11/30/2007, 09:18 PM
@Whiteknox
Sorry for getting ahead of myself. Welcome to the forum, and feel free to ask more questions. Concerning your specific question, it has definitely been considered before, but there are no known "nice" extensions. You can always use Lagrange interpolation to connect the integer values of any function. The problem arises with continuity, differentiability, and whether or not it still satisfies a formula.
For example, if you were to use Lagrange interpolation between all known hyper-operations then you would end up with a function \( a\ R(n)\ b \) as you describe for real a, b, n, BUT doing this would prevent it from satisfying:
\( a\ R(n)\ b = a\ R(n-1)\ (a\ R(n)\ (b - 1)) \)
for all non-integer n, since Lagrange interpolation does not take this into account. A method that works very well for tetration, also works well for hyper-operations in general: choosing an interval of n, for example 2 < n < 3 so that everything between multiplication and exponentiation are approximated by some kind of interpolation (maybe not Lagrange interpolation), and use the formula of hyper-operators to extend the domain of n from this interval to all real n. The problem with this method is that doing this prevents the function from being differentiable (and in some cases continuous).
So in order to find a "nice" extension of hyper-operators to real n, one would need to start with an "unknown" interpolation, apply the formula, then "solve" for the interpolation with respect to the formula in such a way that additional requirements are met, for example: continuity, differentiability, and so on.
Andrew Robbins
Sorry for getting ahead of myself. Welcome to the forum, and feel free to ask more questions. Concerning your specific question, it has definitely been considered before, but there are no known "nice" extensions. You can always use Lagrange interpolation to connect the integer values of any function. The problem arises with continuity, differentiability, and whether or not it still satisfies a formula.
For example, if you were to use Lagrange interpolation between all known hyper-operations then you would end up with a function \( a\ R(n)\ b \) as you describe for real a, b, n, BUT doing this would prevent it from satisfying:
\( a\ R(n)\ b = a\ R(n-1)\ (a\ R(n)\ (b - 1)) \)
for all non-integer n, since Lagrange interpolation does not take this into account. A method that works very well for tetration, also works well for hyper-operations in general: choosing an interval of n, for example 2 < n < 3 so that everything between multiplication and exponentiation are approximated by some kind of interpolation (maybe not Lagrange interpolation), and use the formula of hyper-operators to extend the domain of n from this interval to all real n. The problem with this method is that doing this prevents the function from being differentiable (and in some cases continuous).
So in order to find a "nice" extension of hyper-operators to real n, one would need to start with an "unknown" interpolation, apply the formula, then "solve" for the interpolation with respect to the formula in such a way that additional requirements are met, for example: continuity, differentiability, and so on.
Andrew Robbins