[Paper Review] A note on degenerate stirling polynomials of the second kind

Taekyun Kim|arXiv (Cornell University)|Apr 7, 2017

Advanced Mathematical Identities3 references99 citations

TL;DR

The paper defines degenerate Stirling polynomials of the second kind via generating functions, derives several identities and relations with degenerate and Whitney numbers, and presents recurrence formulas and connections to classical polynomials.

ABSTRACT

In this paper, we consider the degenerate Stirling polynomials of the second kind which are derived from the generating function. In addition, we give some new identities for these polynomials.

Motivation & Objective

• Motivate the study of degenerate Stirling polynomials of the second kind via generating functions.
• Define the degenerate Stirling polynomials of the second kind and related x-parameterized versions.
• Derive new identities and recurrence relations connecting these polynomials with degenerate Stirling, Whitney, and Euler/Carlitz polynomials.

Proposed method

• Introduce degenerate and x-parametrized Stirling polynomials through generating functions (2.4) and (2.1).
• Compute explicit expressions like S_{2,λ}(n,k|x) in terms of S_{2}(l,k) and binomial-type sums.
• Derive recurrence relations such as S_{2,λ}(n+1,k|x) in terms of S_{2,λ}(n,k|x) and S_{2,λ}(n,k-1|x).
• Link degenerate polynomials to Whitney numbers via generating functions and establish closed forms (Theorems 2.1–2.8).
• Obtain degenerate Euler/Carlitz type relations and show limiting behavior as λ → 0 recovering classical counterparts.

Experimental results

Research questions

• RQ1 What are the degenerate Stirling polynomials of the second kind S_{2,λ}(n,k|x) and how can they be generated?
• RQ2 How do these degenerate polynomials relate to classical S_{2}(n,k) and to degenerate Whitney numbers W_{m,r}(n,k|λ)?
• RQ3 What identities and recurrence relations connect S_{2,λ}(n,k|x) with S_{1}(n,m), S_{2}(n,k), and Whitney numbers?
• RQ4 What is the limiting behavior of degenerate quantities as λ → 0, and how do they recover known classical polynomials?
• RQ5 How do higher-order degenerate Euler/Carlitz polynomials interact with degenerate Stirling polynomials?

Key findings

• Defined degenerate Stirling polynomials of the second kind S_{2,λ}(n,k|x) via a specific generating function (2.4).
• Established explicit expression S_{2,λ}(n,k|x) = sum_{l=k}^{n} binom(n,l) S_{2}(l,k) x^{n-l}.
• Derived recurrence: S_{2,λ}(n+1,k|x) = (x+k) S_{2,λ}(n,k|x) + S_{2,λ}(n,k-1|x) - nλ S_{2,λ}(n,k|x).
• Showed limiting case λ → 0 recovers the classical S_{2}(n+1,k|x) recurrence: S_{2}(n+1,k|x) = (x+k) S_{2}(n,k|x) + S_{2}(n,k-1|x).
• Expressed degenerate Euler/Carlitz relations via S_{2,λ}(n,k|x) (Theorem 2.4) and connected to higher-order Whitney numbers (Theorems 2.5–2.8).
• Provided identities linking sums over Δ^k and S_{1}(n,m) to S_{2,λ}(n,k|x) (Theorems 2.2, 2.3, 2.7, 2.8).