In a recent article generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal probability distributions was a key point to allow to give them a statistical interpretation in terms of probabilities. In this article we present an approach based on generating functions that solves the previous difficulties. Our main theorem makes explicit the conditions under which those formal probability distributions are always non-negative. Therefore, the constraints of non-negativeness are automatically fulfilled giving a complete characterization in terms of generating functions. A large number of analytical examples becomes available.
REFERENCES
1.
V. V.
Dodonov
, “‘Nonclassical states’ in quantum optics: A ‘squeezed’ review of the first 75 years
,” J. Opt. B: Quantum Semiclassical Opt.
4
, R1
–R33
(2002
), and references therein.2.
J.-P.
Gazeau
, Coherent States in Quantum Physics
(Wiley
, Weinheim
, 2009
).3.
R. L.
de Matos Filho
and W.
Vogel
, “Nonlinear coherent states
,” Phys. Rev. A
54
, 4560
–4563
(1996
).4.
Z.
Kis
, W.
Vogel
, and L.
Davidovich
, “Nonlinear coherent states of trapped-atom motion
,” Phys. Rev. A
64
, 03401
–03410
(2001
).5.
E. M. F.
Curado
, J.-P.
Gazeau
, and Ligia M. C. S.
Rodrigues
, “Nonlinear coherent states for optimizing quantum information
,” Phys. Scr.
82
, 038108
–038109
(2010
).6.
E. M. F.
Curado
, J.-P.
Gazeau
, and Ligia M. C. S.
Rodrigues
, “On a generalization of the binomial distribution and its Poisson-like limit
,” J. Stat. Phys.
146
, 264
–280
(2011
).7.
R.
Hanel
, S.
Thurner
, and C.
Tsallis
, “Limit distributions of scale-invariant probabilistic models of correlated random variables with the q-Gaussian as an explicit example
,” Eur. Phys. J. B
72
, 263
–268
(2009
).8.
A.
Rodrigues
and C.
Tsallis
, “A dimension scale-invariant probabilistic model based on Leibniz-like pyramids
,” J. Math. Phys.
53
, 023302
–023327
(2012
).9.
W.
Magnus
, F.
Oberhettinger
, and R. P.
Soni
, Formulas and Theorems for the Special Functions of Mathematical Physics
, 3rd ed. (Springer-Verlag
, Berlin
, 1996
).10.
I. S.
Gradshteyn
and I. M.
Ryzhik
, Table of Integrals, Series, and Products
, 5th ed. (Academic
, USA
, 1994
).11.
M. S.
Abramowitz
and I. A.
Stegun
, Handbook of Mathematical Functions
(Dover Publications
, NY
, 1972
), ISBN 978-0-486-61272-0.12.
We remark that, after this manuscript was submitted to arXiv, these conjectures were proved by
H.
Denoncourt
, private communication (March 23, 2012) and by C.
Vignat
and O.
Leveque
, “Proof of a conjecture by Gazeau et al. using Gould Hopper polynomials
,” e-print arXiv:1203.5418v1 [math-ph].13.
R.
Loudon
, The Quantum Theory of Light
, 3rd ed. (Oxford University Press
, New York
, 2000
).14.
C. H.
Bennett
and G.
Brassard
, in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing
(IEEE
, New York
, 1984
), p. 175
.15.
C. A.
Fuchs
, “Distinguishability and accessible information in quantum theory
,” Ph.D. dissertation, University of New Mexico
, 1996
.© 2012 American Institute of Physics.
2012
American Institute of Physics
You do not currently have access to this content.