A Rigidity Property of the Deformed q-Exponential Function

The deformed \(q\)-exponential function \[ e_q(z,u) = \sum_{n=0}^{\infty} u^{\binom{n}{2}} \frac{z^n}{(q;q)_n} \] 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 \[ (u,v)=(1,q) \qquad\text{or}\qquad (u,v)=(q,1). \] Thus the classical Jackson inversion identity is rigid within the deformed family and no new multiplicative inversion identities arise from the deformation parameter.