关于广义Motzkin和Schröder系数多项式的实根估计

On the Real Root Estimation of Generalized Motzkin and Schröder Coefficient Polynomials

为确定更广泛类型的实系数多项式的实数根,利用变系数二阶线性递归数列定义了广义Motzkin数和Schröder数,并研究了以它们为系数的多项式的根的存在性问题,得到“当多项式的次数为偶数时,它无实根;当多项式的次数为奇数时,它只有1个实根”的结论.此外,文献[1-8]中的相同结论均为其特殊情况.

In order to determine the real roots of more general types of real coefficient polynomials,the generalized Motzkin numbers and Schröder numbers are defined by using the second-order linear recursive sequence with variable coefficients,and the existence of polynomial roots with them as coefficients is studied.It is obtained that when the degree of the polynomial is even,it has no real roots;when the degree of the polynomial is odd,it has only one real root.In addition,the same conclusion in references[1-8]is its special case.