AI 中文总结
研究复阶 \(u\)-变形齐次函数，建立其解析和代数性质，证明负指数部分与拉马努金偏theta函数的关系及逆变换，推导 \(q\)-差分方程，还得出该理论在柯西和斯蒂尔杰斯 - 维格特型区域的特殊形式。
AI 中文摘要
我们引入并研究了一类复阶的 \(u\)-变形齐次函数 \(\rr_{\alpha}(x,y;u\,|\,q)\)，它通过将 \(u\)-变形齐次多项式扩展到非负整数范围之外得到。我们建立了它们的基本解析和代数性质，包括收敛准则、递推关系、 \(q\)-导数公式以及通过变形 \(q\)-指数算子的算子理论实现。该理论的一个核心特征出现在负指数部分：对于每个非负整数 \(n\)，函数 \(\rr_{\minus n\minus1}(x,y;u\,|\,q)\) 可以用移位拉马努金偏theta函数进行有限三角分解。我们还证明了逆三角变换，表明相应的偏theta函数可以从负指数 \(u\)-变形齐次函数中恢复。我们还推导了这些函数的受电弓型 \(q\)-差分方程。最后，我们在柯西和斯蒂尔杰斯 - 维格特型区域中推导了负指数理论的特殊形式，包括边界几何情况和真正的偏theta区域。
英文摘要
We introduce and study a class of \(u\)-deformed homogeneous functions of complex order \(\rr_α(x,y;u\,|\,q)\), obtained by extending \(u\)-deformed homogeneous polynomials beyond the nonnegative integer regime. We establish their basic analytic and algebraic properties, including convergence criteria, recurrence relations, \(q\)-derivative formulas, and an operator-theoretic realization through a deformed \(q\)-exponential operator. A central feature of the theory appears in the negative-index sector: for every nonnegative integer \(n\), the functions \(\rr_{\minus n\minus1}(x,y;u\,|\,q)\) admit finite triangular decompositions in terms of shifted Ramanujan partial theta functions. We also prove the inverse triangular transformation, showing that the corresponding partial theta functions can be recovered from the negative-index \(u\)-deformed homogeneous functions. We also derive pantograph-type \(q\)-difference equations for these functions. Finally, we derive specialized forms of the negative-index theory in Cauchy and Stieltjes--Wigert-type regimes, including a boundary geometric case and genuine partial-theta regimes.
