An analytic derivation of a generating function for $k$-alternating permutations

We study the inversion enumerator of permutations whose descent set is fixed to be the set of multiples of a fixed integer $k\ge 2$. For each $n$, let ${\mathfrak{S}}_n^{(k)}$ denote the set of permutations of $\{1,\dots ,n\}$ whose descent set is exactly $\{i:k\mid i\}$, and define the polynomial $$a_n^{(k)}(q)=\sum _{\sigma \in {\mathfrak{S}}_n^{(k)}}{q}^{\mathrm{i}\mathrm{n}\mathrm{v}(\sigma )}\mathrm{.}$$ We prove that the associated $q$-exponential generating function $$F_k(t;q)=\sum _{n\ge 0}\frac{t^n}{[n{]}_q!}{a}_n^{(k)}(q),$$ where $[n{]}_q!$ denotes the $q$-factorial, admits an explicit closed form as a ratio of two $k$-periodically truncated $q$-exponential series. The proof is purely analytic and is based on a functional equation satisfied by $F_k(t;q)$, obtained via a decomposition of the $q$-exponential series into residue classes modulo $k$. Coefficient extraction yields a convolution identity involving Gaussian binomial coefficients, which uniquely determines the inversion enumerator. This provides an analytic alternative to classical inclusion–exclusion and structural combinatorial arguments for permutation classes with periodic descent constraints.