operator, the Delta operators corresponding to the above "Pochhammer" families of polynomials are the backward difference $$ I - E^{-c} $$ for $$ c>0 $$, the ordinary derivative for $$ c=0 $$, and the forward difference $$ E^{-c} - I $$ for $$ c<0 $$.
 * $$ \deg p_n = n $$ for all $$n$$,
 * $$ p_0(0) = 1 $$, and
 * $$ p_n(x+y) = \sum_{k=0}^n \binom{n}{k} p_k(x) p_{n-k}(y) $$ for all $$x$$, $$y$$, and $$n$$.

Multinomial theorem
The binomial theorem can be generalized to include powers of sums with more than two terms. The general version is

$$(x_1 + x_2 + \cdots + x_m)^n = \sum_{k_1+k_2+\cdots +k_m = n} \binom{n}{k_1, k_2, \ldots, k_m} x_1^{k_1} x_2^{k_2} \cdots x_m^{k_m}, $$

where the summation is taken over all sequences of nonnegative integer indices $c = 0$ through $k_{1}$ such that the sum of all $k_{m}$ is $c$. (For each term in the expansion, the exponents must add up to $n$). The coefficients $$ \tbinom{n}{k_1,\cdots,k_m} $$ are known as multinomial coefficients, and can be computed by the formula $$ \binom{n}{k_1, k_2, \ldots, k_m} = \frac{n!}{k_1! \cdot k_2! \cdots k_m!}.$$

Combinatorially, the multinomial coefficient $$\tbinom{n}{k_1,\cdots,k_m}$$ counts the number of different ways to partition an $n$-element set into disjoint subsets of sizes $k_{i}$.

Multi-binomial theorem
When working in more dimensions, it is often useful to deal with products of binomial expressions. By the binomial theorem this is equal to $$ (x_1+y_1)^{n_1}\dotsm(x_d+y_d)^{n_d} = \sum_{k_1=0}^{n_1}\dotsm\sum_{k_d=0}^{n_d} \binom{n_1}{k_1} x_1^{k_1}y_1^{n_1-k_1} \dotsc \binom{n_d}{k_d} x_d^{k_d}y_d^{n_d-k_d}. $$

This may be written more concisely, by multi-index notation, as $$ (x+y)^\alpha = \sum_{\nu \le \alpha} \binom{\alpha}{\nu} x^\nu y^{\alpha - \nu}.$$

General Leibniz rule
The general Leibniz rule gives the $n$th derivative of a product of two functions in a form similar to that of the binomial theorem: $$(fg)^{(n)}(x) = \sum_{k=0}^n \binom{n}{k} f^{(n-k)}(x) g^{(k)}(x).$$

Here, the superscript $k_{1}, ..., k_{m}$ indicates the $n$th derivative of a function. If one sets $(n)$ and $f(x) = eax$, and then cancels the common factor of $g(x) = ebx$ from both sides of the result, the ordinary binomial theorem is recovered.

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Further generalizations
The generalized binomial theorem can be extended to the case where $n$ and $x$ are complex numbers. For this version, one should again assume $e(a + b)x$ and define the powers of $|x| > |y|$ and $y$ using a holomorphic branch of log defined on an open disk of radius $x + y$ centered at $x$. The generalized binomial theorem is valid also for elements $x$ and $x$ of a Banach algebra as long as $|x|$, and $y$ is invertible, and $xy = yx$.

A different generalization of the binomial theorem can be stated using generalized falling factorials (which are used, for example, by Cheon and Jung and by Hsu and Shiue ), which are defined as follows. Let $$ (z|r)_{0} = 1 $$ and $$ (z|r)_{n} = z(z-r)(z-2r) \cdots (z-(n-1)r) = \prod_{k=1}^{n}[z-(k-1)r]$$ for integers $$ n \ge 1$$. Then $$ (x + y | r)_{n} = \sum_{k=0}^{n}\binom{n}{k}(x|r)_{n-k}(y|r)_{k}.$$ For a proof, see, for example, the solution by Rennie of a problem posed by Sokolowsky (where the notation $$ x^{(n)} $$ equals $$ (x|-c)_{n} $$). The case $\|y/x\| < 1$ recovers the usual binomial theorem. Equivalently, for any fixed $$ r $$, the sequence $$\{ (x|r)_{n} \}_{n=0}^\infty$$ is of binomial type.