论文标题
相对于一类差分运算符的正交多项式
On orthogonal polynomials with respect to a class of differential operators
论文作者
论文摘要
我们考虑有关线性差分运算符$$ \ MATHCAL {l}^{（M）} = \ sum_ {k = 0}^{m}ρ_{k}（Z） $ \ {ρ_k\} _ {k = 0}^{m} $是复杂的多项式，使得$ deg [ρ_k] \ leq k，0 \ leq k \ leq m $，具有至少一个index的平等。我们分析了这些多项式的唯一性和零位置。在这种正交性中发生的一个有趣的现象是算子的存在，正交多项式的相关序列将其减少到有限的集合。对于给定的运算符，我们可以根据具有不同系数不同的差异方程式的线性系统来确保其无限序列的量度分类。同样，对于一阶差分运算符的情况，我们定位零并确定这些多项式的强渐近行为。
We consider orthogonal polynomials with respect to a linear differential operator $$\mathcal{L}^{(M)}=\sum_{k=0}^{M}ρ_{k}(z)\frac{d^k}{dz^k}, $$ where $\{ρ_k\}_{k=0}^{M}$ are complex polynomials such that $deg[ρ_k]\leq k, 0\leq k \leq M$, with equality for at least one index. We analyze the uniqueness and zero location of these polynomials. An interesting phenomenon occurring in this kind of orthogonality is the existence of operators for which the associated sequence of orthogonal polynomials reduces to a finite set. For a given operator, we find a classification of the measures for which it is possible to guarantee the existence of an infinite sequence of orthogonal polynomials, in terms of a linear system of difference equations with varying coefficients. Also, for the case of a first-order differential operator, we locate the zeros and establish the strong asymptotic behavior of these polynomials.
