Is there a name for this formula/distribution: $f(i) = ( a / i^k ) b^i$?

As far as I can tell, it doesn't quite have a name, but perhaps the following makes sense? The logarithmic distribution has pmf \begin{align*} p(i) = -\frac{1}{\log(1-p)}\frac{p^i}{i} \end{align*} So this would be a special case when $k = 1$ in your parametrization, and $a$ is then determined to ensure the pmf sums to 1. The reason for this name is because the pmf is derived from the series expansion of the logarithmic function. In your formulation, the pmf would indeed equal \begin{align*} p(i) = \text{Li}_{k}(b) \frac{b^i}{i^k} \end{align*} Hence, if we would to continue the tradition of naming distributions based off series expansions from which they are derived, this could be called the "Polylogarithmic distribution". No results show up when I google this, though.

Assuming that $a$ is a normalization constant, the formula $$ \label{eq:original}\tag{1} f(i) = \frac{ab^i}{i^k} $$ is mentioned by Good (1953, eqn 54) as a generalization of the Zipf distribution $$ f(i) = \frac{a}{i^k}. $$ (Good stipulates $0 < b < 1$ and calls $b^i$ a "convergence factor".) There is precedent in the literature for calling ($\ref{eq:original}$) the "Good distribution": see Zörnig & Altmann (1995) and Eeg-Olofsson (2008).

A more general form, $$ f(i) = \frac{ab^i}{(\nu + i)^k}, $$ has been dubbed the "Lerch distribution" (Zörnig & Altmann 1995; Klar, Parthasarathy & Henze 2010, p. 130), as the normalization constant is related to the Lerch transcendent.

I can't think of an obvious/objective criterion for choosing between "Good distribution" and "polylogarithmic distribution" (Tom Chen's answer; Kemp 1995) – while both names appear in the literature, the references are so few it's not clear that one name should be favoured over the other. Perhaps one could call it the Good–Kemp distribution?

References

Eeg-Olofsson, M. (2008) Why is the Good distribution so good? Towards an explanation of word length regularity. Lund University Department of Linguistics and Phonetics Working Papers, 53, 15–21. https://journals.lub.lu.se/LWPL/article/view/2270/1845

Good, I. J. (1953) The population frequencies of species and the estimation of population parameters. Biometrika, 40, 237–264. https://doi.org/10.1093/biomet/40.3-4.237

Kemp, A. W. (1995) Splitters, lumpers and species per genus. Mathematical Scientist, 20, 107–118. http://www.appliedprobability.org/data/files/TMS%20articles/20_2_4.pdf

Klar, B., Parthasarathy, P. R. & Henze, N. (2010) Zipf and Lerch limit of birth and death processes. Probability in the Engineering and Informational Sciences, 24, 129–144. https://doi.org/10.1017/S0269964809990179

Zörnig, P. & Altmann, G. (1995) Unified representation of Zipf distributions. Computational Statistics & Data Analysis, 19, 461–473. https://doi.org/10.1016/0167-9473(94)00009-8