广义Fibonacci多项式的结式、判别式和导数
The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials
摘要：
一个二阶多项式序列是\emph{Fibonacci-type}（\emph{Lucas-type}），如果它的比内公式具有与Fibonacci（Lucas）数的结构相似的结构.这些类型的序列的已知示例是：Fibonacci多项式、Pell多项式、Fermat多项式、切比雪夫多项式、Morgan-Voyce多项式、Lucas多项式、Pell-Lucas多项式、Fermat-Lucas多项式、切比雪夫多项式。两个多项式的\emph{resultant}是Sylvester矩阵的行列式，多项式$p$的\emph{discriminant}是$p$及其导数的结式。我们研究了Fibonacci型多项式和Lucas型多项式以及这两种类型的组合的结式、判别式和导数。作为推论，我们给出了上述已知多项式的结果、判别式和导数的显式公式。
英文摘要：
A second order polynomial sequence is of \emph{Fibonacci-type} (\emph{Lucas-type}) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Known examples of these type of sequences are: Fibonacci polynomials, Pell polynomials, Fermat polynomials, Chebyshev polynomials, Morgan-Voyce polynomials, Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials, Chebyshev polynomials. The \emph{resultant} of two polynomials is the determinant of the Sylvester matrix and the \emph{discriminant} of a polynomial $p$ is the resultant of $p$ and its derivative. We study the resultant, the discriminant, and the derivatives of Fibonacci-type polynomials and Lucas-type polynomials as well combinations of those two types. As a corollary we give explicit formulas for the resultant, the discriminant, and the derivative for the known polynomials mentioned above.
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