相关论文: $\psi$-Poisson, $q$-Cigler, $\psi$-Dobinski, $\psi…
Husimi distributions and Wigner distributions are well-known quasi-probability distributions which appear in several contexts. In this paper, we show some remarkable aspects of these distribution functions related to geometric structures of…
Extensions of the $Stirling$ numbers of the second kind and $Dobinski$ -like formulas are proposed in a series of exercises for graduates. Some of these new formulas recently discovered by me are to be found in the source paper $ [1]$.…
Reformulated uniform asymptotic expansions are derived for ordinary differential equations having a large parameter and a simple turning point. These involve Airy functions, but not their derivatives, unlike traditional asymptotic…
I study the Lyapunov exponent and the integrated density of states for general Jacobi operators. The main result is that questions about these, can be reduced to questions about ergodic Jacobi operators. Then, I apply this to $a(n) = 1$ and…
We present a new perspective of assessing the rates of convergence to the Gaussian and Poisson distributions in the Erd\"os-Kac theorem for additive arithmetic functions $\psi$ of a random integer $J_n$ uniformly distributed over…
A mapping of nonextensive statistical mechanics into Gibbs' statistical mechanics exists, which leads to a generalization of Einstein's formula for fluctuations. A unified treatment of stability of relaxed states in nonextensive statistical…
The main aim of this article is to characterize and investigate the three parameter exponentiated exponential Poisson probability distribution ${\rm EEP}(\alpha, \beta, \lambda)$ by giving explicit closed form expressions for its…
$C_{\lambda}$-extended oscillator algebras generalizing the Calogero-Vasiliev algebra, where $C_{\lambda}$ is the cyclic group of order $\lambda$, are studied both from mathematical and applied viewpoints. Casimir operators of the algebras…
The asymptotic correspondence between the probability mass function of the $q$-deformed multinomial distribution and the $q$-generalised Kullback-Leibler divergence, also known as Tsallis relative entropy, is established. The probability…
We study different fractional extensions of the Poisson process and generalized counting processes by introducing time-change represented by the inverse to the sums of stable and tempered stable subordinators. We state the governing…
In quantum mechanics, bosonic operators are mathematical objects that are used to represent the creation ($a^\dagger$) and annihilation ($a$) of bosonic particles. The natural power of a linear combination of bosonic operators represents an…
The binomial, the negative binomial, the Poisson, the compound Poisson and the Erlang distribution do all admit integral representations with respect to its (continuous) parameter. We use the Margulis-Russo type formulas for Bernoulli and…
Minkowski's First Theorem and Dirichlet's Approximation Theorem provide upper bounds on certain minima taken over lattice points contained in domains of Euclidean spaces. We study the distribution of such minima and show, under some…
Traditional statistical mechanics is constrained by the binary paradigms of identical/distinguishable and bosonic/fermionic particle statistics, leading to a fundamental logical gap in describing systems with partial distinguishability. We…
By considering generalized logarithm and exponential functions used in nonextensive statistics, the four usual algebraic operators : addition, subtraction, product and division, are generalized. The properties of the generalized operators…
The normal ordering formulae for powers of the boson number operator $\hat{n}$ are extended to deformed bosons. It is found that for the `M-type' deformed bosons, which satisfy $a a^{\dagger} - q a^{\dagger} a = 1$, the extension involves a…
Conventional Bell and Stirling numbers arise naturally in the normal ordering of simple monomials in boson operators. By extending this process we obtain generalizations of these combinatorial numbers, defined as coherent state matrix…
In recent Kwasniewski's papers inspired by O. V. Viskov it was shown that the $\psi$-calculus in parts appears to be almost automatic, natural extension of classical operator calculus of Rota - Mullin or equivalently - of umbral calculus of…
In this paper, subnormal operators, not necessarily bounded, are discussed as generalized observables. In order to describe not only the information about the probability distribution of the output data of their measurement but also a…
In this paper the generalization of the Poisson distribution is derived for the case when each consecutive event changes event rate. A simple formula for the probability of observing of a given number of events for the selected period of…