Hermite Polynomials

[mike3]

I would find that last identity dubious -- that would make the continuum sum just the log of the derivative!

That's an identity for tetration. If \( f \) is tetration its derivative is the continuum product. (at least im pretty certain of that, tommy's always saying it and a little algebra shows it)

So you're suggesting to guarantee the convergence, you should limit \( \alpha \) to be less than \( \pi/2 \)? (Which is not a problem for tetration since it's actually bounded on the strip) is this to try and prove equivalence between this and your fractional calculus continuum sum?

Well I meant if you have alpha less than pi/2 I can give you a continuum sum. It may not be faulbahers but it probably is (since continuum sums are unique if they are exponentially bounded on the real line and of exponential type less than pi/2 on the imaginary line.) It's a triple integral transform that's a little ugly but it works.

The coefficients \( a_n \) depend on whatever function you're trying to continuum sum -- I don't get what you mean by a "more convenient pattern". It'll depend on the function, just as whether or not there's a "convenient pattern" for the Taylor series coefficients depends on the function.

I meant, the following.

if \( f \) is tetration base e, then \( e^{\sum_{j=0}^{z-1} f(j)} = \frac{d}{dz}f(z) \)

With this relationship you can solve for \( a_n \) using what I presume is a lot of summation identities.

Is this what you plan to do? Or do you plan to find another way to solve for \( a_n \). I.e: "a more convenient pattern."

OK, so you're talking about tetration specifically -- that's why I was thrown off. I just took "\( f \)" to mean "arbitrary function".

To try to solve these equations, we would need to find the exponential of the Hermite series, and then equate coefficients with the derivative I am not quite sure how to do this. In particular, the formula for an exponential of a power series is well-known, but a Hermite series?

Now, the Weierstrass transform provides a transformation from Hermite to power series, as was mentioned here. But it's not clear if it can be used to crack what \( \exp(f) \) means when \( f \) is represented by a Hermite series.

The question is whether or not the system of equations is linear or nonlinear, and if linear, is it infinite linear or not (given that the coefficients of the continuum sum are themselves infinite sums, it seems this is likely to be the case). If it is infinite linear, or nonlinear, it may not have a unique solution (which would be interesting -- what would these alternative-continuum-sum tetrationals be, anyways? Would they be the "alternate fixed point" solutions sheldonison has constructed? Something else altogether?), unless additional constraints can be applied.