[Paper Review] Pascal like matrices - an accessible factory of one source identities and resulting applications

A. K. Kwaśniewski|ArXiv.org|Mar 7, 2004

Matrix Theory and Algorithms3 citations

TL;DR

This paper introduces $ψ$-Pascal and $ˇ_q$-Pascal matrices as generalized frameworks extending classical Pascal matrices within the $ψ$-Finite Operator Calculus. By leveraging $ψ$-binomial coefficients and $ψ$-derivative operators, the matrices generate a unified source of combinatorial identities and enable extensions to $q$-calculus and Fibonomial settings, offering a natural factory for deriving identities and applications in special polynomials.

ABSTRACT

The extension of pascalian like matrices depending on a variable from any field of zero characteristics are shown at work for the first time. Their properties appear to be one source factory of identities and resulting foreseen applications

Motivation & Objective

• To generalize Pascal matrices using $ψ$-Finite Operator Calculus to create a systematic source of combinatorial identities.
• To extend classical Pascal matrix properties to $q$-calculus and Fibonomial settings via $ψ$-binomial coefficients.
• To demonstrate that $ψ$-Pascal matrices serve as a natural factory for streams of identities and applications in special polynomials.
• To establish the algebraic structure of $ψ$-Pascal matrices as abelian semigroups, with group structure only under normal sequences.
• To enable applications in $ψ$-basic Bernoulli, Hermite-Ward, and Appell polynomials through $ψ$-integration and $ψ$-Sheffer identities.

Proposed method

• Defining $ψ$-binomial coefficients $\binom{n}{k}_\psi = \frac{n_\psi!}{k_\psi!(n-k)_\psi!}$ for admissible sequences $\psi$.
• Introducing the $ψ$-derivative $\partial_\psi$ via $\partial_\psi x^n = n_\psi x^{n-1}$, generalizing the standard derivative.
• Constructing the $ψ$-Pascal matrix as $P_\psi[x] = \exp_\psi\{xK_\psi\}$, where $K_\psi$ is a lower-shift matrix with entries $(j+1)_\psi \delta_{i,j+1}$.
• Deriving the matrix form $P_\psi[x] = \left(x^{i-j} \binom{i}{j}_\psi\right)_{i,j \in \mathbb{Z}_n}$, showing its exponential generation.
• Establishing the identity $P_\psi[x]P_\psi[y] = P_\psi[x +_\psi y]$ for $\psi$-additivity, valid in the semigroup setting.
• Applying the framework to $q$-calculus and Fibonomial cases via specific choices of $\psi_n = \frac{1}{n_q!}$ or $\frac{1}{F_n!}$.

Experimental results

Research questions

• RQ1 How can Pascal matrix identities be generalized beyond the classical case using a unified $ψ$-calculus framework?
• RQ2 What algebraic structure do $ψ$-Pascal matrices form, and when do they become groups?
• RQ3 How do $ψ$-Pascal matrices generate streams of combinatorial identities through matrix multiplication?
• RQ4 In what ways can $q$-calculus and Fibonomial calculus be embedded within the $ψ$-Pascal framework?
• RQ5 Can the $ψ$-Pascal matrix framework be used to extend classical polynomial families like Bernoulli and Hermite polynomials?

Key findings

• The $ψ$-Pascal matrix $P_\psi[x]$ is defined as $\exp_\psi\{xK_\psi\}$, with $K_\psi$ being a lower-shift matrix, yielding a matrix representation $\left(x^{i-j} \binom{i}{j}_\psi\right)$.
• The matrix satisfies $P_\psi[x]P_\psi[y] = P_\psi[x +_\psi y]$, forming an abelian semigroup under $ψ$-addition, becoming a group only for normal sequences.
• The identity $\sum_{j \leq k \leq i} \binom{i}{k}_\psi \binom{k}{j}_\psi = (1 +_\psi 1)^{i-j} \binom{i}{j}_\psi$ is derived from matrix multiplication, generalizing classical Pascal identities.
• The $q$-Pascal and Fibonomial cases are explicitly realized by choosing $\psi_n = \frac{1}{n_q!}$ and $\psi_n = \frac{1}{F_n!}$, respectively.
• The framework enables extension of Bernoulli, Hermite-Ward, and Appell polynomials via $ψ$-Sheffer identities and $ψ$-integration.
• The $\hat{q}_\psi$-Fermat matrix and $\hat{q}_\psi$-Pascal matrix are identified as natural extensions analogous to the standard Pascal matrix in broader applications.