论文标题
多个与汇合高几何函数相关的正交多项式
Multiple orthogonal polynomials associated with confluent hypergeometric functions
论文作者
论文摘要
我们介绍和分析了一个超同质多项式多项式的新家族，相对于在正真实线上支持的两种措施，可以用第二种汇合的超测量函数来描述。这两种措施形成了尼基辛系统。我们的重点是用于步骤线上的索引的多个正交多项式。相对于这些指数，I型和II型多项式的衍生物的序列再次是多个正交的，它们对应于具有移位参数的原始序列。对于I型多项式，我们提供了Rodrigues公式。我们通过其显式表达将II型多项式表征为终止的广义高几幅序列，是对三阶微分方程的解决方案，并通过其复发关系。后者涉及无限且渐近周期性的复发系数。基于这些信息，我们推断了II型多项式最大零的渐近行为。我们还讨论了这些多项式与多个正交多项式之间的限制关系，相对于修改的贝塞尔权重。在讨论中的II型多项式参数上的特定选择对应于已知的三重对称Hahn-classical多元正交多项式在类似恒星样集中的分数。
We introduce and analyse a new family of multiple orthogonal polynomials of hypergeometric type with respect to two measures supported on the positive real line which can be described in terms of confluent hypergeometric functions of the second kind. These two measures form a Nikishin system. Our focus is on the multiple orthogonal polynomials for indices on the step line. The sequences of the derivatives of both type I and type II polynomials with respect to these indices are again multiple orthogonal and they correspond to the original sequences with shifted parameters. For the type I polynomials, we provide a Rodrigues formula. We characterise the type II polynomials via their explicit expression as a terminating generalised hypergeometric series, as solutions to a third-order differential equation and via their recurrence relation. The latter involves recurrence coefficients which are unbounded and asymptotically periodic. Based on this information we deduce the asymptotic behaviour of the largest zeros of the type II polynomials. We also discuss limiting relations between these polynomials and the multiple orthogonal polynomials with respect to the modified Bessel weights. Particular choices on the parameters for the type II polynomials under discussion correspond to the cubic components of the already known threefold symmetric Hahn-classical multiple orthogonal polynomials on star-like sets.
