I came across a piece of notation that I haven't seen before, and I'm
wondering if it has a name.  It's a function denoted (a,n) where n is
an integer and a is a real number, usually an integer or half-
integer.  (It appears in coefficients for expansions of Bessel
functions.)  A couple of equivalent definitions for it are:

(a,n) = Gamma(a+n+1/2)/(n! Gamma(a-n+1/2))

([1] page 126) where Gamma is the gamma function, and

(a,n) := 2^(-2n) (1/n!) \prod_{j=1}^n [4a^2-(2j-1)^2],

([2] page 493, where n! has been corrected to (1/n!))

(a,n) := (a+1/2)_n (a-n+1/2)_n / n!

([2] page 493) where (x)_n is the rising factorial (x)(x+1)...(x+n-1).

When a is a half-integer, so that a=k+1/2, this reduces to the
suggestive

(k + 1/2, n) = (k+n)!/(n! (k-n)!)

It looks pretty combinatorial to me, but the first couple
combinatorists I asked hadn't seen it before.  If you know what it's
called, or have seen this function in other contexts, I'd be
interested to know.

Thanks!

[1] B. G. Korenev. Bessel functions and their applications, volume 8
of Analytical Methods and Special Functions. Taylor & Francis Ltd.,
London, 2002. ISBN 0-415-28130-X. Translated from the Russian by E. V.
Pankratiev.
http://books.google.com/books?id=qy1GNv2ovHQC&pg=PA126&lpg=PA126&dq=h...

   In Watson's "Theory of Bessel functions" it appears in page 198 and
is introduced with the wording "If, following Hankel, we write (v,m)=
...= Gamma(v+m+1/2)/(m!*Gamma(v-m+1/2))" so that I had suggested 'Hankel
symbol' if... it had not been known this way :-) :
http://mathworld.wolfram.com/HankelsSymbol.html
   (unsimplified formulation by the way...)