论文标题
满足四项连续关系的多项式表的零
Zeros of a table of polynomials satisfying a four-term contiguous relation
论文作者
论文摘要
对于任何$ a（z），b（z），c（z）\ in \ mathbb {c} [z] $，我们研究了多项式$ \ left \ left \ left \ {p_ {m，n}（z）（z）\ right \ right \} _ {m，n \ in \ in \ mathbb {n \ in \ mathy ref in \ in \ n} $ ref in \ n \ n} $ restion的零分布的零分布\ [p_ {m，n}（z）= a（z）p_ {m-1，n}（z）+b（z）p_ {m，n-1}（z）+c（z）+c（z）p_ {m-1，n-1}（z）（z）（z）\]，初始条件$ p_ {0,0,0,（z）= 1 $ p_ {$ p_ {$ p_ = 1 $ p_ = $ p ^ n n n n of m，n \ in \ mathbb {n} $。我们表明，$ p_ {m，n}（z）$的零位于曲线上，其方程用$ a（z），b（z）$和$ c（z）$明确给出。我们还研究了具有一般初始条件的案例的零分布。
For any $A(z),B(z),C(z)\in\mathbb{C}[z]$, we study the zero distribution of a table of polynomials $\left\{ P_{m,n}(z)\right\} _{m,n\in\mathbb{N}_{0}}$ satisfying the recurrence relation \[ P_{m,n}(z)=A(z)P_{m-1,n}(z)+B(z)P_{m,n-1}(z)+C(z)P_{m-1,n-1}(z) \] with the initial condition $P_{0,0}(z)=1$ and $P_{-m,-n}(z)=0$ $\forall m,n\in\mathbb{N}$. We show that the zeros of $P_{m,n}(z)$ lie on a curve whose equation is given explicitly in terms of $A(z),B(z)$, and $C(z)$. We also study the zero distribution of a case with a general initial condition.
