occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

## Thursday, December 27, 2007

### M Theory Lesson 142

The original Hoffman post mentioned an expression in $\Gamma$ functions, similar to that appearing in the relation

$B(a,b)= \frac{\Gamma(a)\Gamma(b)}{\Gamma(a + b)} + \frac{\Gamma(a)\Gamma(c)}{\Gamma(a + c)} + \frac{\Gamma (1 - a - b) \Gamma (b)}{\Gamma (1 - a)}$
$= \frac{\zeta (1 - a)}{\zeta (a) } \frac{\zeta (1 - b)}{\zeta (b) } \frac{\zeta (a + b)}{\zeta (1 - a - b) }$

which appears in Castro's discussion of the zeroes of the Riemann zeta function. The $B$ function is the familiar 4-point amplitude of Veneziano, which we have been expressing in terms of chorded polygons; in this case a square with two diagonals representing the 1 dimensional associahedron, the interval.