Related papers: The negative q-binomial
This paper is mostly a survey, with a few new results. The first part deals with functional equations for q-exponentials, q-binomials and q-logarithms in q-commuting variables and more generally under q-Heisenberg relations. The second part…
An assessment is given as to the extent to which pure unitary evolution, as distinct from environmental decohering interaction, can provide the transition necessary for an observer to interpret perceived quantum dynamics as classical. This…
We identify a positivity property for partition functions in quantum systems with a unitary symmetry group, and we call this "twist positivity." The existence of Feynman-Kac measures and the existence of zero-mass limits are both related to…
We present classification of Q-conditional symmetries for the two-dimensional nonlinear wave equations and the reductions corresponding to these nonlinear symmetries. Classification of inequivalent reductions is discussed.
The continuous big $q$-Hermite polynomials are shown to realize a basis for a representation space of an extended $q$-oscillator algebra. An expansion formula is algebraically derived using this model.
We give a survey of some known and some new results about factors of different sorts of $q-$Fibonacci numbers.
The algebra of observables of $SO_{q}(3)$-symmetric quantum mechanics is extended to include the inverse $\frac{1}{R}$ of the radial coordinate and used to obtain eigenvalues and eigenfunctions of a \q-deformed Coulomb Hamiltonian.
In this paper we state some conjectures about q-Fibonacci polynomials which for q=1 reduce to well-known results about Fibonacci numbers and Fibonacci polynomials.
We consider some $q$-series which depend on a pair of positive integers $(k,m)$. While positivity of these series holds for the first few values of $(k,m)$, the situation is quite unclear for other values of $(k,m)$. In addition, our series…
The polynomial coefficient $\binom {n,q}{k}$ is defined to be the coefficient of $x^{k}$ in the expansion of $(1+x+x^2+... +x^{q-1})^n$. In this note we give an asymptotic estimate for $\binom {n,q}{cn}$ as $n$ tends to infinity, where $c$…
This paper presents a new `partitional' approach to understanding or interpreting standard quantum mechanics (QM). The thesis is that the mathematics (not the physics) of QM is the Hilbert space version of the math of partitions on a set…
Some explanations and implications of the underlying theory approach for quantum theories (QM or QFT) are discussed and suggested. This simple idea seems to have significantly nontrivial effects for our understanding of the quantum…
Examples are given of q-deformed systems that may be interpreted by the standard rules of quantum theory in terms of new degrees of freedom and supplementary quantum numbers.
A q-difference analogue of the fourth Painlev\'e equation is proposed. Its symmetry structure and some particular solutions are investigated.
This article outlines a novel interpretation of quantum theory: the Q-based interpretation. The core idea underlying this interpretation, recently suggested for quantum field theories by Drummond and Reid [2020], is to interpret the phase…
In this paper, we study the binomial sum $S_{n}(q):=% \overset{n}{\underset{k=0}{\sum }}a_{k}\binom{n}{k}\left( 1-q\right) ^{k}q^{n-k}$ for a given sequence $\left( a_{n}\right) $ of real or complex numbers. We express $S_{n}(q)$ in…
Recently, the presence in the literature (also recent) of voluminous discussions about the interpretation of quantum mechanics has been labelled as scandalous. We claim that a weak point of the debate which hosted such a conclusion is a…
Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds…
In this paper, we present the (p; q)-analogues of some inequalities concerning the digamma function. Our results generalize some earlier results.
A systematic procedure for generating certain identities involving elementary symmetric functions is proposed. These identities, as particular cases, lead to new identities for binomial and q-binomial coefficients.