Editing Falling and rising factorials (section)

== Connection coefficients and identities ==

Falling and rising factorials are closely related to [[Stirling number|Stirling numbers]]. Indeed, expanding the product reveals [[Stirling numbers of the first kind]]<math display="block">
\begin{align}
(x)_n & = \sum_{k=0}^n s(n,k) x^k \\
      &= \sum_{k=0}^n \begin{bmatrix}n \\ k \end{bmatrix} (-1)^{n-k}x^k \\
x^{(n)} & = \sum_{k=0}^n \begin{bmatrix} n \\ k \end{bmatrix} x^k \\
\end{align}
</math>

And the inverse relations uses [[Stirling numbers of the second kind]]<math display="block">
\begin{align}
x^n & = \sum_{k=0}^{n} \begin{Bmatrix} n \\ k \end{Bmatrix} (x)_{k} \\
    & = \sum_{k=0}^n \begin{Bmatrix} n \\ k \end{Bmatrix} (-1)^{n-k} x^{(k)} .
\end{align} 
</math>

The falling and rising factorials are related to one another through the [[Lah numbers|Lah numbers <math display="inline">L(n, k) = \binom{n-1}{k-1} \frac{n!}{k!} </math>]]:<ref name="Wolfram_functions">
{{cite web |title=Introduction to the factorials and binomials |url=http://functions.wolfram.com/GammaBetaErf/Factorial/introductions/FactorialBinomials/05/ |website=Wolfram Functions Site}}
</ref><math display="block">
\begin{align}
x^{(n)} & = \sum_{k=0}^n L(n,k) (x)_k \\
(x)_n & = \sum_{k=0}^n L(n,k) (-1)^{n-k} x^{(k)}
\end{align}
</math>

Since the falling factorials are a basis for the [[polynomial ring]], one can express the product of two of them as a [[linear combination]] of falling factorials:<ref>{{cite journal|last1=Rosas|first1=Mercedes H.|year= 2002|title=Specializations of MacMahon symmetric functions and the polynomial algebra |volume=246|number=1–3|journal=Discrete Math.|doi=10.1016/S0012-365X(01)00263-1|pages=285–293|hdl=11441/41678 |hdl-access=free}}</ref>
<math display="block">
(x)_m (x)_n = \sum_{k=0}^m \binom{m}{k} \binom{n}{k} k! \cdot (x)_{m+n-k} \ .</math>

The coefficients <math>\tbinom{m}{k} \tbinom{n}{k} k! </math> are called ''connection coefficients'', and have a combinatorial interpretation as the number of ways to identify (or "glue together") {{mvar|k}} elements each from a set of size {{mvar|m}} and a set of size {{mvar|n}}.

There is also a connection formula for the ratio of two rising factorials given by
<math display="block">
\frac{x^{(n)}}{x^{(i)}} = (x+i)^{(n-i)} ,\quad \text{for }n \geq i .</math>

Additionally, we can expand generalized exponent laws and negative rising and falling powers through the following identities:<ref name="Graham-Knuth-Patashnik-1988">
{{cite book
 |first1=Ronald L. |last1=Graham    |author1-link=Ronald L. Graham
 |first2=Donald E. |last2=Knuth     |author2-link=Donald E. Knuth
 |first3=Oren      |last3=Patashnik |author3-link=Oren Patashnik
 |name-list-style=amp
 |year=1988
 |title=[[Concrete Mathematics]]
 |publisher=Addison-Wesley
 |place=Reading, MA
 |isbn=0-201-14236-8
 |pages=47, 48, 52
}}
</ref>{{rp|style=ama|p= 52}}

<math display="block">
\begin{align} 
(x)_{m+n} & = (x)_m (x-m)_n = (x)_n (x-n)_m \\[6pt]
x^{(m+n)} & = x^{(m)} (x+m)^{(n)} = x^{(n)} (x+n)^{(m)} \\[6pt]
x^{(-n)} & = \frac{\Gamma(x-n)}{\Gamma(x)} = \frac{(x-n-1)!}{(x-1)!} = \frac{1}{(x-n)^{(n)}} = \frac{1}{(x-1)_n} = \frac{1}{(x-1)(x-2) \cdots (x-n)} \\[6pt]
(x)_{-n} & = \frac{\Gamma(x+1)}{\Gamma(x+n+1)} = \frac{x!}{(x+n)!} = \frac{1}{(x+n)_n} = \frac{1}{(x+1)^{(n)}} = \frac{1}{(x+1)(x+2) \cdots (x+n)}
\end{align} 
</math>

Finally, [[duplication formula|duplication]] and [[multiplication formula]]s for the falling and rising factorials provide the next relations:
<math display="block">
\begin{align}
(x)_{k+mn} &= x^{(k)} m^{mn} \prod_{j=0}^{m-1} \left(\frac{x-k-j}{m}\right)_{n}\,,& \text{for } m &\in \mathbb{N} \\[6pt]
x^{(k+mn)} &= x^{(k)} m^{mn} \prod_{j=0}^{m-1} \left(\frac{x+k+j}{m}\right)^{(n)},& \text{for } m &\in \mathbb{N} \\[6pt]
(ax+b)^{(n)} &= x^n \prod_{j=0}^{n-1} \left(a+\frac{b+j}{x}\right),& \text{for } x &\in \mathbb{Z}^+ \\[6pt]
(2x)^{(2n)} &= 2^{2n} x^{(n)} \left(x+\frac{1}{2}\right)^{(n)} .
\end{align}</math>