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The difference equation, as referenced in the title, is a very specific object I'm referring to. If you have a holomorphic function $\phi$ on a domain $G$, then a solution $F$ to the difference equation of $\phi$ is a function holomorphic on $G'$ (for some domain $G'$) such that

$$F(z+1) - F(z) = \phi(z)$$

for all $z \in G$.

This idea is studied frequently and very well approached. First I will explain the general construction of the ways I know how to approach the question. Second, I wonder to ask more complicated ways of solving specific cases of the questions.

Let's take one of the most renowned equations in mathematics and explain its significance in this scenario. The Euler-Maclaurin summation formula is one of the best methods of solving this difference equation. Sadly, I only know what people have said about how Euler abstractly approached the equation; not the literal history; I also know none of the information about how Maclaurin approached it.

Euler, using an incredibly simple identity deduced that if $Tf(x) = f(x+1)$

$$\sum_{j=1}^{n-1} (T-1)f(j) = f(n) - f(1)$$

proposing therefore that inverting the difference operator, $T-1$, taking $\frac{1}{T-1}$ (say expanding it as a power series in $T$) we would arrive at a formula for summation dependent only on $T$. Sadly, there were divergence issues. Therefore this should fall apart, excepting that Euler was smarter than this. He said if $D$ was the differential operator, then $e^D = T$, and therefore

$$\frac{1}{e^{D} -1} = \frac{1}{T-1}$$

but adding the key element of multiplying by $D$ and dividing by $D$ Euler arrived at

$$D^{-1}\frac{D}{e^{D}-1}$$

and taking the power series of $$\frac{x}{e^x-1}$$ we arrived at the typical power series defining Bernoulli numbers

$$\frac{D}{e^{D}-1} \phi(x) = \sum_{m=0}^\infty B_m\frac{D^m \phi(x)}{m!}$$

and multiplied by $D^{-1}$ produces

$$F(x) = \sum_{m=0}^{\infty}B_m \frac{D^{m-1}\phi(x)}{m!}$$

where this expression miraculously converges in many common instances, if we consider Euler rigorously he most likely had no way of distinguishing though. Euler, being slightly modest, wrote the formula as an approximation of the sum by an integral. This was in fact his inspiration, to approximate a sum by an integral. The above was written as

$$\sum_{j=1}^{n} f(j) = \int_0^n f(t)\,dt + \frac{f(n)-f(0)}{2} + \sum_{k=0}^p B_{2k}\frac{f^{2k-1}(n)-f^{2k-1}(0)}{2k!} + R_p$$

where $R_p$ gets smaller as $p$ grows. In a land of numerical approximation, Euler mostly used this as a way of computing complicated sums. But hidden in it is a way of solving the difference equation.

This led to Ramanujan summation. This was essentially the above method except it is rigorously proven to solve the case when $\phi$ is entire and $|\phi(z)| = O(e^{\tau|\Im(z)| + \rho|\Re(z)|})$ as $z$ escapes to infinity where $\tau< \pi$. (By "rigorously proven" and "Ramanujan", I mean that Ramanujan wrote the formula down and later mathematicians proved it.) The solution to $\phi's$ difference equation is

$$F(z) = \int_0^z \phi(x)\,dx + \frac{\phi(z)}{2} + \sum_{k=1}^\infty \frac{B_{2k}}{2k!} \phi^{2k-1}(z)$$

The unbelievable fact about this case is that this solution $F$ also satisfies $|F(z)| = O(e^{\tau|\Im(z)| + \rho |\Re(z)|})$ and is UNIQUE if $F(0) = C$ for fixed $C$. This is the only case I know of such a nice solution satisfying such a nice uniqueness equation.

Now in my own work, I've taken $\phi$ holomorphic on $\mathbb{C}_{\Re(z)>0}$ and $|\phi(z)| = O(e^{\kappa|\Im(z)| + \rho|\Re(z)|})$ for $\kappa < \pi/2$ then a unique solution to the difference equation satisfying the same bounds still exists (better than that, can be constructed).

The best I've gotten at specializing this is if $\phi$ is holomorphic for $a < \Re(z) < b$, and $|\phi(z)| = O(e^{\kappa|\Im(z)|})$ for $\kappa < \pi/2$, then a unique solution to the difference equation exists. Sadly I can't seem to produce better results than these few scenarios. This has led me to the following quandraries.

Is there a general, well documented way of handling the difference equation of entire functions that don't satisfy exponential bounds? The solutions probably won't have a nice uniqueness criterion, but the solution is necessarily unique modulo a 1-periodic function (AKA a solution exists). Similarly, could functions on the half plane which aren't exponentially bounded have a solution to the difference equation?

Are there ways of finding the solution of a difference equation when $\phi$ is holomorphic on a simply connected domain biholomorphic to the unit disk? I have investigated this for the right half plane, and for any vertical strip of the complex plane, but I wonder if the difference equation has been studied on the unit disk or any other common domain. And that regardless of bounds, there exists solutions in common cases.

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