Mike Stay> It seems reasonable to think of continued fractions as generating a series of natural numbers. What's the convolution product on the series? Most generating functions are, well, functions. Is there a standard way of turning a continued fraction into a function? (Flajolet appears to have done a bunch of combinatorics work with continued fractions, but it's a little too dense for me to understand.) Rogers and Ramanujan showed how to q-deform continued fractions, replacing the 1/x step with q^n/x. From looking at continued fraction formulas for e and pi, however, it seems like for certain purposes we'd want linear or quadratic growth of the numerator instead of exponential. For the first case, an h-deformation that replaces the numerators with 1+nh instead of q^n seems promising. Has anyone seen work on that? -- Mike Stay - metaweta@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=metaweta%40gmail.com>http://www.cs.auckland.ac.nz/~mikehttp://reperiendi.wordpress.com This may not be quite what you're after, but a year or so back I posted valuations of some CFs with polynomial numerators and denominators as quotients of hypergeometric series. These were more "interesting" than the usual results where you build the CF from dividing out the middle term of a three term recurrence for the hypergeometric. Instead, my method was to write ODEs for the generating functions of the numerators and denominators of the convergents, solve them, and take limits. This promised to be quite tedious, except that it was possible to take "unconscionable shortcuts" (I think I called them). I think I can dig up some results, if you want. --rwg There was also a minor wrinkle on the Rogers-Ramanujan CF.