AI 中文总结
研究引入广义q-摩根-沃伊斯多项式及其特殊情形，通过特定递推关系定义，利用斐波那契算子和二项式定理建立生成函数与显式表达式，扩展到负指数，给出求和公式、行列式表示，还建立了q-卡西尼公式。
AI 中文摘要
本研究引入并研究了广义q-摩根-沃伊斯多项式及其特殊情形，包括第一类和第二类q-摩根-沃伊斯多项式、q-霍拉达姆-摩根-沃伊斯多项式以及斐波那契型q-摩根-沃伊斯多项式。广义q-摩根-沃伊斯多项式由具有特定q次幂q^{n - 2}和负号的递推关系定义。利用斐波那契算子和二项式定理建立了这些多项式的生成函数和显式表达式，还将其扩展到负指数并推导了相应公式，全面给出了广义多项式及其特殊情形的求和公式和行列式表示，最后通过特定方阵定义及其递推关系建立了广义q-摩根-沃伊斯多项式的q-卡西尼公式。
英文摘要
This study introduces and investigates generalized q-Morgan-Voyce polynomials and their specific cases, including the first and second kinds of q-Morgan-Voyce polynomials, q-Horadam-Morgan-Voyce polynomials, and Fibonacci-type q-Morgan-Voyce polynomials. The generalized q-Morgan-Voyce polynomials are defined by the recurrence relation featuring the specific q-power q^{n-2} and a negative sign, formulated as M_{n}(x,q) = (x+1+q)M_{n-1}(x,q) - q^{n-2}M_{n-2}(x,q) for n >= 2. The generating functions and explicit expressions for these polynomials are established by utilizing the Fibonacci operator and the binomial theorem. Furthermore, the study extends these polynomials to negative indices and derives the corresponding explicit formulas. Summation formulas and determinantal presentations of the generalized polynomials and their special cases are also comprehensively provided. Finally, the q-Cassini's formula for the generalized q-Morgan-Voyce polynomials is established through the definition of specific square matrices and their subsequent recurrence relations.
