09/24/2013, 10:44 PM
Since f(z) = ln( (2sinh(z) - z) / z ) is entire we know there must be a solution to (2sinh(z) - z) / z = (1 + a_1 z)(1 + a_2 z^2)(1+a_3 z^3)... where the a_n are (complex) constants.
(Notice f(0) = 1)
It might then be intresting to consider the size and signs of the a_n.
This is a " Gottfried type " of idea I must say.
Although these products are clearly from the time mathematicians considered the so called q-series and are still investigated by some today (like me) I would like to give a link to Gottried's paper.
Im not so knowledgeable about Witt vectors , but I think this paper explaines some things. ( So that I do not need to repeat the Obvious )
I would also like to say these products have an intresting combinatorical interpretation that can be seen as a competative way against the circle method ( powers of a Taylor series ).
http://go.helms-net.de/math/musings/drea...quence.pdf
Here the importance of the ln and both the partition and divisor function occurence is well illustrated.
The " fake zero's " a_n^(1/n) are both fascinating and puzzling.
In Gottfried's paper it means the " q-product form " of exp(z) has a natural boundary smaller or equal to the unit radius BECAUSE OF EITHER THE LN OR THE A_N ( depending on your viewpoint ).
But other cases of " q-product forms " apart from exp(z) might be more complex.
Since the focus here is on functions that grow about exponential rate and also the focus on fixpoints and expansions this seems like a natural question to me.
Regards
tommy1729
(Notice f(0) = 1)
It might then be intresting to consider the size and signs of the a_n.
This is a " Gottfried type " of idea I must say.
Although these products are clearly from the time mathematicians considered the so called q-series and are still investigated by some today (like me) I would like to give a link to Gottried's paper.
Im not so knowledgeable about Witt vectors , but I think this paper explaines some things. ( So that I do not need to repeat the Obvious )
I would also like to say these products have an intresting combinatorical interpretation that can be seen as a competative way against the circle method ( powers of a Taylor series ).
http://go.helms-net.de/math/musings/drea...quence.pdf
Here the importance of the ln and both the partition and divisor function occurence is well illustrated.
The " fake zero's " a_n^(1/n) are both fascinating and puzzling.
In Gottfried's paper it means the " q-product form " of exp(z) has a natural boundary smaller or equal to the unit radius BECAUSE OF EITHER THE LN OR THE A_N ( depending on your viewpoint ).
But other cases of " q-product forms " apart from exp(z) might be more complex.
Since the focus here is on functions that grow about exponential rate and also the focus on fixpoints and expansions this seems like a natural question to me.
Regards
tommy1729