Number to Power of One Rising is Itself

Theorem

Let $x \in \R$ be a real number.

$x^{\overline 1} = x$

where $x^{\overline 1}$ denotes the rising factorial.

Proof

\(\ds x^{\overline 1}\)

\(=\)

\(\ds \dfrac {\map \Gamma {x + 1} } {\map \Gamma x}\)

Rising Factorial as Quotient of Factorials

\(\ds \)

\(=\)

\(\ds x\)

Gamma Difference Equation

$\blacksquare$