[Paper Review] Analogies of the Qi formula for some Dowling type numbers

Roberto B. Corcino, Jeneveb T. Malusay+4 more|arXiv (Cornell University)|Aug 8, 2018

Advanced Mathematical Identities27 references3 citations

TL;DR

This paper generalizes Qi's formula for Bell numbers—expressing them as sums over Lah and Stirling numbers of the second kind—by deriving analogous explicit formulas for Dowling-type numbers, including Whitney-Lah numbers, generalized Bell numbers, and other variants. The key contribution is a unified framework using generalized Lah-type numbers and unified Stirling numbers, extending the Qi formula to broader classes of combinatorial sequences via inverse relations and generating functions.

ABSTRACT

In the paper, the authors establish explicit formulas for the Dowling numbers and their generalizations in terms of generalizations of the Lah numbers and the Stirling numbers of the second kind. These results gen- eralize the Qi formula for the Bell numbers in terms of the Lah numbers and the Stirling numbers of the second kind.

Motivation & Objective

• To extend Qi's formula for Bell numbers to generalized Dowling-type numbers, including Whitney numbers and generalized Bell numbers.
• To define generalized Lah-type numbers as sums of products of unified Stirling numbers of the first and second kinds.
• To establish explicit formulas for these generalized numbers using inverse relations between unified Stirling numbers.
• To unify and generalize previous results on Whitney-Lah numbers, generalized Bell numbers, and their generating functions.

Proposed method

• Define generalized Lah-type numbers as $ L(n,j;\alpha,\beta,\gamma) = \sum_{k=j}^{n} (-1)^k S^2(n,k) S^1(k,j) $, where $ S^1, S^2 $ are unified Stirling numbers.
• Use the inverse relation between unified Stirling numbers: $ f_n = \sum_k S^1(n,k) g_k \Leftrightarrow g_n = \sum_k S^2(n,k) f_k $, to derive identities.
• Apply the exponential generating function and recurrence relations of unified Stirling numbers to derive explicit expressions.
• Establish connections between generalized Bell numbers $ W_n $, unified Stirling numbers $ S(n,k;\alpha,\beta,\gamma) $, and generalized Lah numbers via $ W_n = \sum_{k=0}^n (-1)^n \left( \sum_{j=0}^n L(k,j;\alpha,\beta,\gamma) \right) S(n,k;\alpha,\beta,\gamma) $.
• Verify consistency with known cases: when $ \alpha=0, \beta=1, \gamma=r $, the formulas reduce to known results for $ \genfrac{[}{]}{0pt}{}{n}{k}_r $, $ \genfrac{\{}{\}}{0pt}{}{n}{k} $, and $ \genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}_r $.
• Extend the framework to higher-order unified Stirling numbers $ S(n,k,\alpha,\beta,\gamma;\epsilon) $, and propose analogous formulas for Bell-type numbers and Lah-type numbers in this generalized setting.

Experimental results

Research questions

• RQ1 Can Qi's formula for Bell numbers be generalized to Dowling-type numbers using generalized Lah and Stirling numbers?
• RQ2 How can unified Stirling numbers $ S^1(n,k;\alpha,\beta,\gamma) $ and $ S^2(n,k;\alpha,\beta,\gamma) $ be used to define generalized Lah-type numbers?
• RQ3 What explicit formula expresses generalized Bell numbers in terms of these generalized Lah and Stirling numbers?
• RQ4 How do the generalized formulas reduce to known cases such as Whitney-Lah numbers, generalized Stirling numbers, and standard Bell numbers?
• RQ5 Can the framework be extended to higher-order unified Stirling numbers $ S(n,k,\alpha,\beta,\gamma;\epsilon) $, and what are the resulting analogues of the Qi formula?

Key findings

• The paper derives a generalized Qi-type formula: $ W_n = \sum_{k=0}^n (-1)^n \left( \sum_{j=0}^n L(k,j;\alpha,\beta,\gamma) \right) S(n,k;\alpha,\beta,\gamma) $, expressing generalized Bell numbers in terms of generalized Lah and Stirling numbers.
• When $ \alpha=0, \beta=1, \gamma=r $, the formula reduces to known identities for $ \genfrac{[}{]}{0pt}{}{n}{k}_r $, $ \genfrac{\{}{\}}{0pt}{}{n}{k} $, and $ \genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}_r $, confirming consistency with prior results.
• The generalized Lah-type numbers $ L(n,j;\alpha,\beta,\gamma) $ are defined via $ \sum_{k=j}^n (-1)^k S^2(n,k) S^1(k,j) $, providing a combinatorial bridge between unified Stirling numbers and generalized Bell numbers.
• The unified Stirling numbers $ S(n,k;\alpha,\beta,\gamma) $ satisfy inverse relations that allow the derivation of explicit formulas for generalized Bell numbers.
• The framework successfully generalizes the classical Qi formula from Bell numbers to Dowling-type numbers, including Whitney-Lah numbers and generalized Stirling numbers.
• The extension to $ S(n,k,\alpha,\beta,\gamma;\epsilon) $ with parameter $ \epsilon $ is proposed, suggesting a path for further generalization of the Qi-type formula to higher-order combinatorial sequences.