I always thought n /===\ k (q; q) | | 1 - q n n! := Gamma(n + 1) = | | ------ = --------, q q | | 1 - q n k = 1 (1 - q) (for integer n), based on the notion that "q-number" n := n 1 - q ------. 1 - q But O. Pavlyk points out that quantum theorists seem to prefer -n n q - q "q-number" n := --------, and thus -1 q - q 1 k n -- - q 2 2 /===\ k (q ; q ) | | q n n! := | | ------- = --------------------, q | | 1 (n - 1) n k = 1 - - q --------- q 2 2 n q (1 - q ) (http://arxiv.org/abs/math.QA/0207244, http://arxiv.org/abs/quant-ph/9506036). Does anyone know the motivation for this? It wouldn't be inconceivable that the boring definition n! = Gamma(n+1) break down in q-land. Physicists tend to bend definitions to their narrow purposes rather than the greater good of mathematics, so let's see what happens to the reflection and tuplication formulae. Using qfac for the traditional(?) function and qfaq for the "quantized" one, Reflection: 2 3 (z - 1/2) (q; q) sqrt(q) (1 - q) inf qfac(- z, q) qfac(z - 1, q) = -----------------------------------, 2 2 %pi ------ 2 %pi log(q) sqrt(- ------) theta (z, %e ) log(q) 1 2 2 3 2 (q ; q ) (1 - q ) inf qfaq(- z, q) qfaq(z - 1, q) = ---------------------------------------, 2 %pi ------ 3/4 %pi log(q) q sqrt(- ------) theta (z, %e ) log(q) 1 so qfaq is nicer! m-Tuplication: m m - 1 m m m n /===\ (q ; q ) qfac(m n, q) (1 - q) | | k m inf | | qfac(n - -, q ) = -----------------------------------, | | m m - 1 k = 0 m n - ----- m 2 (q; q) (1 - q ) inf m - 1 /===\ | | k m | | qfaq(n - -, q ) = | | m k = 0 (m - 1) (12 m n - 5 m + 1) m -------------------------- 2 m 2 m 12 2 m n (q ; q ) qfaq(m n, q) q (1 - q ) inf --------------------------------------------------------------------. m - 1 m n - ----- 2 2 2 m 2 (q ; q ) (1 - q ) inf Gack! I guess quantum hackers don't tuplicate. I haven't looked at the respective q-extensions of Sum x^n/n!, e.g., but it's hard to imagine anything that would redeem that tuplication mess. --rwg