[Paper Review] q-Poisson, q-Dobinski, q-Rota and q-coherent states

A. K. Kwaśniewski|ArXiv.org|Feb 16, 2004

Advanced Mathematical Identities4 references3 citations

TL;DR

This paper generalizes the classical Dobinski formula and Rota's umbral calculus framework to q-analogues, introducing q-Poisson distributions, q-Dobinski formulas, and q-coherent states via q-umbral calculus. It defines q-Bell numbers through q-Stirling numbers of the second kind, particularly using Cigler's q-extension, and extends these concepts to ψ-umbral calculus for broader combinatorial interpretations.

ABSTRACT

q- Dobinski formula may be interpreted as the average of powers of a random variable X_q with the q- Poisson distribution.

Motivation & Objective

• To extend the classical Dobinski formula and Rota's umbral calculus to q-analogues using q-Poisson distributions.
• To define q-Bell numbers as expectation values in q-coherent states, generalizing the standard Bell number interpretation.
• To introduce and analyze Cigler's q-Stirling numbers of the second kind via weighted partition enumeration based on block content.
• To generalize the framework to ψ-umbral calculus, enabling broader combinatorial interpretations beyond q-calculus.
• To unify q-extensions of Stirling and Bell numbers through generating functions and linear functionals in umbral calculus.

Proposed method

• Uses q-umbral calculus with the linear functional $ L_q $ such that $ L_q(X_q^{\underline{n}}) = 1 $, defining $ B_n(q) = L_q(X_q^n) $.
• Applies Jackson's q-derivative to derive the q-Poisson probability generating function $ G(t) = \sum_{k \geq 0} p_k t^k $, where $ p_k = [\exp_q \lambda]^{-1} \frac{\lambda^k}{k_q!} $.
• Introduces Cigler's q-Stirling numbers $ \left\{ \begin{array}{c} n \\ k \end{array} \right\}_q $ via weighted partition enumeration with weight $ q^{\text{cigl}(\pi)} = q^{\sum_{l \in B_0} l} $.
• Derives the q-Dobinski formula as $ B_n(q) = \sum_{k=0}^n \left\{ \begin{array}{c} n \\ k \end{array} \right\}_q $, interpreted as the expectation of the n-th q-power of the number operator in q-coherent states.
• Extends the framework to ψ-umbral calculus by replacing q with a function sequence $ \psi $, defining $ x_\psi $, $ n_\psi $, and $ x_\psi^{\underline{k}} $, generalizing q-extensions.
• Uses mnemonic upside-down notation $ n_\psi \equiv \psi(n) $, $ x_\psi \equiv \psi(x) $, and $ x_\psi^{\underline{k}} = \psi(x)\psi(x-1)\cdots\psi(x-k+1) $ for compact expression of generalized factorials and falling factorials.

Experimental results

Research questions

• RQ1 How can the classical Dobinski formula be generalized to q-analogues using q-coherent states and q-umbral calculus?
• RQ2 What is the combinatorial interpretation of Cigler's q-Stirling numbers of the second kind in terms of weighted partitions?
• RQ3 How do q-Bell numbers emerge as expectation values of the number operator in q-coherent states?
• RQ4 What is the role of the q-Poisson distribution in realizing the q-Dobinski formula via linear functionals?
• RQ5 How can the q-umbral framework be extended to ψ-umbral calculus to generalize q-Stirling and q-Bell numbers?

Key findings

• The q-Dobinski formula is realized as $ B_n(q) = L_q(X_q^n) $, where $ L_q(X_q^{\underline{n}}) = 1 $, and $ X_q^n $ is the q-Pochhammer rising factorial.
• Cigler's q-Stirling numbers $ \left\{ \begin{array}{c} n \\ k \end{array} \right\}_q $ are defined as $ \sum_{\pi \in A_{n,k}} q^{\text{cigl}(\pi)} $, with $ \text{cigl}(\pi) = \sum_{l \in B_0} l $, and are expressed via q-binomial coefficients.
• The q-Bell number $ B_n(q) $ is the sum over all partitions $ \pi \in \Pi $ of $ q^{\text{cigl}(\pi)} $, generalizing the classical Bell number via weighted partition enumeration.
• The q-Poisson distribution is characterized by the generating function $ G(t) = \sum_{k \geq 0} p_k t^k $, with $ p_k = [\exp_q \lambda]^{-1} \frac{\lambda^k}{k_q!} $, and satisfies $ p_n = \left[ \frac{\partial_q^n G(t)}{n_q!} \right]_{t=0} $.
• The ψ-umbral extension generalizes q-umbral calculus by replacing $ q $ with a function sequence $ \psi $, defining $ x_\psi^{\underline{k}} = \psi(x)\psi(x-1)\cdots\psi(x-k+1) $, enabling broader combinatorial generalizations.
• The framework unifies q-extensions and ψ-extensions of Stirling and Bell numbers through generating functions and linear functionals, with the q-Dobinski formula as a central identity.