Editing Binomial theorem (section)

== Generalizations ==

=== Newton's generalized binomial theorem ===
{{Main|Binomial series}}
Around 1665, [[Isaac Newton]] generalized the binomial theorem to allow real exponents other than nonnegative integers. (The same generalization also applies to [[complex number|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 {{mvar|r}}, one can define
<math display="block">{r \choose k}=\frac{r(r-1) \cdots (r-k+1)}{k!} =\frac{(r)_k}{k!},</math><!--This is not the same as \frac{r!}{k!(r−k)!}. Please do not change it.-->
where <math>(\cdot)_k</math> is the [[Pochhammer symbol]], here standing for a [[falling factorial]].  This agrees with the usual definitions when {{mvar|r}} is a nonnegative integer.  Then, if {{mvar|x}} and {{mvar|y}} are real numbers with {{math|{{abs|''x''}} > {{abs|''y''}}}},<ref name=convergence group=Note>This is to guarantee convergence. Depending on {{mvar|r}}, the series may also converge sometimes when {{math|1={{abs|''x''}} = {{abs|''y''}}}}.</ref> and {{mvar|r}} is any complex number, one has
<math display="block">\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}</math>

When {{mvar|r}} is a nonnegative integer, the binomial coefficients for {{math|1=''k'' > ''r''}} are zero, so this equation reduces to the usual binomial theorem, and there are at most {{math|1=''r'' + 1}} nonzero terms. For other values of {{mvar|r}}, the series typically has infinitely many nonzero terms.

For example, {{math|1=''r'' = 1/2}} gives the following series for the square root:
<math display="block">\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.</math>

Taking {{math|1=''r'' = &minus;1}}, the generalized binomial series gives the [[Geometric series#Sum|geometric series formula]], valid for {{math|{{abs|''x''}} < 1}}:
<math display="block">(1+x)^{-1} = \frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - x^5 + \cdots.</math>

More generally, with {{math|1=''r'' = −''s''}}, we have for {{math|{{abs|''x''}} < 1}}:<ref name=wolfram2>{{cite web| url=https://mathworld.wolfram.com/NegativeBinomialSeries.html|title=Negative Binomial Series|website=Wolfram MathWorld|last=Weisstein|first=Eric W.}}</ref>
<math display="block">\frac{1}{(1+x)^s} = \sum_{k=0}^\infty {-s \choose k} x^k = \sum_{k=0}^\infty {s+k-1 \choose k} (-1)^k x^k.</math>

So, for instance, when {{math|1=''s'' = 1/2}},
<math display="block">\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.</math>

Replacing {{mvar|x}} with {{mvar|-x}} yields:
<math display="block">\frac{1}{(1-x)^s} = \sum_{k=0}^\infty {s+k-1 \choose k} (-1)^k (-x)^k = \sum_{k=0}^\infty {s+k-1 \choose k} x^k.</math>

So, for instance, when {{math|1=''s'' = 1/2}}, we have for {{math|{{abs|''x''}} < 1}}:
<math display="block">\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.</math>

=== Further generalizations ===
The generalized binomial theorem can be extended to the case where {{mvar|x}} and {{mvar|y}} are complex numbers. For this version, one should again assume {{math|{{abs|''x''}} > {{abs|''y''}}}}<ref name=convergence group=Note /> and define the powers of {{math|1=''x'' + ''y''}} and {{mvar|x}} using a [[Holomorphic function|holomorphic]] [[complex logarithm|branch of log]] defined on an open disk of radius {{math|{{abs|''x''}}}} centered at {{mvar|x}}.  The generalized binomial theorem is valid also for elements {{mvar|x}} and {{mvar|y}} of a [[Banach algebra]] as long as {{math|1=''xy'' = ''yx''}}, and {{mvar|x}} is invertible, and {{math|{{norm|''y''/''x''}} < 1}}.

A version of the binomial theorem is valid for the following [[Pochhammer symbol]]-like family of polynomials: for a given real constant {{mvar|c}}, define <math> x^{(0)} = 1 </math> and
<math display="block"> x^{(n)} = \prod_{k=1}^{n}[x+(k-1)c]</math>
for <math> n > 0.</math>  Then<ref name="Sokolowsky">{{cite journal| url=https://cms.math.ca/publications/crux/issue/?volume=5&issue=2| title=Problem 352|first1=Dan|last1=Sokolowsky|first2=Basil C.|last2=Rennie|journal=Crux Mathematicorum|volume=5|issue=2|date=1979 | pages=55–56}}</ref>
<math display="block"> (a + b)^{(n)} = \sum_{k=0}^{n}\binom{n}{k}a^{(n-k)}b^{(k)}.</math>
The case {{math|1=''c'' = 0}} recovers the usual binomial theorem.

