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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 domain1 $D\subseteq\mathbb C$, 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 $n$)

1 For a given $x\in D,$ also $x\pm i+1\in D$ for all integers $i=0,\ldots,n$

| | | | | created: 2020-04-03 09:27:05 | modified: 2020-04-04 09:28:14 | by: bookofproofs | references: [1112], [8404], [8405]

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Bibliography (further reading)

[8405] Bool, George: “A Treatise on the Calculus of Finite Differences”, Dover Publications, Inc., 1960

[1112] Graham L. Ronald, Knuth E. Donald, Patashnik Oren: “Concrete Mathematics”, Addison-Wesley, 1994, 2nd Edition

[8404] Miller, Kenneth S.: “An Introduction to the Calculus of Finite Differences And Difference Equations”, Dover Publications, Inc, 1960