[Paper Review] A New Approach to the $r$-Whitney Numbers by Using Combinatorial Differential Calculus

Jósé L. Ramírez, Miguel A. Méndez|arXiv (Cornell University)|Feb 21, 2017

Advanced Combinatorial Mathematics27 references3 citations

TL;DR

This paper introduces a novel combinatorial interpretation of $r$-Whitney numbers of the second kind using combinatorial differential calculus based on the grammar $G = \{y \to yx^m, x \to x\}$. By specializing $m=1$, it provides a new combinatorial model for $r$-Stirling numbers and defines $[m]$-Stirling numbers of the first and second kind, leading to a generalized Touchard polynomial family. The key contribution is establishing the $r$-Dowling polynomials as a Sheffer family relative to the $[m]$-Touchard polynomials and deriving new determinantal identities and combinatorial identities involving Bernoulli and Euler polynomials via Riordan arrays and umbral calculus.

ABSTRACT

In the present article we introduce two new combinatorial interpretations of the $r$-Whitney numbers of the second kind obtained from the combinatorics of the differential operators associated to the grammar $G:=\{ y ightarrow yx^{m}, x ightarrow x\}$. By specializing $m=1$ we obtain also a new combinatorial interpretation of the $r$-Stirling numbers of the second kind. Again, by specializing to the case $r=0$ we introduce a new generalization of the Stirling number of the second kind and through them a binomial type family of polynomials that generalizes Touchard's. Moreover, we show several well-known identities involving the $r$-Dowling polynomials and the $r$-Whitney numbers using the combinatorial differential calculus. Finally we prove that the $r$-Dowling polynomials are a Sheffer family relative to the generalized Touchard binomial family, study their umbral inverses, and introduce $[m]$-Stirling numbers of the first kind. From the relation between umbral calculus and the Riordan matrices we give several new combinatorial identities involving the $r$-Whitney number of both kinds, Bernoulli and Euler polynomials.

Motivation & Objective

• To develop a new combinatorial interpretation of $r$-Whitney numbers of the second kind using combinatorial differential calculus.
• To generalize the $r$-Stirling numbers of the second kind through a new model based on the grammar $G = \{y \to yx^m, x \to x\}$.
• To define $[m]$-Stirling numbers of the first and second kind and introduce the $[m]$-Touchard polynomials as a binomial-type generalization of Touchard polynomials.
• To establish the $r$-Dowling polynomials as a Sheffer family relative to the $[m]$-Touchard polynomials.
• To derive new combinatorial identities involving $r$-Whitney numbers, Bernoulli polynomials, and Euler polynomials using umbral calculus and Riordan arrays.

Proposed method

• The authors use combinatorial differential calculus based on the grammar $G = \{y \to yx^m, x \to x\}$ to generate combinatorial interpretations of $r$-Whitney numbers of the second kind.
• They derive recurrence relations for $r$-Whitney numbers of the second kind using the $A$-sequence and $Z$-sequence of exponential Riordan arrays.
• The $r$-Dowling polynomials are shown to be a Sheffer family relative to the $[m]$-Touchard polynomials via umbral calculus and generating function techniques.
• The $r$-Whitney numbers of the first kind are characterized via the exponential Riordan array $\left\langle (1+mz)^{-r/m}, \ln(1+mz)^{1/m} \right\rangle$, with generating functions involving Bernoulli numbers.
• Determinantal identities are derived by expressing $r$-Dowling polynomials recursively and expanding the resulting matrix determinant.
• The connection between umbral calculus and Riordan matrices is leveraged to derive new identities involving Bernoulli and Euler polynomials.

Experimental results

Research questions

• RQ1 How can the $r$-Whitney numbers of the second kind be reinterpreted combinatorially using differential operators from a specific grammar?
• RQ2 What new combinatorial structure emerges when $m=1$ in the grammar-based model, and how does it relate to $r$-Stirling numbers of the second kind?
• RQ3 Can a generalized Touchard polynomial family be constructed from the $[m]$-Stirling numbers of the second kind, and what are its properties?
• RQ4 Are the $r$-Dowling polynomials a Sheffer family relative to the $[m]$-Touchard polynomials, and how can this be proven using umbral calculus?
• RQ5 What new combinatorial identities involving $r$-Whitney numbers, Bernoulli polynomials, and Euler polynomials can be derived from the Riordan array framework?

Key findings

• The $r$-Whitney numbers of the second kind satisfy a recurrence involving coefficients derived from Cauchy numbers of the first kind: $W_{m,r}(n+1,k+1) = \sum_{j=0}^{\infty} \frac{n+1}{k+1} \binom{k+j}{j} c_j m^j W_{m,r}(n,k+j)$.
• The $r$-Whitney numbers of the first kind are characterized by the exponential Riordan array $\left\langle (1+mz)^{-r/m}, \ln(1+mz)^{1/m} \right\rangle$, with generating function for the $A$-sequence involving Bernoulli numbers.
• The $r$-Dowling polynomials $\mathcal{D}_n^{[m,r]}(x)$ satisfy the determinantal identity ${\mathcal{D}}_{n}^{[m,r]}(x)=(-1)^{n}\begin{vmatrix}1&x&&\cdots&&x^{n-1}&x^{n}\\ 1&w_{m,r}(1,0)&&\cdots&&w_{m,r}(n-1,0)&w_{m,r}(n,0)\\ 0&1&&\cdots&&w_{m,r}(n-1,1)&w_{m,r}(n,1)\\ \vdots&&&\cdots&&&\vdots\\ 0&0&&\cdots&&1&w_{m,r}(n,n-1)\end{vmatrix}$.
• The $r$-Dowling polynomials form a Sheffer family relative to the $[m]$-Touchard polynomials, as established through umbral calculus and generating function analysis.
• The $[m]$-Stirling numbers of the first kind are defined as the umbral inverse of the $[m]$-Stirling numbers of the second kind, leading to a new generalization of Stirling numbers.
• New combinatorial identities are derived involving $r$-Whitney numbers, Bernoulli polynomials, and Euler polynomials via the Riordan array framework and the connection between umbral calculus and Riordan matrices.