(06/22/2010, 10:29 PM)tommy1729 Wrote: i was thinking about " general sums ".

that means continuum iterations of continuum sums.

i believe q-analogues and fourrier series play an important role in this.

its still vague , but i wanted to throw it on the table.

What do you mean? You mean continuous iteration of the sum operator?

Hmmmmm... Well, for

, the basis of Fourier series, we have

.

.

Induction shows that

.

Thus we have the (indefinite!) continuum sum

by setting

, and we can "formally" continuously iterate the summation and difference operator by setting fractional, real, and complex values for

. Iteration of the difference operator seems to have been studied before -- look up "fractional finite differences". The generalization above may remind one of generalizing the derivative to non-integer order.

For a Fourier/exp-series,

,

the fractional forward difference is

which only gives iterations of the formal continuum sum if

. If

, then we get

which is undefined for negative t and even t = 0, meaning we can't even apply the operator 0 times. I'm not sure how to extend it in those cases.