AI 中文总结
本文研究变形q-指数函数的乘法逆问题，通过系数多项式和几何分析证明仅当参数对为(1,q)或(q,1)时恒等式成立，从而确立经典Jackson逆恒等式的刚性。
AI 中文摘要
变形q-指数函数 \\[ e_q(z,u) = \sum_{n=0}^{\infty} u^{\binom{n}{2}} \frac{z^n}{(q;q)_n} \\] 提供了一个包含若干经典q-指数函数作为特殊情形的统一框架，其中包括Jackson q-指数函数 \\(e_q(z)\\) 和 \\(E_q(z)\\)。本文研究乘法逆问题 \\[ e_q(z,u)e_q(-z,v)=1, \\] 并确定使得该恒等式成立的所有变形参数对 \\((u,v)\\)。为此，我们引入一族系数多项式，其公共零点刻画了逆性质。对首个非平凡系数的几何分析将问题归结为两个参数分支。对称分支通过奇偶现象被排除，而非对称分支完全由前两个系数约束确定。作为结果，我们证明了一个刚性定理，表明 \\[ e_q(z,u)e_q(-z,v)=1 \\] 当且仅当 \\[ (u,v)=(1,q) \quad\text{或}\quad (u,v)=(q,1)。 \\] 因此，经典Jackson逆恒等式在变形族内是刚性的，且变形参数不会产生新的乘法逆恒等式。
英文摘要
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.
