12/13/2009, 05:41 PM
this might be slightly off topic , but i think ( wont explain ) that a product analogue of mittag-leffler is more usefull for tetration.
product expansions are not so commenly seen but imho intresting in general.
for instance ( in the spirit of " q " (analogue) ) we have
exp(x) = (1 + a_1 x)(1 + a_2 x^2)...(1 + a_i x^i)
where a_i = 1 , 1/2 , -1/3 , 3/8 , ... = A137852 * (i !)
( and note denominator a_p*q = p^q * q^p for primes p and q )
( see also witt vectors )
of course this product expansion of exp(x) is not valid everywhere
( exp HAS NO ZEROS AND ' hint for the radius ' : a_i ^ (1/i) )
and a general product expansion might be complicated considering that functions have zero's ( or even dont ! ) but there might be a way around that problem perhaps.
im aware that certain product expansions might not be unique for some or all functions.
and even if we only have convergeance in a certain domain , it might still be sufficient to help at tetration.
i dont think there are known ways to find real iterations of a product into a product without converting to integrals and sums (and back again) and other ' non-product forms ' , but i think some specificly chosen aid-functions might work and help in solving the tetration issues.
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as for the continuum sum idea and other types of continuation/interpolation i think Ramanujan's master theorem might be usefull.
since a continuum sum = sum of a continuum up to a constant imho , and clever substitions ( to avoid poles of gamma for instance ) together with the master theorem provide ( when converging ) a kind of continuation/interpolation/summation method ( which may or may not be equivalent to some other ).
convergeance will probably be the biggest obstacle.
you will probably benefit from adding an extra parameter , do substitutions , use master theorem , substitute again and fix parameter to arrive at tetration.
this master theorem idea is not ( at least in my mind ) related to the product idea posted above.
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i had those 2 ideas for a very long time , its about time i posted them
i wish i had more time to investigate , but im to busy with number theory at the moment. ( and i could use better hardware and software :p for christmas ? :p)
regards
tommy1729
product expansions are not so commenly seen but imho intresting in general.
for instance ( in the spirit of " q " (analogue) ) we have
exp(x) = (1 + a_1 x)(1 + a_2 x^2)...(1 + a_i x^i)
where a_i = 1 , 1/2 , -1/3 , 3/8 , ... = A137852 * (i !)
( and note denominator a_p*q = p^q * q^p for primes p and q )
( see also witt vectors )
of course this product expansion of exp(x) is not valid everywhere
( exp HAS NO ZEROS AND ' hint for the radius ' : a_i ^ (1/i) )
and a general product expansion might be complicated considering that functions have zero's ( or even dont ! ) but there might be a way around that problem perhaps.
im aware that certain product expansions might not be unique for some or all functions.
and even if we only have convergeance in a certain domain , it might still be sufficient to help at tetration.
i dont think there are known ways to find real iterations of a product into a product without converting to integrals and sums (and back again) and other ' non-product forms ' , but i think some specificly chosen aid-functions might work and help in solving the tetration issues.
-----------------
as for the continuum sum idea and other types of continuation/interpolation i think Ramanujan's master theorem might be usefull.
since a continuum sum = sum of a continuum up to a constant imho , and clever substitions ( to avoid poles of gamma for instance ) together with the master theorem provide ( when converging ) a kind of continuation/interpolation/summation method ( which may or may not be equivalent to some other ).
convergeance will probably be the biggest obstacle.
you will probably benefit from adding an extra parameter , do substitutions , use master theorem , substitute again and fix parameter to arrive at tetration.
this master theorem idea is not ( at least in my mind ) related to the product idea posted above.
--------------
i had those 2 ideas for a very long time , its about time i posted them
i wish i had more time to investigate , but im to busy with number theory at the moment. ( and i could use better hardware and software :p for christmas ? :p)
regards
tommy1729
