06/30/2008, 09:23 PM

Using a matrix-expression this would be

t°h(x) = W^-1 sum k=0..inf sum j=0..k (-1)^j * binomial(k,j) *diag(1,u^j,u^2j,...) * W

sum j=0..k (-1)^j * binomial(k,j) *dV(u^j) = diag(u^j-1) *PPow(

sum k=0..inf

Hmm, let me try (maybe I didn't get this right yet).

is

which is just a binomial weighting of the coefficients c_n.

In my analyses I got the coefficients

so, for instance, the c_n for the half-iteration are

The rate of growth of the a_n-coefficients for t=0.5 was asymptotically

where m_n are also growing coefficients, if only the leading coefficient

of the polynomials at x^n are taken into account.

Now the quotient of two consecutive binomials

seem to approach -1, so the strong growth of about u^n^2/n!, or the

quotient of two consecutive coefficients of ~ u^2n/n seems to

dominate the characteristic of the c_n-coefficients.

A series with quotient of increasing absolute value u^(2n)/n, u>1 cannot

regularly be Euler-summed; maybe it can be summed with Borel-summation of

higher orders.

To be "not regularly" Euler-summable does not mean, we cannot have

an approximation of a certain degree; however the problem with

this is, that the partial sums may converge up to a certain

index n, from where it "begins to diverge" - and it is not yet

known to me, to what extent we can use the intermediate approximated

value - I'm investigating for verification of some experimental

summation-methods of the required power.

Hmm - i hope this is not more confusing than clarifying - I've my

head not really free today (have to prepare the final lesson tomorrow)

Gottfried

t°h(x) = W^-1 sum k=0..inf sum j=0..k (-1)^j * binomial(k,j) *diag(1,u^j,u^2j,...) * W

sum j=0..k (-1)^j * binomial(k,j) *dV(u^j) = diag(u^j-1) *PPow(

sum k=0..inf

bo198214 Wrote:

Notes:

- Perhaps Gottfried can jump in to provide summability in the divergent case .

Hmm, let me try (maybe I didn't get this right yet).

is

which is just a binomial weighting of the coefficients c_n.

In my analyses I got the coefficients

so, for instance, the c_n for the half-iteration are

The rate of growth of the a_n-coefficients for t=0.5 was asymptotically

where m_n are also growing coefficients, if only the leading coefficient

of the polynomials at x^n are taken into account.

Now the quotient of two consecutive binomials

seem to approach -1, so the strong growth of about u^n^2/n!, or the

quotient of two consecutive coefficients of ~ u^2n/n seems to

dominate the characteristic of the c_n-coefficients.

A series with quotient of increasing absolute value u^(2n)/n, u>1 cannot

regularly be Euler-summed; maybe it can be summed with Borel-summation of

higher orders.

To be "not regularly" Euler-summable does not mean, we cannot have

an approximation of a certain degree; however the problem with

this is, that the partial sums may converge up to a certain

index n, from where it "begins to diverge" - and it is not yet

known to me, to what extent we can use the intermediate approximated

value - I'm investigating for verification of some experimental

summation-methods of the required power.

Hmm - i hope this is not more confusing than clarifying - I've my

head not really free today (have to prepare the final lesson tomorrow)

Gottfried

Gottfried Helms, Kassel