Falling and rising factorials

Summary

The falling factorial $x^{\underline{n}}$ multiplies $n$ consecutive factors going down from $x$, and the rising factorial $x^{\overline{n}}$ multiplies $n$ consecutive factors going up from $x$. Both are polynomials of degree $n$ in $x$, and both reduce to the factorial $n!$ for a suitable choice of $x$. The falling factorial counts ordered selections without repetition, and the rising factorial (the Pochhammer symbol of special functions) appears in hypergeometric series. The node falling factorial treats the falling case alone.