So I just thought of something off hand, and it determines how I parameterize things. All of these cases are identical, but it helps me to choose my language if you can find one more reasonable. We can take \(3\) \(F\) functions. Let's define:
\[
F_1(x) = \sum_{k=0}^\infty b_k \frac{x^k}{k!^2}\\
\]
\[
F_2(x) = \sum_{k=0}^\infty b_k \frac{x^{2k}}{2k!}
\]
Or; we can go full Bessel function:
\[
F_3^v(x) = \sum_{k=0}^\infty b_k \frac{x^{2k + v}}{2^{2k+v}k!\Gamma(1+k+v)}
\]
Each of these forms always produce the same "borel sum"--but they do so with different flavour's of integral transforms. The first one we can write as an entire function, and we can prove that the coefficients, create a new function, which satisfies appropriate bounds; and therefore an integral transform exists. (This is how I wrote my original mellin transform proof that \(b_k = O(c^kk!)\), for Gottfried's problem). The second one is what I just posted. The third one is definitely the most interesting, and it's how I attempted to first solve this problem.
Personally, I think we should go the Bessel function route, because we get a whole bunch of dope ass \(\Gamma\) functions everywhere; and Gamma functions are super well behaved and studied; they're like a comfy home. Plus! We get to work with \(\Gamma\) functions.
This also let's us have the best behaved Mellin transforms. Laplace transforms are great, but they are actually more restrictive than Mellin transforms. Laplace transforms are easier to perform; but they tend to only be interesting when you actually approach the boundary of what converges/what doesn't. Mellin transforms only talk about that boundary. No one Laplace transforms a zeta function (which you can analytically continue the \(\zeta\) function using borel sums). It takes ten minutes to describe the zeta function acted on by a laplace transform. There's a reason we look at the mellin transform. They have more complex rules. Laplace is good for differential stuff--but we have no differential stuff here.
I think we need to be looking at Euler's expression:
\[
\int_0^\infty \frac{e^{-t}}{1+tz}\,dt = \sum_{k=0}^\infty (-1)^k k! z^k\\
\]
And just as much, Euler's expression:
\[
\int_0^\infty e^{-t}t^{z-1}\,dt = \Gamma(z)\\
\]
And I think we need to draw our basis there. Rather than solely relying on Laplace transforms (which are really just the poorman's mellin transform, lol).
So, I think if we want to really discuss this problem mathematically, we're going to have to refer heavily to mellin transforms. Despite Laplace transforms "seeming easier" and additionally being "more efficient".
THE MELLIN TRANSFORM WILL BE NEEDED TO EXPAND THIS THING GLOBALLY AND/OR NEAR THE BOUNDARY!
\[
F_1(x) = \sum_{k=0}^\infty b_k \frac{x^k}{k!^2}\\
\]
\[
F_2(x) = \sum_{k=0}^\infty b_k \frac{x^{2k}}{2k!}
\]
Or; we can go full Bessel function:
\[
F_3^v(x) = \sum_{k=0}^\infty b_k \frac{x^{2k + v}}{2^{2k+v}k!\Gamma(1+k+v)}
\]
Each of these forms always produce the same "borel sum"--but they do so with different flavour's of integral transforms. The first one we can write as an entire function, and we can prove that the coefficients, create a new function, which satisfies appropriate bounds; and therefore an integral transform exists. (This is how I wrote my original mellin transform proof that \(b_k = O(c^kk!)\), for Gottfried's problem). The second one is what I just posted. The third one is definitely the most interesting, and it's how I attempted to first solve this problem.
Personally, I think we should go the Bessel function route, because we get a whole bunch of dope ass \(\Gamma\) functions everywhere; and Gamma functions are super well behaved and studied; they're like a comfy home. Plus! We get to work with \(\Gamma\) functions.
This also let's us have the best behaved Mellin transforms. Laplace transforms are great, but they are actually more restrictive than Mellin transforms. Laplace transforms are easier to perform; but they tend to only be interesting when you actually approach the boundary of what converges/what doesn't. Mellin transforms only talk about that boundary. No one Laplace transforms a zeta function (which you can analytically continue the \(\zeta\) function using borel sums). It takes ten minutes to describe the zeta function acted on by a laplace transform. There's a reason we look at the mellin transform. They have more complex rules. Laplace is good for differential stuff--but we have no differential stuff here.
I think we need to be looking at Euler's expression:
\[
\int_0^\infty \frac{e^{-t}}{1+tz}\,dt = \sum_{k=0}^\infty (-1)^k k! z^k\\
\]
And just as much, Euler's expression:
\[
\int_0^\infty e^{-t}t^{z-1}\,dt = \Gamma(z)\\
\]
And I think we need to draw our basis there. Rather than solely relying on Laplace transforms (which are really just the poorman's mellin transform, lol).
So, I think if we want to really discuss this problem mathematically, we're going to have to refer heavily to mellin transforms. Despite Laplace transforms "seeming easier" and additionally being "more efficient".
THE MELLIN TRANSFORM WILL BE NEEDED TO EXPAND THIS THING GLOBALLY AND/OR NEAR THE BOUNDARY!