Let be the number of integral points in the -dimensional octahedron of radius :

A standard combinatorial argument shows that for positive integers and

and, in particular,

Is it possible to give a “geometric” interpretation (or any other “conceptual” interpretation) of identity (1)?

**Update**

A satisfactory answer was given by Allison and Mark Lewko, see the comments.

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June 20, 2010 at 1:19 pm |

I wrote a blog entry about this identity some time ago (see: http://lewko.wordpress.com/2009/07/31/lattice-points-in-l1-balls/). There I gave a bijective proof of the fact that O(m,n)=O(n,m). This, however, is little more than a reinterpretation of the identity you give for O(m,n) above. This relation can also be seen via the symmetry of the generating function:

I first learned of this identity from a 2000 paper of Bump, Choi, Kurlberg and Vaaler (see: http://www.ams.org/mathscinet-getitem?mr=1738343), where the proof of the generating relation is given. Are you aware of an earlier appearance of this identity?

June 20, 2010 at 3:35 pm |

Dear Mark,

many thanks!

I learned on this identity from the article

P. Kirschenhofer, A. Pethő, R.F. Tichy, On

analytical and Diophantine properties of a family of counting

polynomials, Acta Sci. Math. (Szeged) 65 (1999), 47-59,

but I do not think it was discovered therein. I believe it was known long ago.