[Paper Review] Cauchy type identities and corresponding fermatian matrices via non-comuting variables of extended finite operator calculus

Andrzej Kwaśniewski|ArXiv.org|Mar 5, 2004

Matrix Theory and Algorithms3 citations

TL;DR

This paper introduces a generalized Cauchy identity using $ψ$-labeled $ψ$-extended $ψ$-commuting variables within the framework of extended finite operator calculus. By defining a $ψ$-deformed $ψ$-mutator operator $ψ_{\hat{q}}$, the authors derive a non-commutative $ψ$-Cauchy identity and construct a new class of $ψ$-Fermat matrices with operator entries, unifying $q$-Cauchy and Fibonomial-Cauchy cases as special instances.

ABSTRACT

New family of extended Cauchy type identities is found and related Fermat type matrices are provided ready for applications in extended scope. This is achieved due to the use specifically non-commuting variables of extended finite operator calculus introduced by the author few years ago.

Motivation & Objective

• To extend the classical Cauchy identity to non-commuting $ψ$-deformed variables within the framework of extended finite operator calculus.
• To resolve the failure of standard binomial expansion in non-commutative settings by introducing a $ψ$-mutator operator $ψ_{\hat{q}}$.
• To generalize the $q$-Pascal and Fibonomial-Cauchy identities into a unified $ψ$-extended framework via $ψ$-Gaussian coefficients.
• To define and characterize a new class of symmetric matrices—$ψ$-Fermat matrices—with operator-valued entries.
• To explore the potential of $ψ_{\hat{q}}$-quantum planes as a foundation for a Rota-style operator formulation of extended umbral calculus.

Proposed method

• Introduces $ψ$-labeled sequences as admissible extensions of the standard factorial sequence, including $q$-Gaussian and Fibonomial cases.
• Defines $ψ$-commuting variables via the $ψ_{\hat{q}}$-mutator relation $yx = \hat{q}_{\psi}xy$, generalizing $q$-commutation.
• Constructs $ψ$-binomial coefficients $\binom{n}{k}_{\hat{q}_{\psi}}$ using the $ψ$-deformed factorial $n_{\hat{q}_{\psi}}! = \frac{1 - \hat{q}_{\psi}^n}{1 - \hat{q}_{\psi}}$.
• Derives the $ψ$-Cauchy identity: $\sum_{k \geq 0} \hat{q}_{\psi}^{(r-k)(j-k)} \binom{r}{k}_{\hat{q}_{\psi}} \binom{s}{j-k}_{\hat{q}_{\psi}} = \binom{r+s}{j}_{\hat{q}_{\psi}}$.
• Applies the identity to define the $ψ$-Fermat matrix via $\sum_{k \geq 0} \hat{q}_{\psi}^{(r-k)(j-k)} \binom{i}{k}_{\hat{q}_{\psi}} \binom{j}{k}_{\hat{q}_{\psi}} = \binom{i+j}{j}_{\hat{q}_{\psi}}$.
• Introduces $ψ$-Pascal and $ψ$-Fermat matrices with entries $x^{i-j} \binom{i}{j}_{\hat{q}_{\psi}}$ and $\binom{i+j}{j}_{\hat{q}_{\psi}}$, respectively.

Experimental results

Research questions

• RQ1 Can a non-commutative generalization of the Cauchy identity be constructed using $ψ$-deformed $ψ$-commuting variables?
• RQ2 How does the $ψ_{\hat{q}}$-mutator operator enable a consistent extension of binomial-type identities beyond the $q$-case?
• RQ3 What is the structure of the $ψ$-Fermat matrix, and how does it unify $q$-Pascal and Fibonomial-Cauchy cases?
• RQ4 Can the $ψ_{\hat{q}}$-quantum plane serve as a foundation for a Rota-style operator formulation of extended umbral calculus?
• RQ5 What are the implications of the $ψ$-Cauchy identity for the representation theory of operator algebras in combinatorics?

Key findings

• A new $ψ$-Cauchy identity is derived: $\sum_{k \geq 0} \hat{q}_{\psi}^{(r-k)(j-k)} \binom{r}{k}_{\hat{q}_{\psi}} \binom{s}{j-k}_{\hat{q}_{\psi}} = \binom{r+s}{j}_{\hat{q}_{\psi}}$, valid under the $ψ_{\hat{q}}$-commutation relation $[y,x]_{\hat{q}_{\psi}} = 0$.
• The $ψ$-Fermat matrix is defined via $\sum_{k \geq 0} \hat{q}_{\psi}^{(r-k)(j-k)} \binom{i}{k}_{\hat{q}_{\psi}} \binom{j}{k}_{\hat{q}_{\psi}} = \binom{i+j}{j}_{\hat{q}_{\psi}}$, generalizing the $q$-Fermat matrix.
• The $ψ$-binomial coefficients $\binom{n}{k}_{\hat{q}_{\psi}}$ are constructed using $n_{\hat{q}_{\psi}}! = \frac{1 - \hat{q}_{\psi}^n}{1 - \hat{q}_{\psi}}$, extending $q$-binomial and Fibonomial coefficients.
• The $ψ$-Pascal matrix $P[x]$ has entries $x^{i-j} \binom{i}{j}_{\hat{q}_{\psi}}$, and the $ψ$-Fermat matrix $F[1]$ has entries $\binom{i+j}{j}_{\hat{q}_{\psi}}$, both with operator-valued entries.
• The identity holds for any admissible $ψ$-sequence, including $q$-Gaussian and Fibonomial cases, demonstrating broad applicability.
• The framework unifies standard, $q$-, and Fibonomial-Cauchy identities under a single $ψ$-extended operator calculus formalism.