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.

-----------------

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