Generating-function representation for scalar products

U. Rae Kim, Dong Won Jung, Dohyun Kim, Jungil Lee, Chaehyun Yu

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)429-437
Number of pages9
JournalJournal of the Korean Physical Society
Volume79
Issue number5
DOIs
Publication statusPublished - 2021 Sept

Bibliographical note

Funding Information:
As members of the Korea Pragmatist Organization for Physics Education (KPOP), the authors thank the remaining members of KPOP for useful discussions. The work is supported in part by the National Research Foundation of Korea (NRF) under the BK21 FOUR program at Korea University, Initiative for science frontiers on upcoming challenges. The work of JL is supported in part by grants funded by the Korea government (MSIT) under Contract Nos. NRF-2020R1A2C3009918 and NRF-2017R1E1A1A01074699. The work of DWJ and CY is supported in part by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education 2018R1D1A1B07047812 (DWJ) and 2020R1I1A1A01073770 (CY), respectively. All authors contributed equally to this work.

Publisher Copyright:
© 2021, The Korean Physical Society.

Keywords

  • Generating function
  • Scalar product
  • String vibration
  • n-dimensional vector

ASJC Scopus subject areas

  • General Physics and Astronomy

Fingerprint

Dive into the research topics of 'Generating-function representation for scalar products'. Together they form a unique fingerprint.

Cite this