论文标题
贝塞尔多项式的正交公式
Quadrature formulas for Bessel polynomials
论文作者
论文摘要
正交公式是一个公式计算通过有限点评估确定集成的公式。正交多项式的某些正交公式的存在与有趣的问题有关，例如Waring在数字理论中的问题和代数组合中的球形设计。 Sawa和Uchida证明了某些经典正交多项式的重量功能的某些有理正交公式的存在和不存在。经典的正交多项式属于Askey-Scheme，这是超几何正交多项式的层次结构。因此，很自然地将Sawa和Uchida的工作扩展到Askey-Scheme中的其他多项式。在本文中，我们将Sawa和Uchida的工作扩展到Bessel多项式的重量函数。在证明中，我们使用riesz-shohat定理和牛顿多边形。也具有数量理论的兴趣，将某些结果的证据降低为确定椭圆曲线上的理性点的集合。
A quadrature formula is a formula computing a definite integration by evaluation at finite points. The existence of certain quadrature formulas for orthogonal polynomials is related to interesting problems such as Waring's problem in number theory and spherical designs in algebraic combinatorics. Sawa and Uchida proved the existence and the non-existence of certain rational quadrature formulas for the weight functions of certain classical orthogonal polynomials. Classical orthogonal polynomials belong to the Askey-scheme, which is a hierarchy of hypergeometric orthogonal polynomials. Thus, it is natural to extend the work of Sawa and Uchida to other polynomials in the Askey-scheme. In this article, we extend the work of Sawa and Uchida to the weight function of the Bessel polynomials. In the proofs, we use the Riesz--Shohat theorem and Newton polygons. It is also of number theoretic interest that proofs of some results are reduced to determining the sets of rational points on elliptic curves.
