Normal ordering for deformed boson operators and operator-valued deformed Stirling numbers

The normal ordering formulae for powers of the boson number operator n are extended to deformed bosons. It is found that for the 'M-type' deformed bosons, which satisfy aaDagger -qa Dagger a=1, the extension involves a set of deformed Stirling numbers which replace the Stirling numbers occurring in the conventional case. On the other hand, the deformed Stirling numbers which have to be introduced in the case of the 'P-type' deformed bosons, which satisfy aa Dagger -qa Dagger a=q-n are found to depend on the operator n. This distinction between the two types of deformed bosons is in harmony with earlier observations made in the context of a study of the extended Campbell-Baker-Hausdorff formula.

[1]  John Riordan,et al.  Introduction to Combinatorial Analysis , 1959 .

[2]  Stephen C. Milne,et al.  A $q$-analog of restricted growth functions, Dobinski’s equality, and Charlier polynomials , 1978 .

[3]  John Riordan,et al.  Introduction to Combinatorial Analysis , 1958 .

[4]  Michelle L. Wachs,et al.  p, q-Stirling numbers and set partition statistics , 1990, J. Comb. Theory, Ser. A.

[5]  Leonard Carlitz,et al.  $q$-Bernoulli numbers and polynomials , 1948 .

[6]  Jeffrey B. Remmel,et al.  Q-counting rook configurations and a formula of frobenius , 1986, J. Comb. Theory A.

[7]  A. R. Edmonds.  Angular Momentum in Quantum Mechanics , 1957 .

[8]  S. Berman,et al.  Nuovo Cimento , 1983 .

[9]  Ramaswamy Jagannathan,et al.  A (p, q)-oscillator realization of two-parameter quantum algebras , 1991 .

[10]  J. M. Eisenberg,et al.  Excitation mechanisms of the nucleus , 1976 .

[11]  W. Louisell,et al.  Radiation and noise in quantum electronics , 1964 .

[12]  L. C. Biedenharn,et al.  The quantum group SUq(2) and a q-analogue of the boson operators , 1989 .

[13]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[14]  Harold Exton,et al.  q-hypergeometric functions and applications , 1983 .

[15]  J. Katriel,et al.  A q-analogue of the Campbell-Baker-Hausdorff expansion , 1991 .

[16]  Jacob Katriel,et al.  Combinatorial aspects of boson algebra , 1974 .

[17]  R. Wilcox.  Exponential Operators and Parameter Differentiation in Quantum Physics , 1967 .

[18]  H. Gould.  The $q$-Stirling numbers of first and second kinds , 1961 .

[19]  Leonard Carlitz,et al.  On abelian fields , 1933 .

[20]  A. J. Macfarlane,et al.  On q-analogues of the quantum harmonic oscillator and the quantum group SU(2)q , 1989 .

[21]  N. D. Mermin,et al.  Special Functions: A Group Theoretic Approach , 1970 .