Generating-function representation for scalar products

다국어 초록 (Multilingual 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.

참고문헌 (Reference)

1 G.N. Watson, S1–8 : 189-, 1933

2 G.N. Watson, S1–8 : 194-, 1933

3 G.N. Watson, S1–8 : 289-, 1933

4 A. K. Svinin, 63 : 25-, 2019

5 A.L. Fetter, "Theoretical Mechanics of Particles and Continua" McGraw-Hill 1980

6 J. Beery, "Sums of Powers of Positive Integers-Introduction" Mathematical Association of America Publications 2009

7 Kim U-Rae, "Solving an eigenproblem with analyticity of the generating function" 한국물리학회 79 (79): 113-124, 2021

8 G.B. Arfken, "Mathematical Methods for Physicists" Elsevier 2012

9 S.K. Lando, "Lectures on Generating Functions"

10 M. Abramowitz, "Handbook of Mathematical Functions" Dover 785-, 1965

11 D.-W. Jung, "Finding normal modes of loaded string with Lagrange multipliers"

12 P.D. Ritger, "Differential Equations with Applications" McGraw-Hill 367-372, 1968

13 J. Brown, "Complex Variables and Applications" McGraw-Hill Education 2013

14 R.A. Matzner, "Classical Mechanics" Prentice-Hall 232-239, 1991

15 J.B. Marion, "Classical Dynamics of Particles and Systems" Cengage Learning 2003