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与三维Ambisonic指向性设计中的应用，提出了一种替代性数学形式体系，与笔者及Matthias Frank合著的Ambisonics专著、Jérôme Daniel的博士论文、Gary Elko的差分阵列著作章节、以及Boaz Rafaely的球面传声器阵列专著中的描述形成互补。超球面/盖根鲍尔多项式在轴对称波束设计与球面t设计理解中具有极高价值，本报告将阐明圆环型、球面型及超球面型轴对称多项式的本质特征。除其自身数学趣味性外，这些多项式在球面波束成形（如球面与差分传声器阵列文献所述）中具有重要应用价值。通过超球面/盖根鲍尔多项式，本报告将统一推导任意维度D下的各类指向性设计或Ambisonic阶次加权方案（包括最大指向性指数/基础型、最大rE型、超心型、心型/同相型），这些方案均源自现有文献。高阶心型与超心型之间存在何种关联？如何定义具有轴向平坦性约束的指向性模式？