This report on axisymmetric ultraspherical/Gegenbauer polynomials and their use in Ambisonic directivity design in 2D and 3D presents an alternative mathematical formalism to what can be read in, e.g., my and Matthias Frank's book on Ambisonics or J\'er\^ome Daniel's thesis, Gary Elko's differential array book chapters, or Boaz Rafaely's spherical microphone array book. Ultraspherical/Gegenbauer polynomials are highly valuable when designing axisymmetric beams and understanding spherical t designs that this report will shed some light on what circular, spherical, and ultraspherical axisymmetric polynomials are. While mathematically interesting by themselves already, they can be useful in spherical beamforming as described in the literature on spherical and differential microphone arrays. In this report, these ultraspherical/Gegenbauer polynomials will be used to uniformly derive for arbitrary dimensions D the various directivity designs or Ambisonic order weightings known from literature: max-DI/basic, max-rE , supercardioid, cardioid/inphase. Is there a way to relate higher-order cardioids and supercardioids? How could one define directivity patterns with an on-axis flatness constraint?

翻译：本报告论述了轴对称超球面/盖根鲍尔多项式及其在2D与3D声场合成指向性设计中的应用，提出了与传统表述（例如本人与Matthias Frank合著的声场合成著作、Jérôme Daniel的博士论文、Gary Elko的差分阵列专著章节、以及Boaz Rafaely的球阵列专著）不同的数学形式体系。超球面/盖根鲍尔多项式在轴对称波束设计与理解球面t-设计方面具有重要价值，本报告将阐明圆形、球面及超球面轴对称多项式的本质。这些多项式本身具有数学趣味性，同时可应用于文献所述的球面波束成形与差分传声器阵列。本报告将利用超球面/盖根鲍尔多项式，统一推导文献中已知的任意维度D下的各类指向性设计或声场合成阶次加权方案：最大指向性指数/基础型、最大rE型、超心型、心型/同相型。是否存在高阶心型与超心型之间的关联方法？如何定义具有轴向平坦度约束的指向性图案？