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Normal ordering for nonlinear deformed ladder operators and the

f-generalization of the Stirling and Bell numbers

### Abstract

We resolve the normal ordering problem for symmetric and asymmetric strings of the nonlinear deformed ladder operators for supersymmetric and shape-invariant potential systems, where r and n are positive integers. We provide exact and explicit expressions for their normal forms and , where in all are at the right side. We find that the solutions involve sequence of expansion coefficients which, for r, n > 1, corresponds to the f-deformed generalization of the classical Stirling and Bell numbers of the second kind. We apply the general formalism for the translational shape-invariant potential systems as well as for the particular case of the harmonic oscillator potential system. We show that these numbers are obtained for families of polynomial expressions generated with the deformations parameters and the parameters related to the forms of the supersymmetric partner potentials.

© 2015 AIP Publishing LLC

Received 10 December 2014
Accepted 20 November 2015
Published online 11 December 2015

Acknowledgments:
This work was supported in part by U.S. National Science Foundation Grant No. PHY-1205024 at the University of Wisconsin, and in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation. A.N.F.A. acknowledges partial support from CNPq and FAPERJ/CNPq through the PRONEX program.

Article outline:

I. INTRODUCTION
II. THE LADDER OPERATORS’ ALGEBRAIC CONSTRUCTION
A. Undeformed ladder operators and their algebra
1. Definitions and algebraic relations
2. Eigenstates and eigenvalues
B. Nonlinear deformed ladder operators and their algebra
1. Definitions
2. Eigenstates and eigenvalues
3. Algebraic relations
III. THE NORMAL ORDERING OF THE SYMMETRIC STRING
A. The deformed ladder operators’ normal ordering process
B. The *f*-deformed generalization of the Stirling and Bell numbers
C. The *f*-deformed Stirling numbers recurrence relation
D. The analog of the Dobiński relation for the *f*-deformed Bell function
E. A generating function for the *f*-deformed Bell function
IV. THE NORMAL ORDERING OF ASYMMETRIC STRINGS AND
A. An auxiliary relation and some ladder relations
B. The (*r*, 1)-generalized Stirling and Bell numbers of the second kind for *f*-deformed systems
C. The recurrence relation for the *f*-deformed Stirling numbers
V. PHYSICAL APPLICATIONS
A. Quantum statistical relations
1. Motivation
2. Density operator and partition function
3. The observable expectation values
4. The pot of gold at the end of the rainbow
VI. SOME ILLUSTRATIVE EXAMPLES OF NONLINEAR QUANTUM DEFORMED SYSTEMS
A. The exponential analog of the Stirling and Bell numbers
1. Exponentially deformed algebra
2. The exponentially deformed Stirling and Bell numbers of the second kind
3. Exponentially deformed form of the (*r*, 1)-generalized Stirling and Bell numbers of the second kind
4. The example of the translational shape-invariant systems
B. The hyperbolic analog of the Stirling and Bell numbers
1. Hyperbolically deformed algebra
2. The hyperbolically deformed Stirling and Bell numbers of the second kind
3. The hyperbolically deformed form of the (*r*, 1)-generalized Stirling and Bell numbers of the second kind
4. The example of the translational shape-invariant systems
C. Arik-Coon multiparameter analog of the Stirling and Bell numbers
1. Multiparameter deformed algebra
2. Arik-Coon deformed Stirling and Bell numbers of the second kind
3. Arik-Coon deformed form of the (*r*, 1)-generalized Stirling and Bell numbers of the second kind
4. The example of the translational shape-invariant systems
VII. FINAL REMARKS

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2015-12-11

2016-10-26

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