October 2010 (UTC) 

 ::::I've updated the motivation again. The recurrence relation needs to be satisfied for positive real inputs, as mentioned in Bohr-Mollerup theorem. If I wrote complex last night (not worth looking in the history), oops, my bad; I've changed the variable to x since z tends to imply complex. I'd be happy to hear a motivation for logarithmic convexity, or for an entirely different true uniqueness condition to be used. The one you wrote below, if it's true, would be fine, so long as you motivated the final condition. Of the two conditions under consideration, I prefer logarithmic convexity, but I almost always prefer mental pictures to algebra. 
 ::::The main point of my edit was that the motivation section didn't motivate the gamma function, it only motivated one of infinitely many alternativeswhere the alternatives are nontrivial. Any text which accomplishes the goal of giving a brief uniqueness argument is exactly what I had in mind.   08:23, 27