Tetration Forum - general sums

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.

(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 $e^{ux}$ , the basis of Fourier series, we have

$\Delta e^{ux} = \Sigma^{-1} e^{ux} = (e^u - 1) e^{ux}$ .
$\Delta^2 e^{ux} = \Delta \Delta e^{ux} = \Delta (e^u - 1) e^{ux} = (e^u - 1) \Delta e^{ux} = (e^u - 1)(e^u - 1) e^{ux} = (e^u - 1)^2 e^{ux}$ .

Induction shows that

$\Delta^n e^{ux} = (e^u - 1)^n e^{ux}$ .

Thus we have the (indefinite!) continuum sum $\frac{e^{ux}}{e^u - 1}$ by setting $n = -1$ , and we can "formally" continuously iterate the summation and difference operator by setting fractional, real, and complex values for $n$ . 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,

$f(x) = \sum_{n=-\infty}^{\infty} a_n e^{nux}$ ,

the fractional forward difference is

$\Delta^t f(x) = \sum_{n=-\infty}^{\infty} a_n (e^{nu} - 1)^t e^{nux}$

which only gives iterations of the formal continuum sum if $a_0 = 0$ . If $a_0 \ne 0$ , then we get $0^t$ 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.

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

There exists a formula for iterated sums.

If

$Sum_n({ a_n }) = S_n$
$S_0 = 0$
$S_n = S_{n - 1} + a_n$

Then

$Sum_n^k({ a_n }) = \sum_{j=1}^{n} { {k + (n - j) - 1}\choose{k - 1} } \cdot a_j$

(06/24/2010 06:53 PM)kobi_78 Wrote: [ -> ]There exists a formula for iterated sums.

If

$Sum_n({ a_n }) = S_n$
$S_0 = 0$
$S_n = S_{n - 1} + a_n$

Then

$Sum_n^k({ a_n }) = \sum_{j=1}^{n} { {k + (n - j) - 1}\choose{k - 1} } \cdot a_j$

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

Hi, I discovered it myself $Smile$

To derive it, use the rule that the multiplication of two power series is the discrete convolution of their coefficients and that $(1 - x)^{-1} = \sum_{n=0}^{\infty}{x^n}$ .

Now try to multiply a formal power series with $(1 - x)^{-1} = \sum_{n=0}^{\infty}{x^n}$ .

Repeat it k times and use the formula of $(1 - x)^{-k}$ to derive the formula.

It however reminds Cauchy's formula for repeated integration (with all umbral calculus variations).