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：本报告探讨了轴对称超球面/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，提出了一种替代性的数学形式体系。与本人及Matthias Frank合著的Ambisonics专著、Jérôme Daniel的博士论文、Gary Elko的差分阵列专著章节，以及Boaz Rafaely的球面传声器阵列专著中的相关论述不同，本报告揭示了超球面/盖根鲍尔多项式在设计轴对称波束和理解球面t设计中的核心价值，并系统阐释了圆形、球面及超球面轴对称多项式的内涵。这些多项式本身即具有数学研究价值，更可有效应用于文献中所述的球面波束形成技术（如球面与差分传声器阵列）。本报告将利用超球面/盖根鲍尔多项式，统一推导任意维度D下的各类指向性设计及Ambisonic阶加权方案，包括最大指向性指数/基本型、最大rE型、超心形、心形/同相型等经典形式。进一步探讨高阶心形与超心形的关联性，并提出了具有轴向平坦度约束的指向性模式定义方法。