A Rigidity Property of the Deformed q-Exponential Function

The deformed $q$-exponential function $$e_q(z,u)=\sum _{n=0}^{\mathrm{\infty }}{u}^{\left(\genfrac{}{}{0px}{}{n}{2}\right)}\frac{z^n}{(q;q{)}_n}$$ provides a common framework containing several classical $q$-exponential functions as particular cases, including the Jackson $q$-exponentials $e_q(z)$ and $E_q(z)$. In this paper we investigate the multiplicative inversion problem $$e_q(z,u)e_q(-z,v)=1,$$ and determine all pairs of deformation parameters $(u,v)$ for which this identity holds. To this end, we introduce a family of coefficient polynomials whose common zeros characterize the inversion property. A geometric analysis of the first nontrivial coefficients reduces the problem to two parameter branches. The symmetric branch is excluded through a parity phenomenon, while the nonsymmetric branch is completely determined by the first two coefficient constraints. As a consequence, we prove a rigidity theorem showing that $$e_q(z,u)e_q(-z,v)=1$$ if and only if $$(u,v)=(1,q)\text{or}(u,v)=(q,1)\mathrm{.}$$ Thus the classical Jackson inversion identity is rigid within the deformed family and no new multiplicative inversion identities arise from the deformation parameter.