Preamble

Sequences $a_n$ defined on the natural numbers are clearly not uniquely interpolated by only one function. In particular, given an interpolation $f(n) = a_n$, then $f(n) + \sin(2\pi n)$ is another interpolation, and so is $f(n) + \frac{1}{(-n-1)!}$. The situation is not completely hopeless though; Carlson's theorem says that under growth conditions in the complex plane there is a unique complex function that interpolates a given sequence on the integers. However, when our sequence grows too fast, there doesn't appears to be any easy way to avoid the non-uniqueness problem.

Despite this reality, I often find that the solution to certain problems depends on evaluating some interpolating function at non-integer values. So, I end up in a situation where the interpolating function is not unique, but the answer to the problem is unique. Since the solution to these problems is unique, these problems, in some sense, induce a unique definition of the interpolating function $f(n)$ within the context of the problem.

The following question is one such context where we need to evaluate a sequence $a_n$ at non-natural values in order to obtain the correct value.

Question Background

The context is a bit bizarre, so bear with me.

If we take the series $$ F(a,x)=\sum_{n=0}^\infty (-1)^n x^{a^n}$$ Then we can rewrite this series as $$\sum_{n=0}^\infty (-1)^n x^{a^n}=\sum_n \sum_k (-1)^n \frac{\ln(x)^k a^{nk}}{k!} = \sum_k \frac{\ln(x)^k}{k!} \sum_n (-1)^n a^{nk} = \sum_{k=0}^\infty \frac{\ln(x)^k}{k!}\frac{1}{1+a^k }$$ Let $$ \widetilde{F}(a,x) = \sum_{k=0}^\infty \frac{\ln(x)^k}{k!}\frac{1}{1+a^k }$$ Then we have $\widetilde{F} = F$ for $|a|<1$. Well, actually, the first series doesn't quite converge, but it's Cesaro mean does (or, we can apply almost any other weak divergent series technique), and using this instead makes $\widetilde{F} = F$ (for $|a|<1$). Now, note that when $|a|>1$, the argument I've given above is no longer valid, since $\sum (-1)^n a^{nk}$ doesn't converge anymore, and I'm replacing a divergent series by its analytical continuation, so nothing yet can be said about when $|a|>1$.

Moreover, it is clear by looking at the series representation $\widetilde{F}$ that both series possess a natural boundary at $|a|=1$. However, one can observe that for $x<1$, both series actually still converge even for $a>1$ (but, they only converge on the real line). So, even though there is a natural boundary, both $F$ and $\widetilde F$ are defined at some points outside this boundary.

A natural question one could ask is "What is the relationship between $F$ and $\widetilde F$ for $a>1$". More specifically, what is $$\sum_{n=0}^\infty (-1)^n x^{a^n} - \sum_{k=0}^\infty \frac{\ln(x)^k}{k!}\frac{1}{1+a^k }$$ For convenience, let's set $x = \frac{1}{e}$ and $a=e$. Then let $a_k = \frac{1}{k!}$. Then we are looking at $$\sum_{n=0}^\infty (-1)^n e^{-e^n} - \sum_{k=0}^\infty a_k \frac{(-1)^k}{1+e^k }$$ A residue calculation (which I won't get into here), suggests that the answer should $$-\pi \sum_{n=-\infty}^\infty a_{\pi i + 2\pi i n} \csc(\pi (\pi i + 2\pi i n)) $$

But, of course, it makes no sense to evaluate $a_{\pi i + 2\pi i n}$, since we are evaluating a sequence defined on the naturals at complex values. However, if we take $a_k = \frac{1}{k!}$ and extend this series naturally into $a_z = \frac{1}{\Gamma(z+1)}$, then we actually obtain the correct answer. In other words $$\sum_{n=0}^\infty (-1)^n x^{a^n} - \sum_{k=0}^\infty \frac{\ln(x)^k}{k!}\frac{1}{1+a^k } = -\pi \sum_{n=-\infty}^\infty a_{\pi i + 2\pi i n} \csc(\pi (\pi i + 2\pi i n)) =$$ $$-\pi \sum_{n=-\infty}^\infty \frac{1}{\Gamma(\pi i + 2 \pi i n + 1)} \csc(\pi (\pi i + 2\pi i n)) $$ This same type of computation works for other values of $a_k$, for instance, we again obtain the correct answer by redoing this computation with $a_n = \frac{\sin(\frac{\pi}{2}n)}{n!}$ and evaluating this function at the imaginary numbers in the natural way.

Question

The above example is only one one many contexts where I tend to find the evaluation of $a_n$ at non-integer values to be crucial piece in finding a solution to a problem. Therefore, I'm interested in natural approaches to extend integer sequences to complex functions. My hope is that this is a question someone before me has thought deeply about and written on. I believe Ramanujan sometimes used ideas in this realm to generate identities. For instance, some of his thoughts are captured in the Ramanujan interpolation and his Master theorem, but often the applications I'm interested in require something stronger than what Mellin Transforms can support. Therefore, I ask are there any systematic approaches to extending sequences defined on the integers into the whole complex plane?