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：本报告探讨轴对称超球面/盖根鲍尔多项式及其在二维和三维Ambisonic指向性设计中的应用，提出了一种替代数学形式，有别于我本人与Matthias Frank合著的Ambisonic专著、Jérôme Daniel的博士论文、Gary Elko的差分阵列专著章节以及Boaz Rafaely的球面麦克风阵列专著中的论述。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、球面和超球面对称多项式的本质。这些多项式本身具有数学趣味性，更可用于文献中所述的球面波束成形技术（如球面及差分麦克风阵列相关文献）。本报告将利用这些超球面/盖根鲍尔多项式，统一推导任意维度D下文献中已知的各类指向性设计或Ambisonic阶次加权方法：最大指向性指数/基本型、最大rE型、超心型、心型/同相型。高阶心型与超心型之间是否存在关联？如何定义具有轴向平坦度约束的指向性模式？