On 5/17/08, Dan Asimov <dasimov@earthlink.net> wrote:

Here's a question that probably has an obvious answer, but I have to ask it:

Of course, you can always just interpolate --- though it's not obvious whether the result would converge as you increase the number of lattice points --- but even if it did, it would depend on the interpolation algorithm. What you presumably are interested in is a continuous extension which is in some way "natural", as in factorial vis-a-vis Gamma function.

Does the Fibonacci multiplication (which Knuth denotes with just a circle: x o y) have a continuous extension to the positive reals that remains commutative and associative ?

It might be useful to examine a 1-D row to start with. We have explicit expressions for 1(*)y = [tau^2 y + 1/tau] ; 2(*)y = [tau^3 y + 1/tau^3] + 2{tau y + 1/(2 tau^3)} - {2 tau y + 1/tau^3} where [...] and {...} denote the usual integer and fractional parts. [I haven't yet checked whether latter expression is in OEIS!] The fractional part discarded in the expression for 1(*)y is explicitly expressible only in terms of the base-F coefficients of y, which ain't too promising --- you need somehow to substitute tau^n for F_n to yield a continuous function. Proving expressions such as above involves a sort of "Fib-adic" analysis, analogous to p-adic analysis, where two integers are considered nearby when their difference equals a big Fibonacci number. I haven't explored how much of conventional analysis is possible under this regime --- Ostrowski's theorem says that something is going to have to be ditched! Fred Lunnon