arXiv subjects

Matthias Schork

Publications and source records attributed to Matthias Schork.

12 recordsLinked to original sources

arXiv2026

Normal ordering in the $(p,q)$-deformed generalized Weyl algebra. III: The binomial formula

We study the $(p, q)$-deformed generalized Weyl algebra generated by variables $X, Y$ and $Z_p$ satisfying the $(p, q)$-commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, with $s\in \mathbb{N}_0$. Within this framework, we investigate the noncommutative binomial formula $(X+Y)^n$ and related identities. In particular, we show how the associated normal ordering coefficients can be expressed in terms of $(p,q)$-deformed $s$-rook numbers. We treat several special cases explicitly, recovering known results from literature as well as deriving new ones.

math.CO↗

arXiv2026

Normal ordering in the $(p,q)$-deformed generalized Weyl algebra. II: Interpretation in terms of rook placements

In this paper, we investigate the combinatorial structure arising from the $(p, q)$-deformed generalized Weyl algebra generated by variables $X, Y$, and $Z_p$, satisfying the $(p, q)$-commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, where $s\in \mathbb{N}_0$. Our primary objective is to use the normal ordering process defined by these relations to develop a novel model of $(p, q)$-deformed rook theory. Specifically, we introduce a new framework of $(p, q)$-deformed $s$-rook numbers derived from this normal ordering process. Utilizing these combinatorial models, we provide explicit combinatorial interpretations for the associated $(p, q)$-generalized Stirling numbers via rook placements on staircase boards. Our results extend several classical and recent formulations in the literature to the general $p\neq 1$ setting.

math.CO↗

arXiv2026

Normal ordering in the $(p,q)$-deformed generalized Weyl algebra. I: Algebraic Framework and Combinatorial Identities

The $(p,q)$-deformed generalized Weyl algebra is generated by variables $X, Y$ and $Z_p$ which satisfy the commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, with $s\in \mathbb{N}_0$. We investigate the problem of normal ordering arbitrary words in these letters with the help of Young diagrams, and we treat certain special cases explicitly. In particular, the connection to generalized Stirling numbers is considered in detail.

math.CO↗

arXiv2026

Rook theory, normal ordering in the $q$-deformed Ore algebra and the polynomial generalization

For words in the variables $X$ and $Y$ satisfying the commutation relation of the $q$-deformed generalized Ore algebra, $XY-qYX= \mu I + \nu Y$, we show that the corresponding normal ordering coefficients can be given an interpretation in terms of mixed placements of rooks and files. In particular, the associated $q$-deformed Ore-Stirling and Ore-Lah numbers are treated in detail. We show that the $q$-deformed Ore-Stirling numbers (resp., $q$-deformed Ore-Lah numbers) are given as mixed placement numbers of rooks and files on the staircase board (resp., Laguerre board). Using this combinatorial interpretation, their recurrence relations are derived. In addition, the normal ordered form of the binomial $(X+Y)^m$ in the $q$-deformed generalized Ore algebra is determined. These considerations are then extended to the $q$-deformed polynomial Weyl algebra generated by $X$ and $Y$ satisfying $XY-qYX=f(Y)$ for some polynomial $f\in \mathbb{C}[Y]$. In particular, associated $q$-deformed polynomial Stirling and Lah numbers are introduced and their properties studied. The normal ordered form of the binomial is also extended to the $q$-deformed polynomial Weyl algebra.

math.CO↗

arXiv2013

Normal ordering problem and the extensions of the Stirling grammar

The purpose of this paper is to investigate the connection between context-free grammars and normal ordering problem, and then to explore various extensions of the Stirling grammar. We present grammatical characterizations of several well known combinatorial sequences, including the generalized Stirling numbers of the second kind related to the normal ordering problem and the $r$-Dowling polynomials. Also, possible avenues for future research are described.

math.CO↗

arXiv2009

On Linear Differential Equations Involving a Para-Grassmann Variable

As a first step towards a theory of differential equations involving para-Grassmann variables the linear equations with constant coefficients are discussed and solutions for equations of low order are given explicitly. A connection to n-generalized Fibonacci numbers is established. Several other classes of differential equations (systems of first order, equations with variable coefficients, nonlinear equations) are also considered and the analogies or differences to the usual (''bosonic'') differential equations discussed.

math-ph↗

arXiv2007

Motzkin numbers of higher rank: Generating function and explicit expression

The generating function and an explicit expression is derived for the (colored) Motzkin numbers of higher rank introduced recently. Considering the special case of rank one yields the corresponding results for the conventional colored Motzkin numbers for which in addition a recursion relation is given.

math.CO↗

arXiv2007

Noncrossing normal ordering for functions of boson operators

Normally ordered forms of functions of boson operators are important in many contexts in particular concerning Quantum Field Theory and Quantum Optics. Beginning with the seminal work of Katriel [Lett. Nuovo Cimento, 10(13):565--567, 1974], in the last few years, normally ordered forms have been shown to have a rich combinatorial structure, mainly in virtue of a link with the theory of partitions. In this paper, we attempt to enrich this link. By considering linear representations of noncrossing partitions, we define the notion of noncrossing normal ordering. Given the growing interest in noncrossing partitions, because of their many unexpected connections (like, for example, with free probability), noncrossing normal ordering appears to be an intriguing notion. We explicitly give the noncrossing normally ordered form of the functions (a^{r}(a^†)^{s})^{n}) and (a^{r}+(a^†)^{s})^{n}, plus various special cases. We are able to establish for the first time bijections between noncrossing contractions of these functions, k-ary trees and sets of lattice paths.

quant-ph↗

arXiv2007

On the normal ordering of multi-mode boson operators

In this article combinatorial aspects of normal ordering annihilation and creation operators of a multi-mode boson system are discussed. The modes are assumed to be coupled since otherwise the problem of normal ordering is reduced to the corresponding problem of the single-mode case. To describe the normal ordering in the multi-mode case for each mode a colour is introduced and coloured contractions are considered. A depiction for coloured contractions via coloured linear representations is given. In analogy to the single-mode case associated coloured Stirling numbers are defined as coefficients appearing in the process of normal ordering powers of the number operators. Several properties of these coloured Stirling numbers are discussed.

quant-ph↗

arXiv2007

Wick's theorem for q-deformed boson operators

In this paper combinatorial aspects of normal ordering arbitrary words in the creation and annihilation operators of the q-deformed boson are discussed. In particular, it is shown how by introducing appropriate q-weights for the associated ``Feynman diagrams'' the normally ordered form of a general expression in the creation and annihilation operators can be written as a sum over all q-weighted Feynman diagrams, representing Wick's theorem in the present context.

quant-ph↗

arXiv2007

Generalized Heisenberg algebras and k-generalized Fibonacci numbers

It is shown how some of the recent results of de Souza et al. [1] can be generalized to describe Hamiltonians whose eigenvalues are given as k-generalized Fibonacci numbers. Here k is an arbitrary integer and the cases considered by de Souza et al. corespond to k=2.

math-ph↗

arXiv2006

A generalization of boson normal ordering

In this paper we define generalizations of boson normal ordering. These are based on the number of contractions whose vertices are next to each other in the linear representation of the boson operator function. Our main motivation is to shed further light onto the combinatorics arising from algebraic and Fock space properties of boson operators.

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