论文标题
在离散的连贯对措施上
On discrete coherent pairs of measures
论文作者
论文摘要
在[Castillo \＆Mbouna，Indag。数学。 {\ bf 31}（2020）223-234]，引入了$π_n$ - coherent对$（m，k）$（m，k）$的概念，并引入了索引$ m $。该定义与标准导数操作员隐式相关，会自动排除所谓的离散正交多项式。本注的目的是双重的：首先，我们使用（离散的）Hahn差异操作员并在此框架中重写已知结果；其次，作为一个应用程序，我们在这种情况下详尽地描述了（离散的）自相关对，无论$ m = 0 $，$ n \ leq2 $和$（m，k）=（1,0）$。这是通过以统一的方式描述杰克逊运营商的经典正交多项式作为特殊或限制$ q $ $ q $ - 多种物质的特殊或限制案例的描述。这给了M. E. H Ismail在其专着[一个变量中的古典和量子正交多项式中提出的猜想的部分答案，剑桥大学出版社，2005年]。
In [Castillo \& Mbouna, Indag. Math. {\bf 31} (2020) 223-234], the concept of $π_N$-coherent pairs of order $(m,k)$ with index $M$ is introduced. This definition, implicitly related with the standard derivative operator, automatically leaves out the so-called discrete orthogonal polynomials. The purpose of this note is twofold: first we use the (discrete) Hahn difference operator and rewrite the known results in this framework; second, as an application, we describe exhaustively the (discrete) self-coherent pairs in the situation whether $M=0$, $N\leq2$, and $(m,k)=(1,0)$. This is proved by describing in a unified way the classical orthogonal polynomials with respect to Jackson's operator as special or limiting cases of a four parametric family of $q$-polynomials. This gives a partial answer to a conjecture posed by M. E. H Ismail in his monograph [Classical and quantum orthogonal polynomials in one variable, Cambridge University Press, 2005].
