Editing Binomial theorem (section)

=== 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>.