Gottfried Wrote:Yes- this is a very beautiful and far-reaching fact, I think ...I've just received an answer in the newsgroup sci.math by Prof G.A.Edgar who states a numerical discrepancy between my matrix-based conjecture and termwise evaluation of the series.

I cannot resolve the problem completely - the problem doesn't affect the Mb-matrix related conjectures (also of earlier date) but the problem of representation of the alternating series of powers of the reciprocal of Bb by the analoguous expression. I don't have an idea currently, how to cure this and how to correctly adapt my conjecture. So - sigh - I have to retract it for the moment.

[update] I should mention, that this concerns only the Bb-matrix, which is not simply invertible. The application of the idea of the formula to other matrix-operators may be still valid; especially for triangular matrices like P the observation is still valid; I assume, it is also valid for the U-iteration x->exp(x)-1 , since the matrix-operator is the triangular Stirling-matrix. I'll check that today [/update]

[update2] The problem occurs also with the U-iteration and its series of negative heights. Looks like the reciprocal matrix needs some more consideration [/update2]

Gottfried

P.s. I'll add the conversation here later as an attachment.

[update3] A graph which shows perfect match between serial and matrix-method-summation for the Tb-series; and periodic differences between the Rb-series [/update3]

Gottfried Helms, Kassel