A finite difference approach to degenerate Bernoulli and Stirling polynomials
type: Item
instance of: scholarly article
title: A finite difference approach to degenerate Bernoulli and Stirling polynomials
zbMATH Open document ID: 0841.11010
DOI: 10.1016/0012-365X(93)E0188-A
author: Arnold Adelberg
published in: Discrete Mathematics
publication date: 1996-07-16T00:00:00Z
review text: For a polynomial \(f(x, y)\), define the divided difference as \[ \nabla_y f(x,y)= (f(x+ y,y)- f(x,y))/ y. \] The author studies the polynomials \(A_{n,s} (x,y)= \nabla^s_y {x\choose {s+n}}\) and \(B_{n,s} (y)= A_{n,s} (0,y)\) for \(n,s= 0, 1, 2,\dots\;\). He finds various symmetries and proves results about zeros, divisibility and irreducibility (over \(\mathbb{Q}\) and over \(\mathbb{C}\)). Many of the results have a natural combinatorial interpretation. Particular values of the polynomials yield binomial identities, both old and (apparently) new. The author finds the generating functions of \(A_{n,s}\) and \(B_{n,s}\) and applies them, e.g., to provide some new symmetries. He shows how his polynomials are related to Bernoulli polynomials and to Stirling numbers and polynomials. In fact, there is a relationship to the Stirling numbers of the first and second kind and more generally to the degenerate Stirling numbers defined as special values of Carlitz's degenerate Bernoulli polynomials of arbitrary order.
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11B73
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 769303
zbMATH Keywords: Stirling numbers
zbMATH Keywords: binomial identities
zbMATH Keywords: generating functions
zbMATH Keywords: Bernoulli polynomials
zbMATH Keywords: Stirling polynomials
MaRDI profile type: Publication
cites work: On the degrees of irreducible factors of higher order Bernoulli polynomials
cites work: Q4696664
cites work: Note on irreducibility of the Bernoulli and Euler polynomials
cites work: Q4189282
cites work: Review of the stirling numbers, their generalizations and Statistical Applications
cites work: Q4769056
cites work: Stirling polynomials
cites work: Q4058966
cites work: Operational formulas connected with two generalizations of Hermite polynomials
cites work: Degenerate weighted Stirling numbers
cites work: Q4053680
cites work: Q5535148
cites work: Enumeration of combinations with restricted differences and cospan
cites work: Q5589310
cites work: Q3320678
cites work: Q3748279
description: scientific article; zbMATH DE number 769303