My current work revolves around studying functions from the $p$-adic integers to the $q$-adic rationals, where $p$ and $q$ are distinct primes ("$(p,q)$-adic functions", as I call them). I've been familiarizing myself with the work of W.M Schikhof, A.C.M van Rooij, and their school. This has been something of a mixed blessing, at best, since, on the whole, the literature in this area leans heavily toward the abstract, highly general, algebraified "soft analysis" style of functional analysis, which is difficult for me to parse, not just because of the abstraction, but because I am concerned with doing "hard analysis" with specific $(p,q)$-adic functions.
So far, the most useful resource that I've found is Schikhof's classic Ultrametric Calculus, however, even there, the content directly germane to my $(p,q)$-adic approach is barely mentioned at all; there's a single exercise in the final chapter covering $(p,q)$-adic integration, as well as a brief discussion in the appendices of how to use the linear functional approach to construct a theory of integration on ultrametric spaces, where—unlike the rest of the book—the characteristics of the residue fields of the domain and range are not mandated to be the same.
My immediate goal is get a better understanding of the properties of the zeroes of $(p,q)$-adic functions. To give an enlightening non-example of what I want, Cherry - Lectures on Non-Archimedean Function Theory gives an excellent exposition of the theory of analytic and meromorphic $(p,p)$-adic functions, including the likes of critical radii and p-adic Nevanlinna theory. I'm asking to see if anyone knows of comparable material applicable to the $(p,q)$-adic case. These could be in the form of books or papers on this specialized case of more general considerations of non-archimedean analysis (though my research has not stumbled on much so far), or, they could be books and papers in the abstract, generalized style which just so happen to include my $(p,q)$-adic situation as a particular special case. I strongly suspect that if any work has already been done on these topics, it is most likely subsumed in one of those more abstract approaches. Consequently, I'd really appreciate it if someone more well-versed in those generalities could point me toward the relevant literature.
Aside from mere assistance with my research, part of the reason why I ask is because I am working on my PhD dissertation (without, alas, the benefit of an advisor whose research areas overlap with my own), and, just the other day, I discovered something interesting about $(p,q)$-adic functions' zeroes, and would like to know whether this result (and, more generally, this research direction) is actually novel. Specifically, let $f(x)$ be a $(p,q)$-adic function, possibly non-continuous, which admits a representation by a van der Put series which, at minimum, converges point-wise for every $p$-adic integer $x$. Let $c$ be a $q$-adic rational number and let $x$ be a $p$-adic integer which is not a natural number. Then, $f(x)=c$ occurs if and only if:
$$\liminf_{n–>∞}\frac{|f([x]_{p^n})-c|_{q}}{\sqrt{|f([x]_{p^n})-f([x]_{p^{n-1}})|_{q}}}<∞$$
where $[x]_{p^n}$ is the unique non-negative integer congruent to $x$ mod $p^n$. That is to say, there is a lower bound on the rate at which $f(y)-c$ decays to zero as $y$ tends $p$-adically to $x$.
