We present a generating-function representation of a vector defined in either Euclidean or Hilbert space with arbitrary dimensions. The generating function is constructed as a power series in a complex variable whose coefficients are the components of a vector. As an application, we employ the generating-function formalism to solve the eigenproblem of a vibrating string loaded with identical beads. The corresponding generating function is an entire function. The requirement of the analyticity of the generating function determines the eigenspectrum all at once. Every component of the eigenvector of the normal mode can be easily extracted from the generating function by making use of the Schläfli integral. This is a unique pedagogical example with which students can have a practical contact with the generating function, contour integration, and normal modes of classical mechanics at the same time. Our formalism can be applied to a physical system involving any eigenvalue problem, especially one having many components, including infinite-dimensional eigenstates.

我们提出了在欧几里得或希尔伯特空间中定义的任意维度向量的生成函数表示。生成函数被构造为复变量中的幂级数，其系数是向量的分量。作为一个应用，我们采用生成函数形式来解决装有相同珠子的振动弦的特征问题。对应的生成函数是整函数。生成函数的解析性要求一下子决定了本征谱。通过使用 Schläfli 积分，可以很容易地从生成函数中提取正常模式特征向量的每个分量。这是一个独特的教学示例，学生可以通过它实际接触生成函数、轮廓积分、和经典力学的正常模式同时进行。我们的形式主义可以应用于涉及任何特征值问题的物理系统，尤其是具有许多组件的物理系统，包括无限维特征值。