Editing Double factorial (section)

==Generalizations==

===Definitions===

In the same way that the double factorial generalizes the notion of the [[Factorial|single factorial]], the following definition of the integer-valued multiple factorial functions (multifactorials), or {{mvar|α}}-factorial functions, extends the notion of the double factorial function for positive integers <math>\alpha</math>:

<math display="block">
n!_{(\alpha)} = \begin{cases}
     n \cdot (n-\alpha)!_{(\alpha)} & \text{ if } n > \alpha \,; \\
     n & \text{ if } 1 \leq n \leq \alpha \,; \text{and} \\
     (n+\alpha)!_{(\alpha)} / (n+\alpha) & \text{ if } n \leq 0 \text{ and is not a negative multiple of } \alpha \,;
\end{cases}
</math>

===Alternative extension of the multifactorial===
Alternatively, the multifactorial {{math|''z''!<sub>(''α'')</sub>}} can be extended to most real and complex numbers {{mvar|z}} by noting that when {{mvar|z}} is one more than a positive multiple of the positive integer {{mvar|α}} then

<math display="block">\begin{align} z!_{(\alpha)} &= z(z-\alpha)\cdots (\alpha+1) \\
&= \alpha^\frac{z-1}{\alpha}\left(\frac{z}{\alpha}\right)\left(\frac{z-\alpha}{\alpha}\right)\cdots \left(\frac{\alpha+1}{\alpha}\right) \\
&= \alpha^\frac{z-1}{\alpha} \frac{\Gamma\left(\frac{z}{\alpha}+1\right)}{\Gamma\left(\frac{1}{\alpha}+1\right)}\,.
\end{align}</math>

This last expression is defined much more broadly than the original.  In the same way that {{math|''z''!}} is not defined for negative integers, and {{math|''z''‼}} is not defined for negative even integers, {{math|''z''!<sub>(''α'')</sub>}} is not defined for negative multiples of {{mvar|α}}.  However, it is defined and satisfies {{math|1=(''z''+''α'')!<sub>(''α'')</sub> = (''z''+''α'')·''z''!<sub>(''α'')</sub>}} for all other complex numbers&nbsp;{{mvar|z}}. This definition is consistent with the earlier definition only for those integers {{mvar|z}} satisfying&nbsp;{{math|''z'' ≡ 1 mod ''α''}}.

In addition to extending {{math|''z''!<sub>(''α'')</sub>}} to most complex numbers&nbsp;{{mvar|z}}, this definition has the feature of working for all positive real values of&nbsp;{{mvar|α}}. Furthermore, when {{math|1=''α'' = 1}}, this definition is mathematically equivalent to the {{math|Π(''z'')}} function, described above. Also, when {{math|1=''α'' = 2}}, this definition is mathematically equivalent to the [[#Complex arguments|alternative extension of the double factorial]].

===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>

===Exact finite sums involving multiple factorial functions===

Suppose that {{math|''n'' ≥ 1}} and {{math|''α'' ≥ 2}} are integer-valued. Then we can expand the next single finite sums involving the multifactorial, or {{mvar|α}}-factorial functions, {{math|(''αn'' − 1)!<sub>(''α'')</sub>}}, in terms of the [[Pochhammer symbol]] and the generalized, rational-valued [[binomial coefficients]] as

<math display="block">
\begin{align} 
(\alpha n-1)!_{(\alpha)} & = \sum_{k=0}^{n-1} \binom{n-1}{k+1} (-1)^k \times \left(\frac{1}{\alpha}\right)_{-(k+1)} \left(\frac{1}{\alpha}-n\right)_{k+1} \times \bigl(\alpha(k+1)-1\bigr)!_{(\alpha)} \bigl(\alpha(n-k-1)-1\bigr)!_{(\alpha)} \\ 
& = \sum_{k=0}^{n-1} \binom{n-1}{k+1} (-1)^k \times \binom{\frac{1}{\alpha}+k-n}{k+1} \binom{\frac{1}{\alpha}-1}{k+1} \times \bigl(\alpha(k+1)-1\bigr)!_{(\alpha)} \bigl(\alpha(n-k-1)-1\bigr)!_{(\alpha)}\,, 
\end{align} 
</math>

and moreover, we similarly have double sum expansions of these functions given by

<math display="block"> 
\begin{align}
(\alpha n-1)!_{(\alpha)} & = \sum_{k=0}^{n-1} \sum_{i=0}^{k+1} \binom{n-1}{k+1} \binom{k+1}{i} (-1)^k \alpha^{k+1-i} (\alpha i-1)!_{(\alpha)} \bigl(\alpha(n-1-k)-1\bigr)!_{(\alpha)} \times (n-1-k)_{k+1-i} \\ 
& = \sum_{k=0}^{n-1} \sum_{i=0}^{k+1} \binom{n-1}{k+1} \binom{k+1}{i} \binom{n-1-i}{k+1-i} (-1)^k \alpha^{k+1-i} (\alpha i-1)!_{(\alpha)} \bigl(\alpha(n-1-k)-1\bigr)!_{(\alpha)} \times (k+1-i)!.
\end{align} 
</math>

The first two sums above are similar in form to a known ''non-round'' combinatorial identity for the double factorial function when {{math|1=''α'' := 2}} given by {{harvtxt|Callan|2009}}.

<math display="block">(2n-1)!! = \sum_{k=0}^{n-1} \binom{n}{k+1} (2k-1)!! (2n-2k-3)!!.</math>

Similar identities can be obtained via context-free grammars.<ref>{{Cite journal|last1=Triana |first1=Juan |last2=De Castro |first2=Rodrigo |year=2019 |title=The formal derivative operator and multifactorial numbers|journal=Revista Colombiana de Matemáticas|volume=53 |issue=2 |pages=125–137 |doi=10.15446/recolma.v53n2.85522 |issn=0034-7426 |doi-access=free }}</ref> Additional finite sum expansions of congruences for the {{mvar|α}}-factorial functions, {{math|(''αn'' − ''d'')!<sub>(''α'')</sub>}}, modulo any prescribed integer {{math|''h'' ≥ 2}} for any {{math|0 ≤ ''d'' < ''α''}} are given by {{harvtxt|Schmidt|2018}}.<ref>{{cite journal | last = Schmidt | first = Maxie D. | arxiv = 1701.04741 | journal = Integers | mr = 3862591 | pages = A78:1–A78:34 | title = New congruences and finite difference equations for generalized factorial functions | url = https://math.colgate.edu/~integers/s78/s78.pdf | volume = 18 | year = 2018}}</ref>