Czechoslovak Journal of Physics

, Volume 56, Issue 10–11, pp 1093–1098 | Cite as

Dobiński relations and ordering of boson operators

  • P. Blasiak
  • A. Gawron
  • A. Horzela
  • K. A. Penson
  • A. I. Solomon


We introduce a generalization of the Dobiński relation, through which we define a family of Bell-type numbers and polynomials. Such generalized Dobiński relations are coherent state matrix elements of expressions involving boson ladder operators. This may be used in order to obtain normally ordered forms of polynomials in creation and annihilation operators, both if the latter satisfy canonical and deformed commutation relations.


02.10.Ox 03.65.Fd 05.30.Jp 

Key words

boson normal ordering coherent states combinatorics Dobiński relations 


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  1. [1]
    A.M. Navon: Nuovo Cimento B16 (1973) 324.CrossRefADSGoogle Scholar
  2. [2]
    J. Katriel: Lett. Nuovo Cimento10 (1974) 565.Google Scholar
  3. [3]
    P. Blasiak, K.A. Penson, and A.I. Solomon: Phys. Lett. A309 (2003) 198.MATHCrossRefADSMathSciNetGoogle Scholar
  4. [4]
    For a review and exhaustive list of references see: P. Blasiak:Combinatorics of boson normal ordering and some applications, PhD Thesis, INP Kraków and Université Paris VI; quant-ph/0507206; Concepts of Physics1 (2004) 177.Google Scholar
  5. [4]a
    P. Blasiak, A. Gawron, A. Horzela, K.A. Penson, and A. I. Solomon: Czech. J. Phys.55 (2005) 1335.CrossRefADSMathSciNetGoogle Scholar
  6. [5]
    N. Fleury and A. Turbiner: J. Math. Phys.35 (1994) 6144.MATHCrossRefADSMathSciNetGoogle Scholar
  7. [6]
    P. Blasiak, A. Horzela, K.A. Penson, and A.I. Solomon: J. Phys. A: Math. Gen.39 (2006) 4999.MATHCrossRefADSMathSciNetGoogle Scholar
  8. [7]
    G. Dobiński: Grunert Archiv (Arch. für M. und Physik)61 (1877) 333.Google Scholar
  9. [8]
    L. Comtet:Advanced Combinatorics, Reidel, Dordrecht, 1974.MATHGoogle Scholar
  10. [9]
    H.S. Wilf:Generatingfunctionology, Academic Press, New York, 1994.MATHGoogle Scholar
  11. [10]
    W.H. Louisell:Quantum Statistical Properties of Radiation, J. Wiley, New York, 1990.MATHGoogle Scholar
  12. [11]
    H.-Y. Fan: J. Opt. B: Semiclass. Opt.5 (2003) R147.Google Scholar
  13. [12]
    J.R. Klauder and E.C.G. Sudarshan:Fundamentals of Quantum Optics, Benjamin, New York, 1968.Google Scholar
  14. [13]
    G. Dattoli, P.E. Ricci, and I. Khomasuridze: Int. Transforms and Special Functions15 (2004) 309.MATHCrossRefMathSciNetGoogle Scholar
  15. [14]
    A.I. Solomon: Phys. Lett. A196 (1984) 29.CrossRefADSGoogle Scholar
  16. [15]
    K.A. Penson and A.I. Solomon: J. Math. Phys.40 (1999) 2354.MATHCrossRefADSMathSciNetGoogle Scholar
  17. [16]
    J. Katriel and M. Kibler: J. Phys A25 (1992) 2683.MATHCrossRefADSMathSciNetGoogle Scholar
  18. [16]a
    J. Katriel: Phys. Lett. A273 (2000) 159.MATHCrossRefADSMathSciNetGoogle Scholar
  19. [17]
    P. Blasiak, A. Horzela, K.A. Penson, and A.I. Solomon: Czech. J. Phys.54 (2004) 1179.CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • P. Blasiak
    • 1
  • A. Gawron
    • 1
  • A. Horzela
    • 1
  • K. A. Penson
    • 2
  • A. I. Solomon
    • 2
    • 3
  1. 1.H. Niewodniczański Institute of Nuclear PhysicsPolish Acad. Sci.KrakówPoland
  2. 2.Laboratoire de Physique Théorique de la Matière CondenséeUniversité P. et M. CurieParis Cedex 05France
  3. 3.Physics and Astronomy DepartmentThe Open UniversityMilton KeynesUnited Kingdom

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