three binomial(n,k) definitions

Michael Somos somos at
Mon Dec 16 20:47:51 CET 2002


   Jim Propp wrote :

> could say that the natural domain of the binomial coefficient function 
> is actually a double cover of Z x Z 

which is an intriguing suggestion. One might develop a discrete analog
of analytic functions with a discrete analog of analytic continuation
and discrete analog of Riemann surfaces. Just as the sqrt() function is
only defined up to sign and its Riemann surface is a double cover of the
complex plane, perhaps B(m,n) also has similar extension.

In fact, I had been thinking along these lines without looking at the
connection to complex analysis, however, the cover idea resolves one
of the problems I encountered with non-unique extensions. It would be
nice to see how this idea develops with interesting examples of its use.
Shalom, Michael

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