Bispherical harmonics are the solutions to Laplace’s equation in bispherical coordinates. We investigate the relationships between spherical harmonics and bispherical harmonics in terms of radial inversion and derive new series expansions between the harmonics. The series coefficients are the Delannoy numbers encountered in combinatorics, and we exploit the series to derive for them a new second order homogeneous recurrence relation. We use the T-matrix/null field method to solve for the potential of two different spheres in an arbitrary static external electric field, as a series of bispherical harmonics where the coefficients are related via matrix transformations. The matrix elements are expressed as surface integrals of bispherical harmonics, for which analytic expressions are derived in terms of Legendre functions. The resonant values of the dielectric function are computed via an eigenvalue problem which is found to be more stable than the solution using difference equations. The rate of convergence of the matrix formulation is compared to the re-expansion method using spherical harmonics for a uniform field an a dipole in the gap; each series converges faster in a different region of space, which we discuss in terms of the boundaries of convergence of each solution and the image singularities of the scattered field.

双球面谐波是双球面坐标中拉普拉斯方程的解。我们根据径向反演研究球谐函数和双球谐函数之间的关系，并推导出谐波之间的新级数展开式。级数系数是组合学中遇到的德兰诺伊数，我们利用该级数为它们推导出一个新的二阶齐次递推关系。我们使用 T 矩阵/零场方法来求解任意静态外部电场中两个不同球体的电势，作为一系列双球面谐波，其中系数通过矩阵变换相关。矩阵元素表示为双球谐函数的表面积分，其解析表达式是根据勒让德函数导出的。介电函数的谐振值是通过特征值问题计算的，发现该问题比使用差分方程的解更稳定。将矩阵公式的收敛速度与使用球谐函数的再扩展方法进行比较，以用于均匀场和间隙中的偶极子；每个系列在不同的空间区域收敛得更快，我们根据每个解的收敛边界和散射场的图像奇点来讨论。