论文标题
正交多项式序列的所有高几何体和基本高体积家族的统一结构
A unified construction of all the hypergeometric and basic hypergeometric families of orthogonal polynomial sequences
论文作者
论文摘要
我们构建了一个$ h $的正交多项式序列，其中包含ASKEY计划中的所有家庭和$ Q $ -ASKEY计划。 $ H $中的多项式序列是通用一阶差方程的解决方案，该方程由三个线性复发序列确定。其中两个序列是第三顺序的差方程的解，另一个序列满足了第五阶的相关差方程。 我们获得了正交多项式系数的显式表达式，以及有关多项式牛顿类型的基础的广义矩。我们还获得了$ h $中多项式序列满足的三项复发关系系数的显式公式。 集合$ h $包含ASKEY计划中的所有15个家庭和$ Q $ -ASKEY计划中的所有29个家庭。这些家族中的每一个都是通过直接替换适当的价值来获得我们一般公式中参数的。唯一需要一些限制的情况是Hermite和连续的$ Q $ hermite多项式。我们介绍了一些家庭的参数值。
We construct a set $H$ of orthogonal polynomial sequences that contains all the families in the Askey scheme and the $q$-Askey scheme. The polynomial sequences in $H$ are solutions of a generalized first-order difference equation which is determined by three linearly recurrent sequences of numbers. Two of these sequences are solutions of a difference equation of order three and the other sequence satisfies a related difference equation of order five. We obtain explicit expressions for the coefficients of the orthogonal polynomials and for the generalized moments with respect to a basis of Newton type of the space of polynomials. We also obtain explicit formulas for the coefficients of the three-term recurrence relation satisfied by the polynomial sequences in $H$. The set $H$ contains all the 15 families in the Askey scheme and all the 29 families in the $q$-Askey scheme. Each of these families is obtained by direct substitution of appropriate values for the parameters in our general formulas. The only cases that require some limits are the Hermite and continuous $q$-Hermite polynomials. We present the values of the parameters for some of the families.
