Editing Double factorial (section)

===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>