eclark>There are people who make a living creating "q-analogs" of well-known functions, identities, etc. You can find some leads here: http://mathworld.wolfram.com/q-Analog.html That page is OK, except it mentions the inferior Exton book and not the bible: Gaspar & Rahman's Basic Hypergeometric Series. You could regard q-(basic hypergeometric) series as sums of q-factorials (there is even a "q-Gosper" algorithm, not mine), but the q-pochhammer notation is handier: n-1 (a;q) := (1-a)(1-qa)...(1-q a), n which can be used to define q-factorial at complex z: (q;q) -z oo QF(z) := (1-q) ----------, z+1 (q ;q) oo (using Rich's notation). Unanalogously to ordinary factorials and pochhammers, you'd need a log to define (a;q)_z in terms of q-factorials: a z QF(z + a - 1) (q ; q) = (1 - q) -------------, z QF(a - 1) but all you need is (a; q) oo (a; q) := -----------. z z (a q ; q) oo If this is news to anyone, rereading some of my old emails should make more sense now, e.g., Subject: the q Filbert matrix [long] [...] I tried substituting a couple of candidates for q-Fibonaccis to see if inverting made huge q-polynomials. It is tempting to define qib(0):=0, qib(1):=1, n - 2 qib(n) = qib(n - 1) + q qib(n - 2)), for then n 1 floor(- - -) 2 2 ==== 2 \ k qib(n) = > qbn(n - k - 1, k) q / ==== k = 0 where qbn is the q-binomial coefficient (q; q) n qbn(n, k) = -------------------. (q; q) (q; q) k n - k [...] and Subject: a month in the laboratory ... the probability that an nxn bitmatrix will have (mod 2) rank = k is 2 n - k + 1 k + 1 (n - k) (q ; q) (q ; q) q k n - k P(n,k) := -------------------------------------------, (q; q) n - k if the entries are 1 with probability p = 1-q . Thus the peculiar base q identity Sum P(n,k) = 1. k In particular, the probability of being nonsingular is just (q;q)_n. rcs>There's a set of functional equations similar to X! (X+1/2)! = (2X+1)! * stuff. n n n n z (q ; q ) (1 - q) QF(n z, q) n oo QF(z, q ) = ------------------------------------------------, n - 1 n - 1 n z - ----- /===\ n 2 | | k n (q; q) (1 - q ) | | QF(z - -, q ) oo | | n k = 1 whose q->1 limit is the usual n - 1 n - 1 ----- ----- 2 2 2 %pi (n z)! z! = ------------------------- n - 1 /===\ n z + 1/2 | | i n | | (z - -)! | | n i = 1 rcs>Presumably there's also a reflection equation with QF(Z) QF(-Z) = more-stuff that looks like pi Z / sin(pi Z). A q-factorial reflection formula was the motivation for my q-sin, q-cos definitions in "Experiments and discoveries in q-Trigonometry" in proceedings of 2000 U. Florida (Gainesville) conference: Symbolic Computation, Number Theory, Special Functions, Physics & Combinatorics, (Garvan & Ismail, eds.) pp79-106. The expression (q;q)_oo persists in the reflection formula, hence my excitement at reexpressing it as a simple Theta_2. (The paper gives it as a q-extension of pi.) Unfortunately(?), there are several other plausible q-extensions of sin. --rwg Whale oil is renewable energy.