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It is useful to define a generalization of the falling and rising factorials as follows:
Definition: Falling and Rising Factorial Powers of Functions
Obviously, the falling and rising factorials $x^\underline{n}$ and $x^\overline{n}$ equal $g^\underline{n}(x)$ and $g^\overline{n}(x)$ for the complex identity function $g(x):=x$ for all $x\in\mathbb C.$ If $f:D\to\mathbb C$ is a function with a suitable domain $D\subseteq\mathbb C$, then the falling $f^\underline{n}$ (and respectively) the rising factorials $f^\overline{n}$ are defined as the compositions of $$(g\circ f)(x)=f^\underline{n}(x),\quad f^\overline{n}(x).$$
Examples (for a positive $n$)
- $f(x)=a+bx,$ $$\begin{align}f^{\underline{n}}(x)&=(a+bx)(a+b(x-1))\cdots(a+b(x-n+1))\nonumber\\f^{\underline{-n}}(x)&=\frac 1{(a+b(x+1))(a+b(x+2))\cdots(a+b(x+n))}\nonumber\end{align}$$
- $f(x)=a^x,$ $$\begin{align}f^{\underline{n}}(x)&=a^x\cdot a^{x-1}\cdots a^{x-n+1}=a^{x^{\underline n}}\nonumber\\f^{\underline{-n}}(x)&=\frac 1{a^{x+1}a^{x+2}\cdots a^{x+n}}=\frac 1{a^{(x+1)^{\underline n}}}\nonumber\end{align}$$
- $f(x)=\sin(x),$ $$\begin{align}f^{\overline{n}}(x)&=\sin(x)\sin(x+1)\cdots\sin(x+n-1))\nonumber\\f^{\overline{-n}}(x)&=\frac 1{\sin(x-n)\sin(x-2)\cdots\sin(x-n)}\nonumber\end{align}$$
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References
Bibliography
- Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition
- Miller, Kenneth S.: "An Introduction to the Calculus of Finite Differences And Difference Equations", Dover Publications, Inc, 1960
- Bool, George: "A Treatise on the Calculus of Finite Differences", Dover Publications, Inc., 1960
Footnotes
