Generating-function representation for scalar products

Abstract

We employ the generating-function representation for an n-dimensional vector in Euclidean or Hilbert space to evaluate scalar products. The generating function is constructed as a power series in a complex variable weighted by the components of a vector. The scalar product is represented by a convolution of the generating functions for the vectors integrated over a closed contour in the complex plane. The analyticity of the generating functions associated with the Laurent theorem reduces the evaluation of the scalar product into counting combinatoric multiplicity factors. As applications, we provide two exemplary computations: the sum of the squares of integers and the normalization of normal modes in a vibrating loaded string. As a byproduct of the latter example, we find a new alternative proof of a famous trigonometric identity that is essential for Fourier analyses.