We provide here some probabilistic interpretations of the generalized binomial distributions proposed by Gazeau et al. [“Generating functions for generalized binomial distributions,” J. Math. Phys. 53, 103304 (2012)] https://doi.org/10.1063/1.4757601. In the second part, we prove the “strong conjecture” expressed by Gazeau et al. about the coefficients of the Taylor expansion of the exponential of a polynomial. The proof relies mainly on properties of the Gould-Hopper polynomials.
REFERENCES
1.
H.
Bergeron
, E. M. F.
Curado
, J.-P.
Gazeau
, and L. M. C. S.
Rodrigues
, “Generating functions for generalized binomial distributions
,” J. Math. Phys.
53
, 103304
(2012
).2.
E. M. F.
Curado
, J.-P.
Gazeau
, and L. M. C. S.
Rodrigues
, “On a generalization of the binomial distribution and its Poisson-like limit
,” J. Stat. Phys.
146
, 264
–280
(2011
).3.
H. W.
Gould
and A. T.
Hopper
, “Operational formulas connected with two generalizations of Hermite polynomials
,” Duke Math. J.
29
(1
), 51
–63
(1962
).4.
C.
Vignat
, “A probabilistic approach to some results by Nieto and Truax
,” J. Math. Phys.
51
, 123505
(2010
).5.
M. M.
Nieto
and D. R.
Truax
, “Arbitrary-order Hermite generating functions for obtaining arbitrary-order coherent and squeezed states
,” Phys. Lett. A
208
, 8
–16
(1995
).6.
P.
Carmona
, F.
Petit
, and M.
Yor
, “On exponential functionals of certain Lévy processes
,” in Exponential Functionals of Brownian Motion and Related Processes
, edited by M.
Yor
(Springer
, 2001
).7.
F.
Spitzer
, “On a class of random variables
,” Proc. Am. Math. Soc.
6
(3
), 494
–505
(1955
).8.
S.
Janson
, Gaussian Hilbert Spaces
(Cambridge University Press
, 1997
).© 2013 AIP Publishing LLC.
2013
AIP Publishing LLC
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