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, and 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：这份关于轴对称超球面/盖根鲍尔多项式及其在2D和3D声场直接度设计中应用的报告，提出了一种替代性数学形式，不同于例如我与马蒂亚斯·弗兰克合著的声场著作、杰罗姆·丹尼尔的博士论文、加里·埃尔科的差分阵列著作章节，或博阿兹·拉法埃利的球面麦克风阵列著作中的阐述。超球面/盖根鲍尔多项式在设计轴对称波束和理解球面t设计时极具价值，本报告将阐明圆形、球面和超球面轴对称多项式的本质。除其数学趣味性外，这些多项式在球面波束成形中具有实用价值——正如关于球面和差分麦克风阵列的文献所述。本报告将利用超球面/盖根鲍尔多项式统一推导任意维度D下已知文献中的各类直接度设计或声场阶次加权方案：最大指向性指数/基本型、最大rE型、超心型、心型/同相型。高阶心型与超心型是否存在关联？如何定义具有轴向平坦度约束的直接度模式？