central factorial symbols

src/GithubLatexTemplate.tex

@@ -39,6 +39,8 @@
 \newcommand \qpowerDerivativeHO [2] {\mathcal{D}_{q}^{#2} #1}                % 1 - function; 2 - order
 \newcommand \finiteDifferenceHO [2] {\Delta^{#2} #1}                         % 1 - function; 2 - order
 \newcommand \pTsDerivativeHO [3] {\frac{\partial^{#3}}{\Delta {#2}^{#3}} #1} % 1 - function; 2 - variable;
+
+% central factorials and related symbols
 \newcommand \centralFactorial [2] {#1^{[#2]}}
 \newcommand \fallingFactorial [2] {\left(#1 \right)^{\underline{#2}}}
 \newcommand{\stirlingii}{\genfrac{\{}{\}}{0pt}{}}

src/sections/02_introduction.tex

@@ -20,4 +20,20 @@
 \end{equation*}
 \begin{equation*}
     \llceilCoefficient{a}{b}{m}
+\end{equation*}
+
+And for any natural $m$ we have polynomial identity
+\begin{equation}
+    x^m = \sum_{k=1}^{m} T(m, k) \centralFactorial{x}{k}\label{eq:knuth-power-identity}
+\end{equation}
+where $\centralFactorial{x}{k}$ denotes central factorial defined by
+\begin{equation*}
+    \centralFactorial{x}{n} = x \fallingFactorial{x+\frac{n}{2}-1}{n-1}
+\end{equation*}
+where $\fallingFactorial{n}{k} = n (n-1) (n-2) \cdots (n-k+1)$ denotes falling factorial in Knuth's notation.
+In particular,
+\begin{equation*}
+    \centralFactorial{x}{n}
+    = x \left( x+\frac{n}{2}-1 \right) \left( x+\frac{n}{2}-1 \right) \cdots \left (x+\frac{n}{2}-n-1 \right)
+    = x \prod_{k=1}^{n-1} \left( x+\frac{n}{2}-k \right)
 \end{equation*}