Editing Double factorial (section)

===Generalized Stirling numbers expanding the multifactorial functions===

A class of generalized [[Stirling numbers of the first kind]] is defined for {{math|''α'' > 0}} by the following triangular recurrence relation:

<math display="block">\left[\begin{matrix} n \\ k \end{matrix} \right]_{\alpha} 
     = (\alpha n+1-2\alpha) \left[\begin{matrix} n-1 \\ k \end{matrix} \right]_{\alpha} + \left[\begin{matrix} n-1 \\ k-1 \end{matrix} \right]_{\alpha} + \delta_{n,0} \delta_{k,0}\,. 
</math>

These generalized ''{{mvar|α}}-factorial coefficients'' then generate the distinct symbolic polynomial products defining the multiple factorial, or {{mvar|α}}-factorial functions, {{math|(''x'' − 1)!<sub>(''α'')</sub>}}, as

<math display="block">
\begin{align} 
(x-1|\alpha)^{\underline{n}} & := \prod_{i=0}^{n-1} \left(x-1-i\alpha\right) \\
& = (x-1)(x-1-\alpha)\cdots\bigl(x-1-(n-1)\alpha\bigr) \\ 
& = \sum_{k=0}^n \left[\begin{matrix} n \\ k \end{matrix} \right] (-\alpha)^{n-k} (x-1)^k \\ 
& = \sum_{k=1}^n \left[\begin{matrix} n \\ k \end{matrix} \right]_{\alpha} (-1)^{n-k} x^{k-1}\,.
\end{align} 
</math>

The distinct polynomial expansions in the previous equations actually define the {{mvar|α}}-factorial products for multiple distinct cases of the least residues {{math|''x'' ≡ ''n''<sub>0</sub> mod ''α''}} for {{math|''n''<sub>0</sub> ∈ {0, 1, 2, ..., ''α'' − 1<nowiki>}</nowiki>}}.

The generalized {{mvar|α}}-factorial polynomials, {{math|''σ''{{su|b=''n''|p=(''α'')}}(''x'')}} where {{math|''σ''{{su|b=''n''|p=(1)}}(''x'') ≡ ''σ''<sub>''n''</sub>(''x'')}}, which generalize the [[Stirling polynomial#Stirling convolution polynomials|Stirling convolution polynomials]] from the single factorial case to the multifactorial cases, are defined by

<math display="block">\sigma_n^{(\alpha)}(x) := \left[\begin{matrix} x \\ x-n \end{matrix} \right]_{(\alpha)} \frac{(x-n-1)!}{x!}</math>

for {{math|0 ≤ ''n'' ≤ ''x''}}. These polynomials have a particularly nice closed-form [[ordinary generating function]] given by

<math display="block">\sum_{n \geq 0} x \cdot \sigma_n^{(\alpha)}(x) z^n = e^{(1-\alpha)z} \left(\frac{\alpha z e^{\alpha z}}{e^{\alpha z}-1}\right)^x\,. </math>

Other combinatorial properties and expansions of these generalized {{mvar|α}}-factorial triangles and polynomial sequences are considered in {{harvtxt|Schmidt|2010}}.<ref>{{cite journal|last1=Schmidt|first1=Maxie D.|title=Generalized ''j''-Factorial Functions, Polynomials, and Applications|journal=J. Integer Seq.|date=2010|volume=13|url=https://cs.uwaterloo.ca/journals/JIS/VOL13/Schmidt/multifact.html}}</ref>