In this paper two-thirds of the way down the first page, it defines a function:
$\iota_q : \Bbb Z_p\to\Bbb Z_p$
And goes on to say
Each $\iota_q$ is an interpolation of the arithmetic function on $\Bbb N\cup\{0\}$ given by $\iota_q(n)=1+ q + q^2 + · · · + q^{n−1}$.
I don't understand this statement. I don't know what an "interpolation" is and I can't identify how $\iota_q$ uses its $p$-adic inputs.
$q$ is already defined as a parameter of the function, and is drawn from $\Bbb N$. And $n$ appears to be shown as the input to the function but the form of the function which terminates at an $n-1$ exponent, would only seem to terminate if $n$ is a standard positive integer, as my understanding of p-adic numbers is that they have whole integer numbers of digits, or possibly infinitely many.
I tried to understand some of the surrounding terms I don't understand in the hope that would clear it up, in particular $q$-numbers and interpolation but was unable to find anything that made it clear.
The paper appears to go on to use $n=-\frac12$ as an input so I'm pretty sure locked up in the phrase "an interpolation of" is a way of interpreting non-natural numbers $n$.
If $q=1\bmod p$ then the $\Bbb{Z\to Q}$ function $n\to q^n$ is continuous in the $p$-adic metric and it has a natural (unique) extension to a continuous function $\Bbb{Z}_p\to \Bbb{Z}_p$.
Then $\iota_q(n)= \frac{1-q^n}{1-q}$ (assuming $q\ne 1$) extends uniquely/naturally to a continuous function $\Bbb{Z}_p\to \Bbb{Z}_p$ as well.
The word "$p$-adic interpolation" is just a weird way to say that.