Result: A finite difference approach to degenerate Bernoulli and Stirling polynomials

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.