Definition:Stirling Numbers of the Second Kind/Complex Numbers

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Donald E. Knuth, in his The Art of Computer Programming: Volume 1: Fundamental Algorithms, 3rd ed. of $1997$, suggests an extension of the Stirling numbers of the second kind $\displaystyle {r \brace r - m}$ to the real and complex numbers.

However, beyond stating that such a number is a polynomial in $r$ of degree $2 m$, and providing a few examples, he goes no further than that, and the details of this extension are unclear.