An Introduction to Special Functions with Some Applications to Quantum Mechanics

  • Sergei K. SuslovEmail author
  • José M. Vega-Guzmán
  • Kamal Barley
Conference paper
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)


A short review on special functions and solution of the Coulomb problems in quantum mechanics is given. Multiparameter wave functions of linear harmonic oscillator, which cannot be obtained by the standard separation of variables, are discussed. Expectation values in relativistic Coulomb problems are studied by computer algebra methods.


Nonrelativistic and relativistic Coulomb problems Schrödinger equation Dirac equation Laguerre polynomials Spherical harmonics Bessel functions Hahn polynomials Chebyshev polynomials of a discrete variable Generalized hypergeometric series 

Mathematics Subject Classification (2000)

Primary 33A65 81C05; Secondary 81C40 



We are very grateful to the organizers of the AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications held in Douala, Cameroon, in October 2018, without their help this publication would be impossible. Sincere thanks to our co-authors Christoph Koutschan, Sergey Kryuchkov, Raquel M. López, Peter Paule and Erwin Suazo; some of our joint results were included in this chapter. We are very grateful to the referee for an outstanding job, her/his numerous comments helped us to improve the manuscript. Special thanks are directed to Jeremy Alm, Valentin Andreev, Al Boggess and Michael Laidacker for support and continuous encouragement. This work was partially supported by the National Institute of General Medical Sciences (NIGMS) at the National Institutes of Health (NIH; Grant # K12GM102778) and by the Lamar University Research Enhancement Grant (REG # 420266).


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Authors and Affiliations

  1. 1.Arizona State UniversityTempeUSA
  2. 2.Lamar UniversityBeaumontUSA
  3. 3.Applied Mathematics and StatisticsStony Brook UniversityStony BrookUSA

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