On 6/8/06, Schroeppel, Richard <rschroe@sandia.gov> wrote:
I've been playing with Q-factorials: There's a really nice little expository book by Kac and Cheung, _Quantum Calculus_, that covers all of this stuff. Mathworld also has a bunch of pages on it.
Perhaps we can use some of our standard bag of tricks for defining QF(N) for non-integer N.
The q-Gamma function is \int_0^{\infty} x^{t-1} E_q^{qx} d_q x where E_q = \sum_{j=0}^{\infty} q^{j(j-1)/2} x^j / [j]! [j]! = [j] [j-1] [j-2] ... [1] [j] = (q^j - 1) / (q - 1) \int f(x) d_q x = \sum_{n=0}^{\infty} a_n x^{n+1} / [n+1]! + C and f(x) = \sum_{n=0}^{\infty} a_n x^n / [n]! We can integrate suitably nice functions with the Jackson integral \int f(x) d_q x = (1-q) x \sum_{j=0}^{\infty} q^j f(q^j x)
Rich
-- Mike Stay metaweta@gmail.com http://math.ucr.edu/~mike