More generally, a sequence <math>\{p_n\}_{n=0}^\infty</math> of polynomials is said to be '''of binomial type''' if
* <math> \deg p_n = n </math> for all <math>n</math>,
* <math> p_0(0) = 1 </math>, and
* <math> p_n(x+y) = \sum_{k=0}^n \binom{n}{k} p_k(x) p_{n-k}(y) </math> for all <math>x</math>, <math>y</math>, and <math>n</math>.
An operator <math>Q</math> on the space of polynomials is said to be the ''basis operator'' of the sequence <math>\{p_n\}_{n=0}^\infty</math> if <math>Qp_0 = 0</math> and <math> Q p_n = n p_{n-1} </math> for all <math> n \geqslant 1 </math>. A sequence <math>\{p_n\}_{n=0}^\infty</math> is binomial if and only if its basis operator is a [[Delta operator]].<ref>{{cite book |last=Aigner |first=Martin |author-link=Martin Aigner |title=Combinatorial Theory |url=https://archive.org/details/combinatorialthe0000aign |url-access=limited |date=1979 |publisher=Springer |isbn=0-387-90376-3 |page=105 }}</ref> Writing <math> E^a </math> for the shift by <math> a </math> operator, the Delta operators corresponding to the above "Pochhammer" families of polynomials are the backward difference <math> I - E^{-c} </math> for <math> c>0 </math>, the ordinary derivative for <math> c=0 </math>, and the forward difference <math> E^{-c} - I </math> for <math> c<0 </math>.

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

<math display="block">(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}, </math>

where the summation is taken over all sequences of nonnegative integer indices {{math|''k''<sub>1</sub>}} through {{math|''k''<sub>''m''</sub>}} such that the sum of all {{math|''k''<sub>''i''</sub>}} is&nbsp;{{mvar|n}}. (For each term in the expansion, the exponents must add up to&nbsp;{{mvar|n}}). The coefficients <math> \tbinom{n}{k_1,\cdots,k_m} </math> are known as multinomial coefficients, and can be computed by the formula
<math display="block"> \binom{n}{k_1, k_2, \ldots, k_m} = \frac{n!}{k_1! \cdot k_2! \cdots k_m!}.</math>

Combinatorially, the multinomial coefficient <math>\tbinom{n}{k_1,\cdots,k_m}</math> counts the number of different ways to [[Partition of a set|partition]] an {{mvar|n}}-element set into [[Disjoint sets|disjoint]] [[subset]]s of sizes {{math|1=''k''<sub>1</sub>, ..., ''k''<sub>''m''</sub>}}.

=== {{anchor|multi-binomial}} 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
<math display="block"> (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}. </math>

This may be written more concisely, by [[multi-index notation]], as
<math display="block"> (x+y)^\alpha = \sum_{\nu \le \alpha} \binom{\alpha}{\nu} x^\nu y^{\alpha - \nu}.</math>

=== General Leibniz rule ===
{{Main|General Leibniz rule}}

The general Leibniz rule gives the {{mvar|n}}th derivative of a product of two functions in a form similar to that of the binomial theorem:<ref>{{cite book |last=Olver |first=Peter J. |author-link=Peter J. Olver |year=2000 |title=Applications of Lie Groups to Differential Equations |publisher=Springer |pages=318–319 |isbn=9780387950006 |url=https://books.google.com/books?id=sI2bAxgLMXYC&pg=PA318 }}</ref> 
<math display="block">(fg)^{(n)}(x) = \sum_{k=0}^n \binom{n}{k} f^{(n-k)}(x) g^{(k)}(x).</math>

Here, the superscript {{math|(''n'')}} indicates the {{mvar|n}}th derivative of a function, <math>f^{(n)}(x) = \tfrac{d^n}{dx^n}f(x)</math>.  If one sets {{math|1=''f''(''x'') = ''e''{{sup|''ax''}}}} and {{math|1=''g''(''x'') = ''e''{{sup|''bx''}}}}, cancelling the common factor of {{math|''e''{{sup|(''a'' + ''b'')''x''}}}} from each term gives the ordinary binomial theorem.<ref>{{cite book |last1=Spivey |first1=Michael Z. |title=The Art of Proving Binomial Identities |date=2019 |publisher=CRC Press |isbn=978-1351215800 |page=71}}</ref>