110,994 research outputs found
The q-Binomial Coefficient for Negative Arguments and Some q-Binomial Summation Identities
Using a property of the q-shifted factorial, an identity for q-binomial
coefficients is proved, which is used to derive the formulas for the q-binomial
coefficient for negative arguments. The result is in agreement with an earlier
paper about the normal binomial coefficient for negative arguments. Some new
q-binomial summation identities are derived, and the formulas for negative
arguments transform some of these summation identities into each other. A known
q-binomial summation identity is transformed into a new q-binomial summation
identity
A q-rious positivity
The -binomial coefficients
\qbinom{n}{m}=\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i), for integers ,
are known to be polynomials with non-negative integer coefficients. This
readily follows from the -binomial theorem, or the many combinatorial
interpretations of \qbinom{n}{m}. In this note we conjecture an
arithmetically motivated generalisation of the non-negativity property for
products of ratios of -factorials that happen to be polynomials.Comment: 6 page
Non-classical properties and algebraic characteristics of negative binomial states in quantized radiation fields
We study the nonclassical properties and algebraic characteristics of the
negative binomial states introduced by Barnett recently. The ladder operator
formalism and displacement operator formalism of the negative binomial states
are found and the algebra involved turns out to be the SU(1,1) Lie algebra via
the generalized Holstein-Primarkoff realization. These states are essentially
Peremolov's SU(1,1) coherent states. We reveal their connection with the
geometric states and find that they are excited geometric states. As
intermediate states, they interpolate between the number states and geometric
states. We also point out that they can be recognized as the nonlinear coherent
states. Their nonclassical properties, such as sub-Poissonian distribution and
squeezing effect are discussed. The quasiprobability distributions in phase
space, namely the Q and Wigner functions, are studied in detail. We also
propose two methods of generation of the negative binomial states.Comment: 17 pages, 5 figures, Accepted in EPJ
Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle
Certain fluctuations in particle number at fixed total energy lead exactly to
a cut-power law distribution in the one-particle energy, via the induced
fluctuations in the phase-space volume ratio. The temperature parameter is
expressed automatically by an equipartition relation, while the q-parameter is
related to the scaled variance and to the expectation value of the particle
number. For the binomial distribution q is smaller, for the negative binomial q
is larger than one. These results also represent an approximation for general
particle number distributions in the reservoir up to second order in the
canonical expansion. For general systems the average phase-space volume ratio
expanded to second order delivers a q parameter related to the heat capacity
and to the variance of the temperature. However, q differing from one leads to
non-additivity of the Boltzmann-Gibbs entropy. We demonstrate that a deformed
entropy, K(S), can be constructed and used for demanding additivity. This
requirement leads to a second order differential equation for K(S). Finally,
the generalized q-entropy formula contains the Tsallis, Renyi and
Boltzmann-Gibbs-Shannon expressions as particular cases. For diverging
temperature variance we obtain a novel entropy formula.Comment: Talk given by T.S.Biro at Sigma Phi 2014, Rhodos, Greec
Nonextensive Statistics and Multiplicity Distribution in Hadronic Collisions
The multiplicity distribution of particles in relativistic gases is studied
in terms of Tsallis' nonextensive statistics. For an entropic index q>1 the
multiplicity distribution is wider than the Poisson distribution with the same
average number of particles, being similar to the negative binomial
distribution commonly used in phenomenological analysis of hadron production in
high-energy collisions
Negative -Stirling numbers
International audienceThe notion of the negative -binomial was recently introduced by Fu, Reiner, Stanton and Thiem. Mirroring the negative -binomial, we show the classical -Stirling numbers of the second kind can be expressed as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in and . We extend this enumerative result via a decomposition of the Stirling poset, as well as a homological version of Stembridge’s phenomenon. A parallel enumerative, poset theoretic and homological study for the -Stirling numbers of the first kind is done beginning with de Médicis and Leroux’s rook placement formulation. Letting we give a bijective combinatorial argument à la Viennot showing the -Stirling numbers of the first and second kind are orthogonal.La notion de la -binomial négative était introduite par Fu, Reiner, Stanton et Thiem. Réfléchissant la -binomial négative, nous démontrons que les classiques -nombres de Stirling de deuxième espèce peuvent être exprimés comme une paire de statistiques sur un sous-ensemble des mots de croissance restreinte. Les expressions résultantes sont les polynômes en et . Nous étendons ce résultat énumératif via une décomposition du poset de Stirling, ainsi que d’une version homologique du phénomène de Stembridge. Un parallèle énumératif, poset théorique et étude homologique des -nombres de Stirling de première espèce se fait en commençant par la formulation du placement des tours par suite des auteurs de Médicis et Leroux. On laisse et on donne les arguments combinatoires et bijectifs à la Viennot qui démontrent que les -nombres de Stirling de première et deuxième espèces sont orthogonaux
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