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general sums - tommy1729 - 06/22/2010
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. RE: general sums - mike3 - 06/23/2010
(06/22/2010, 10:29 PM)tommy1729 Wrote: i was thinking about " general sums ". 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. RE: general sums - tommy1729 - 06/23/2010
an intresting paper related to continuum sum ( but not continuum iterations of it ) is this : http://www.math.tu-berlin.de/~mueller/HowToAdd.pdf especially " 3. Basic Algebraic Identities " where the geometric part is what mike3 uses (together with fourrier expansion) to get his continuum sum. the idea of ' removing the period ' is also known and the origin of this 'geometric part equation' is as old as " q-math " ( q-series and q-analogues and fourrier series ) i knew id seen it before ... in fact i used it myself even way before that paper was written , although probably similar papers have been written much earlier. not to mention eulers example given in the paper. intresting is the continuum product product x ; sin(x) + 5/4. or equivalent the continuum sum sum x ; ln(sin(x) + 5/4). and the question if these sums resp products are periodic themselves. and the question if these sums resp products are divergent ( lim x -> oo does not equal +/-oo or 0) ( it is known that integral 0,2pi log(sin(x) + 5/4) = 0 ) regards tommy1729 RE: general sums - kobi_78 - 06/24/2010
There exists a formula for iterated sums. If Then RE: general sums - tommy1729 - 06/24/2010
(06/24/2010, 06:53 PM)kobi_78 Wrote: There exists a formula for iterated sums. thats great ! now all we need to do is take the continuum sum of that. where did you get that formula btw ? newton ? gauss ? thanks regards tommy1729 RE: general sums - kobi_78 - 06/24/2010
Hi, I discovered it myself To derive it, use the rule that the multiplication of two power series is the discrete convolution of their coefficients and that . Now try to multiply a formal power series with . Repeat it k times and use the formula of to derive the formula. It however reminds Cauchy's formula for repeated integration (with all umbral calculus variations). |