Hello everyone.
James phi function reminded me of earlier ideas I had for solving tetration.
I got stuck with those ideas but phi has given me new courage.
James his phi function is part of the generalization \( \phi(s,a,b,c) \) that I considered many years ago.
\( \phi(s+1,a,b,c) = \exp(a s + b + c \phi(s,a,b,c)) \)
Notice the parameters a,b,c are very closely related !
In fact these functions seem very related.
\( \phi(s+1,0,0,1) =\exp(0s+0+1\phi(s,0,0,1)) \) is clearly tetration.
i tried to take limits towards zero for the parameters to arrive at tetration.
Also the derivatives act similar like those of tetration. Hence ideas of continu sum and products arose. See also many threads including mike3 and others.
\( \phi(s+1,1,0,1) =\exp(as+b+c \phi(s,1,0,1)) \) Is James Nixon's phi and as said the derivative is very much like that of tetration.
James his phi had a closed form.
Does this generalization - apart from tetration itself perhaps - have other closed forms ? And are those entire or analytic ?
regards
tommy1729
James phi function reminded me of earlier ideas I had for solving tetration.
I got stuck with those ideas but phi has given me new courage.
James his phi function is part of the generalization \( \phi(s,a,b,c) \) that I considered many years ago.
\( \phi(s+1,a,b,c) = \exp(a s + b + c \phi(s,a,b,c)) \)
Notice the parameters a,b,c are very closely related !
In fact these functions seem very related.
\( \phi(s+1,0,0,1) =\exp(0s+0+1\phi(s,0,0,1)) \) is clearly tetration.
i tried to take limits towards zero for the parameters to arrive at tetration.
Also the derivatives act similar like those of tetration. Hence ideas of continu sum and products arose. See also many threads including mike3 and others.
\( \phi(s+1,1,0,1) =\exp(as+b+c \phi(s,1,0,1)) \) Is James Nixon's phi and as said the derivative is very much like that of tetration.
James his phi had a closed form.
Does this generalization - apart from tetration itself perhaps - have other closed forms ? And are those entire or analytic ?
regards
tommy1729